Spero Consulting

‘Technical’ Category

Update: if you’re look­ing for an aca­d­e­mic dis­tinc­tion, see: The Dif­fer­ence between Risk and Uncer­tainty. In the post below, we crit­i­cize firms for not chang­ing their prac­tices despite the recent fail­ures of their esti­ma­tions, method­olo­gies, and mod­els. Of course, we think that both are worth reading.

Every Mon­day morn­ing for the past sev­eral years, we’ve received an e-​mail from http://​jobs​.phds​.org that lists avail­able posi­tions accord­ing to our spec­i­fi­ca­tions, which are:

“Send Weekly emails con­tain­ing jobs…
…for PhDs in: Busi­ness /​Finance /​Eco­nom­ics
…of types: Con­tract /​Project /​Tem­po­rary, Employee, Non-​tenure-​track fac­ulty, Post­doc­toral researcher, Tenure-​track /​tenured fac­ulty
…in sec­tors: all
…located: in United States
…with key­words: none

Gen­er­ally, there’s 20 — 40 posi­tions listed each week, and most of those involve quan­ti­ta­tive finance, usu­ally in the NYC area. For the past year or so, we’ve been par­tic­u­larly inter­ested to see if the job descrip­tions would change given the fail­ure of many quan­ti­ta­tive trad­ing strate­gies, mod­el­ing tech­niques and risk mea­sures. (Yeah, we know they didn’t actu­ally “fail.” Recent results were just plain bad luck that no one could have pre­dicted. The mod­els worked per­fectly, except when they didn’t.)

Unfor­tu­nately, our par­en­thet­i­cal sar­casm seems to be the implicit posi­tion of many finan­cial firms – with­out the sar­casm, of course. We say because we haven’t observed any change in the posted job descrip­tions in the jobs​.phds​.org emails or any of the other ones that we receive from recruiters who reg­u­larly send sim­i­lar descriptions.

Now, we’ve been mean­ing to write about this obser­va­tion for a few months but were finally moti­vated to do so because of sev­eral other items we read this morn­ing, includ­ing two opin­ion columns and one article.

The arti­cle, Computer-​Trading Mod­els Meet Match in The Wall Street Jour­nal, describes how sev­eral algorithmic-​based hedge funds have lost money recently because of “the recent high volatil­ity.” So, we guess their mod­els aren’t flawless.

One of the op-​ed pieces is by L. Gor­don Crovitz, and it is also in the Journal: In Finance, Too, Learn­ing Entails Risk. In it, Mr. Crovitz attempts to relate “finan­cial engi­neer­ing” to other types of engi­neer­ing, e.g., mechan­i­cal engi­neer­ing, and he seems to imply that it’s still a young dis­ci­pline; so, give it time, but we think that his argu­ment ulti­mately fails and is unconvincing.

That’s because “finan­cial engi­neer­ing” isn’t really engi­neer­ing, which we’d define as the thought­ful appli­ca­tion of sci­ence or tech­nol­ogy to (or in) a well-​understood, phys­i­cal envi­ron­ment. Finance is a sub­set of a social “science.”

Mr. Crovitz writes in his last para­graph that: “The mea­sure of inno­va­tors is not in the mis­takes they make, but in the lessons they learn. We now know that our com­plex mar­kets need bet­ter mod­els, which should include more humil­ity, acknowl­edg­ing that some risks are still too uncer­tain to mea­sure and should be avoided.” We’d argue with the “still too” in the last sen­tence as we doubt that such social uncer­tainty can be resolved or pre­cisely mea­sured. (By the way, we also dis­agree with his con­clu­sion in that sen­tence that “some risks…should be avoided.” We have no prob­lem with folks tak­ing wild or uncer­tain gam­bles; how­ever, we see no rea­son that we should sub­si­dize their losses when those gam­bles go bad.)

To his main point, how­ever, we don’t see much learnin’ goin’ on. It seems to be busi­ness as usual at many firms and funds.

A much more crit­i­cal op-​ed piece is by Michael Barone, and it’s enti­tled ‘For­mu­las’ for cer­tain fail­ure, and his first sen­tence is “Beware of geeks bear­ing for­mu­las.” He dis­cusses (and crit­i­cizes) finan­cial mod­els, global warming/​climate change mod­els, and health-​care mod­els, and it reads much like our post from six months ago, Global Warm­ing and the Mort­gage Cri­sis. Remem­ber that this is Michael Barone, who is very well-​known for using sta­tis­ti­cal data in the analy­sis of pol­i­tics and demographics.

As usual, we point new read­ers to our essay, Uncer­tainty Man­age­ment, which details our per­spec­tive and phi­los­o­phy on these issues as well as any num­ber of related posts: see our blog archives. The main point is that not all uncer­tainty is mea­sur­able, i.e., that mea­sur­able uncer­tainty, or risk, is a proper sub­set of uncer­tainty and unknow­ing. (In other words, spe­cific math­e­mat­i­cal con­di­tions must be met for uncer­tainty to be risk. So, uncer­tainty is a more gen­eral term, i.e., all risk involves uncer­tainty, but not every­thing that is uncer­tain is risky because not all uncer­tainty is mea­sur­able, which a spe­cific math­e­mat­i­cal definition.)

As we read the evi­dence, many insti­tu­tions and their ‘quants’ will con­tinue to solve mis-​specified risk prob­lems, because they don’t know how to treat more dif­fuse and dif­fi­cult uncer­tainty prob­lems; so, they assume them away and treat them as risk prob­lems. We’re clearly not under­es­ti­mat­ing the dif­fi­cul­ties these folks face nor the neces­sity of mak­ing trade-​offs, but we’re not sure if they under­stand the nature of the prob­lem or trade-​off. As we’ve writ­ten many times before, if they don’t under­stand them, then they are igno­rant, and if they do, then they are cyn­i­cal., e.g., Our Eter­nal Ques­tion: Cyn­i­cal or Naïve? Nei­ther char­ac­tis­tic is appeal­ing or useful.

Ignor­ing the larger epis­te­mo­log­i­cal issues and the prob­lem of induc­tion, here’s a sim­ple exam­ple of the dif­fi­culty of mak­ing infer­ences and find­ing use­ful infor­ma­tion. Even when a dis­tri­b­u­tion can be per­fectly known, it’s moments – like the mean and vari­ance – need not exist. (Look a Cauchy dis­tri­b­u­tions and, more gen­er­ally, cer­tain sta­ble dis­tri­b­u­tions. While one can cal­cu­late his­tor­i­cal means and vari­ances from a time series, those “esti­mates” may be non­sen­si­cal. (They can’t esti­mate some­thing that doesn’t exist.) The arith­metic can be per­formed, but the notion is empty.

As we see it, too often if one has a (risk) ham­mer, then every­thing looks like a (risk) nail, and it’s easy to pound away, espe­cially when the alter­nate is to admit that a solu­tion doesn’t exist, which too often sounds like, “I don’t know.” So, while var­i­ous num­bers can be cal­cu­lated – even cal­cu­lated very pre­cisely, earnestly, and dili­gently – to do so is to apply tech­nol­ogy, but it’s not engi­neer­ing nor is it very smart and it can be very harmful.

Implied Risk Neu­tral Default Rates Ver­sus His­tor­i­cal Default Rates

For some prob­lems, there is no good or true solu­tion, but some­thing must be done or esti­mated. Such is the case with cal­cu­lat­ing credit reserves because real default rates can never be known, but risk-​neutral implied or his­tor­i­cal default rates can be cal­cu­lated and used, but both are flawed.

Gen­er­ally, when we dis­cuss this topic, we have reduced-​form mod­els in mind (as opposed to struc­tural ones, but there’s no short­age of assump­tions in struc­tural mod­els, either).

We’ve writ­ten about implied default rates on sev­eral occa­sions, and we recently had a con­ver­sa­tion with some­one who men­tioned that for trad­ing credit reserve cal­cu­la­tions, reg­u­la­tors are requir­ing firms to use implied default rates from risk-​neutral pric­ing mod­els rather than his­tor­i­cal default rates. To be pre­cise, these implied default rates would be derived/​inferred from CDS (credit default swap) prices using risk-​neutral mod­els and any num­ber of quite arbi­trary assumptions.

Pre­sum­ably, given recent high prices for pro­tec­tion – the credit default swaps – implied defaults rates are sub­stan­tially higher than his­tor­i­cal rates, and the reg­u­la­tors are just try­ing to be “con­ser­v­a­tive.” Oh well, so much for our motto of thought before cal­cu­la­tion.

Now, it’s true that for all else equal, if investors are risk-​averse, implied, risk neu­tral default rates will always be greater than actual default rates, and the “more” risk-​averse investors are, the higher the implied, risk-​neutral, prob­a­bil­ity of default. There’s a vari­ety of ways it can be stated, but using CDS prices, we can roughly say that for a fixed gam­ble with a fixed prob­a­bil­ity of default, the more risk-​averse the insurer, the higher the price required to com­pen­sate him for bear­ing that risk, the higher the price of guar­an­tee­ing default, the higher the risk-​neutral prob­a­bil­ity of default.

As we men­tioned above, in the real-​world, actual (future) rates are never known, but some­times, his­tor­i­cal default rates can be used as prox­ies for actual rates, espe­cially if the ana­lyst believes that the envi­ron­ment is unchanged. Below, we’ll briefly explain why we think that is prefer­able to use implied, risk-​neutral default rates.¹

Esti­mat­ing a Credit Reserve

Like a loan-​loss reserve, which is a bank’s esti­mate of the expected loss of default by its bor­row­ers, a trad­ing orga­ni­za­tion must also cal­cu­late a credit reserve for its trades. For trad­ing, that means mak­ing a guess or esti­mate of the expected loss asso­ci­ated with default (by the coun­ter­party) when the trade is in one’s favor.²

Con­di­tional Expected Values

Roughly, that means to cal­cu­late a credit reserve, it’s nec­es­sary to deter­mine when the trade is in one’s favor and then assume or esti­mate the prob­a­bil­i­ties of that occur­ring over the life of the trade.³

By know­ing that range of win­ning val­ues and using esti­mates of the prob­a­bil­i­ties of those val­ues, one can then cal­cu­late the con­di­tional expected gain from the trade – the aver­age trad­ing gain given that one has gained (and not lost). (We’re think­ing of a a discrete-​time, single-​period prob­lem here.)

Ignor­ing col­lat­eral agree­ments for the moment, which would reduce the poten­tial credit expo­sure when one is ahead, the con­di­tional expected value rep­re­sents the aver­age amount that the other party will owe at the end of the trade (if it owes anything).

So, that con­di­tional expected value is a rea­son­able esti­mate of the credit expo­sure at, say, the end of the account­ing period. It’s very sim­i­lar to the esti­mated uti­liza­tion of a credit line at a future date, which is needed to cal­cu­late a loan-​loss pro­vi­sion and reserve. Whether for loans or trades, one needs to esti­mate the expo­sure at default, which those in the indus­try abbre­vi­ate as EAD.

EADs, LGDs, and PDs

Once that expected expo­sure at default is esti­mated, one needs two more esti­mated val­ues to cal­cu­late the coun­ter­party credit reserve (for a sin­gle trade): the prob­a­bil­ity of default (PD) and the loss given default (LGD) rate.

The prod­uct of the three – the (EAD × LGD rate) × PD – is the reserve for that trade.⁴ Also, note that the prod­uct of the first two terms is the (expected) lost given default.

For com­pletely col­lat­er­al­ized trades, the loss given default is nearly zero. There are some tim­ing issues, so sharp changes in val­ues and the lags in post­ing col­lat­eral could cre­ate a small chance of loss, but that’s a rel­a­tively small level of expo­sure com­pared to a sim­i­lar but uncol­lat­er­al­ized trade.⁵

In the past, insti­tu­tional LGD rates were par­tic­u­larly dif­fi­cult to esti­mate because there were so few obser­va­tions of bank­rupt­cies of firms within par­tic­u­lar indus­tries and of par­tic­u­lar sizes. (By the way, it’s worth not­ing that there is some evi­dence that banks recover more than bond-​holders – banks are bet­ter orga­nized for the con­tin­gency – so they have lower LGD rates.)

Now, because LGD rates are dif­fi­cult to esti­mate, they’re usu­ally assumed. With an assump­tion about the LGD rate, one can then solve for the default rate in most risk-​neutral mod­els. For more on this topic, please see our post from last sum­mer: Implied Default Prob­a­bil­i­ties and Risk Neu­tral Mod­els, par­tic­u­larly the graph that shows a sim­ple rela­tion­ship between the LGD rate and the implied default rate (in a sim­ple model).

Now we have enough to com­pare two options for prox­ies of default rates that firms can use to cal­cu­late credit reserves.

Of course, and to reit­er­ate, we’d like to know the actual, true default rate dur­ing some future period for a firm, but we can never know that for sure. (Actu­ally, we’d like to know if the trade is a win­ner and whether the coun­ter­party will default.) In fact, for a sin­gle firm, after that future period, we’ll know if the firm defaulted or not, but we won’t know the true prob­a­bil­ity of default, and even with a cross-​section of firms, we’ll be able to cal­cu­late a real­ized aver­age rate, but that will be only one pos­si­ble aver­age rate, not the true aver­age rate so-​to-​speak. So, we can esti­mate (1) a his­tor­i­cal aver­age default rate across firms, which may or may not be sta­tion­ary and/​or mean­ing­ful, or (2) an implied default rate, which depends upon any num­ber of assump­tions, includ­ing an assump­tion about the LGD rate, and which–by def­i­n­i­tion–is hypo­thet­i­cal and does not reflect reality.

Which One is Better?

Of the two, we’d pre­fer to use actual observed rates rather than implied, risk-​neutral rates. Why? For a few rea­sons. First, there are set­tings in which the observed his­tor­i­cal default rate is a rea­son­ably proxy of the unknown, “true” default rates. That’s never the case with risk-​neutral, implied default rates, which could only equal “true” rates if investors were risk-​neutral, which they are not. (Some of our related posts pro­vide exam­ples of this dif­fer­ence. They are stark and easy to follow.)

Sec­ond, the credit reserve is sup­posed to rep­re­sent the expected loss – or dis­counted expected loss in a multi-​period set­ting – not the price of the expected loss. (Risk neu­tral mod­els per­mit prices to be viewed as expected val­ues, rather than as expected util­i­ties (of unknown util­ity func­tions). That’s the ben­e­fit of risk-​neutral mod­els; they “sim­plify” the math.) By def­i­n­i­tion, the reserve is the bank’s best guess of its expected loss over some time hori­zon. It’s not the price the bank would pay to elim­i­nate default risk. Those are two clearly sep­a­rate notions, and the dif­fer­ence would be the risk premium.

Third, as we men­tioned two para­graphs above, using observed or his­tor­i­cal rates does require assump­tions about the valid­ity of the past rep­re­sent­ing the future. That’s a huge prob­lem – the Prob­lem of Induc­tion – but in our mind that’s cleaner – and more likely to remain in one’s mind – than are the many addi­tional, spe­cific, math­e­mat­i­cal assump­tions to derive/​infer risk-​neutral default rates, the LGD.

As we men­tioned at the begin­ning of the post, there’s no good solu­tion, but we think that using his­tor­i­cal rates is the bet­ter solu­tion. We think that when oth­ers dis­agree it’s because they think that implied, risk-​neutral rates are more than the really are, i.e., the “market’s” esti­mate of default rates – not the “market’s” esti­mate of default rates under a risk-​neutral mea­sure, which means that they’re hypo­thet­i­cal, not real.

We’ll likely edit this post dur­ing the next few days.

Copy­right © 2009 Spero Consulting.

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Foot­notes:

Implied Under the Assump­tion of Risk Neutrality

We have sev­eral posts related to the cal­cu­la­tion of price-​implied default rates under the assump­tion of risk neu­tral­ity and sev­eral posts related to sim­ple CDS calculations.

Those posts have involved dis­crete, single-​period prob­lems, where there are only two dates of inter­est: today and a future date where an uncer­tain claim or cash flow will be real­ized, i.e., when bank­ruptcy would occur.

We’ve focused on binary mod­els and will con­tinue to do so here. In fact, to ana­lyze a two-​period prob­lem, we’ll just build upon our lat­est post from Decem­ber 2: Price Implied Default Rates.

We think that need­less detail obfus­cates the cen­tral points while pro­vid­ing no mar­ginal explana­tory power: either in a sta­tis­ti­cal or ped­a­gog­i­cal sense. So, we like to keep things simple.

Note that we’re pro­vid­ing exam­ples of sim­ple, reduced-​form mod­els à la Jar­row and Turn­bull (1995) or Hull and White (2000), not a struc­tural Mer­ton model like KMV. We’ll do that when we have the time.

In our Decem­ber 2^nd post, we con­sid­ered a risky, one-​year, zero-​coupon bond. We assumed a face value of $1,000, a risk-​free rate of 5%, and the risky bond’s yield to be 8%. We could have stated that last assump­tion as the bond has a price of $925.93.

From those assump­tions, and the addi­tional assump­tion that the owner of the bond would recover 60% of the face value, we cal­cu­lated the risk-​neutral-​model-​implied default rate of 6.94%.

Now the cal­cu­la­tion of that default rate depends upon all of the assump­tions, and obvi­ously the answer will vary with changes in any of the assumed vari­ables: the bond’s price or yield, the risk-​free rate, and the loss given default rate.

Obvi­ously, it also depends upon the applic­a­bil­ity of risk-​neutral val­u­a­tion, which allows us to impose two very impor­tant con­sid­er­a­tions (ver­sus real­ity). It allows us to (1) treat the bond’s price as the expected value of its cash flows, which is only valid if the cred­i­tor (in the model, not in real life) is risk-​neutral, and (2) use the risk-​free rate as the proper dis­count rate for a risk-​neutral per­son. Those assump­tions allow us to work with expected cash flows, rather than curvy pref­er­ences. We’ll focus on cal­cu­la­tions in this post and not on applicability.

Finally, the answer also depends upon our choice of prob­a­bil­ity func­tions. Here, the only uncer­tainty involves full pay­ment or not; so, that credit risk is eas­ily mod­eled as a binary func­tion, but it is impor­tant to note that risk-​neutrality does not imply a par­tic­u­lar prob­a­bil­ity func­tion. Once the ana­lyst has cho­sen from a fam­ily of dis­tri­b­u­tion func­tions, the assump­tion of risk neu­tral­ity will deter­mine (imply) par­tic­u­lar para­me­ter val­ues, but that is all. For the more math­e­mat­i­cally inclined, that is the change-​of-​measure that is referred to in the texts. (Prob­a­bil­i­ties are weights. Dif­fer­ent para­me­ter val­ues within a dis­tri­b­u­tion cause pos­si­ble events to be weighed dif­fer­ently; ergo, the mea­sure is changed.)

In this prob­lem, we’ll keep the same assump­tions as in our pre­vi­ous post for the first of our two peri­ods. So, here is the set­ting: We have two zero-​coupon, risky bonds issued by the same firm and each with a face value of $1,000: one matures in one-​year and the other matures in two years. Imag­ine that there are two risk-​free bonds, too.

The one-​year risky bond is described as above; so, it will have a price of $925.93. If that bond were risk-​free, it would have a price of $952.93. In a risk-​neutral model, the dif­fer­ence in prices is the present value of the expected loss (of the risky bond, of course).

The risk-​free rate in the sec­ond period is 7%. Note that there is no mar­ket risk – that is, no inter­est rate risk – so there is no evo­lu­tion of inter­est rates or any type of rate process in our hum­ble, lit­tle exam­ple. (We’re just mak­ing up num­bers to illus­trate a few basic ideas.)

The bond that matures in two years has a yield-​to-​maturity of 9.982%, which for all intents and pur­poses – and for every­one except the truly anal – is 10%.¹

As an aside, with our two sets of inter­est rates, we can cal­cu­late an over­all yield-​to-​maturity from our term struc­ture of for­ward, risk-​free rates, and for risky rates, we can deter­mine the struc­ture of for­ward rates from our risky yield curve.

Risk-​free yield-​to-​maturity: we don’t really need to cal­cu­late this, so you can skip it is you want, but if the risk-​free bonds are priced to earn 5% in the first year, and a two-​year bond is priced to earn 7% in the sec­ond year, then the geo­met­ric aver­age return for the zero-​coupon, risk-​free bond bet­ter be close to the arith­metic mean of 6%. That yield-​to-​maturity is simply:

[(1 + r₁)·(1 + r₂)]^1⁄₂ — 1 = [1.05·1.07]^1⁄₂ — 1 = 5.995%

So, the yield on a two-​year, zero-​coupon, risk­less bond is about 6%: just like we knew before we did the calculation.

Risky for­ward rate: now, given the risky yield-​to-​maturity is about 10% on the two-​year, zero coupon, bond, and given a first-​year risky rate of 8%, then the implied for­ward rate for the sec­ond period must be:

[(1 + 0.08)·(1 + r₂)]^1⁄₂ — 1 ≈ 10% implies r₂ = 1.1² /1.08 - 1 = 12%

So, if (and only if) the two-​year, risky bond yields (about) 10%, then its price is:

$1,000 ÷ 1.1² = $826.45 ≈ $826.72.

By the way, we’re off by 26¢ by using the easy 10% instead of the more pre­cise 9.982%, but the les­son is free; so, the reader really shouldn’t complain.

Notice that credit spread increased from 3% (8% — 5%) in the first year to 5% (12% — 7%) in the sec­ond. All things equal, we should expect that the risk-​neutral, price-​implied, default rate will increase, too. Let’s see if that happens.

Three Prob­a­bil­i­ties of Default (or default rates): when we move to a multi-​period prob­lem, we have to be care­ful to spec­ify the default rate to which we’re refer­ring. There are con­di­tional, mar­ginal, and cumu­la­tive prob­a­bil­i­ties of default, and that is true whether we’re dis­cussing actual (but unknown) prob­a­bil­i­ties of default or risk-​neutral-​implied prob­a­bil­i­ties of default like we’re doing here.

The con­di­tional prob­a­bil­ity of default for a period, t, is the eas­i­est notion to under­stand: given that the firm has sur­vived until the begin­ning of that period, it is the prob­a­bil­ity that the firm can’t pay its bills dur­ing the next inter­val of time; here, we’re using one year as the time inter­val. We’ll denote con­di­tional prob­a­bil­i­ties as p_t for every period t.

The mar­ginal prob­a­bil­ity of default is the prob­a­bil­ity that the firm will default in period t. Now, the firm only has the oppor­tu­nity to default in period t, if it hasn’t already defaulted; so, the mar­ginal prob­a­bil­ity con­sid­ers the prob­a­bil­ity of sur­viv­ing until that point and the con­di­tional prob­a­bil­ity of default. If p₁ is the (mar­ginal) prob­a­bil­ity of default in the first period, the (1 — p₁), then the mar­ginal prob­a­bil­ity of default is:

(1 — p₁)·p₂,

For our lit­tle prob­lem, we won’t intro­duce any spe­cial nota­tion for the mar­ginal prob­a­bil­i­ties of default.

Finally, the cumu­la­tive prob­a­bil­ity of default is the sum of all the mar­gin­als: p₁ + (1 — p₁)·p₂ in a two-​period prob­lem. We wrote about longer term cumu­la­tive prob­a­bil­i­ties of events in this post, Good Col­umn, Bad Math, where we talk about 100-​year floods.

So, let’s find the con­di­tional prob­a­bil­ity of default in the sec­ond period. Given that there was no default at the end of the first period, what is the prob­a­bil­ity of default in the sec­ond period implied by the bond’s price?

Well, with one period remain­ing, the price of the only remain­ing bond is:

$1,000 ÷ 1.12 = $892.86.

So, we can find the con­di­tional prob­a­bil­ity of default in the second-​period, p₂, the same way that we found the prob­a­bil­ity in our one-​period prob­lem.²

price = $892.86= (1 — p₂) × ($1,000 ÷ (1 + 0.07)) + p₂ × (600 ÷ (1 + 0.07))

$892.86= (1 — p₂) × $934.58 + p₂ × 560.75.

So, if the firm sur­vives the first period, there is an 11.16% con­di­tional prob­a­bil­ity of default in the sec­ond period. That means that the mar­ginal prob­a­bil­ity of default for the sec­ond period is the prob­a­bil­ity that the firm sur­vives the first period mul­ti­plied by the con­di­tional prob­a­bil­ity of default in the second:

(1 — p₁) ·p₂ = (1 — 0.0694) · 0.1116 = 10.385%

The cumu­la­tive prob­a­bil­ity of default is the sum of the two mar­gin­als: 6.94% + 10.39 = 17.33%.

Note that at the end of the first period the dif­fer­ence between the risk-​free bond’s price of $934.58 and the risky bond’s price of $892.86 is $41.72. The $41.72 rep­re­sents the risk-​neutral, “present value” at the start of the sec­ond period of the con­di­tional expected loss in the sec­ond period of the two-​period bond. So, the $41.72 is related to the con­di­tional prob­a­bil­ity of loss and the poten­tial loss of $400:

($400 × 11.16%) ÷ 1.07.

But the sec­ond period will be expe­ri­enced only if there was no default in the first period! So, in a risk-​neutral world, a cred­i­tor will only expe­ri­ence the oppor­tu­nity to lose (a dis­counted aver­age) of $41.72 if there is no default in the first period: with prob­a­bil­ity (1 — 6.944%).

And the value of that today – at the start of it all – must be dis­counted by the first period’s risk-​free rate of 5%. So, the present value of that expected loss that

$41.72 × (1 — 0.06944) ÷ 1.05 = $36.97.

Is our analy­sis cor­rect? Let’s see. A two-​year, risk-​free, zero-​coupon bond would have a price of $890.08. Our risky bond has a price of $826.45. That means that in a risk-​neutral world – given all of our assump­tions – the present value of the sum of the expected losses is the dif­fer­ence: $890.08 — $826.45 = $63.63.

In the first year, the present value of the expected loss on debt with a face value of $1,000 is $26.67. That means that the present value of the expected loss in the sec­ond period must be: $63.63 — $26.67 ≈ $36.97. Hey, where did we see that num­ber before? That’s right — a few inches above where we dis­counted the expected present value of the second-​period loss.

What about CDS?

To pro­tect against loss, the CDS should pro­vide $400 in case of default at the end of each period.

If the CDS pol­icy were sold period-​by-​period, i.e., one-​year terms, the first year’s pre­mium would have to be at least $26.67 and the sec­ond year’s if sold today would cost at least $36.97. The actual cost, like every­thing else in the real world, would depend upon how badly cred­i­tors want to pro­tect against loss, but those val­ues are actu­ar­i­ally fair in a risk-​neutral setting.

Also note that if the CDS pol­icy were sold at the start of the sec­ond period, the pre­mium would have be to at least $41.72 to be actu­ar­i­ally fair in a risk-​neutral world. So, if pur­chased con­sec­u­tively, the insur­ance pre­mi­ums would need to $26.67 today and $41.72 next year in our risk-​neutral world.

What if the insur­ance were pur­chased for two peri­ods? What would the con­stant pre­mium be? In that case, there is a chance that one or both pre­mi­ums will be received (or paid). If there is no bank­ruptcy in the first period, then the pre­mium will be paid twice; so, we need:

pre­mium + (1 — 0.06944) pre­mium ÷ 1.05 = $63.63

pre­mium (1.0 +0.93056 ÷ 1.05) = $63.63

pre­mium = $33.74

We assumed that the pre­mium was paid at the begin­ning of each period; so, it is like an “annu­ity due” and actu­ally is like a ran­dom, annu­ity due. It’s ran­dom because it is a con­stant stream of cash flows, but the end­ing date is unknown. In this sim­ple two-​period exam­ple, the “stream” could be one or two payments.

Also remem­ber that risk-​averse cred­i­tors should be will­ing to pay more than that, i.e., a risk pre­mium, too.

And remem­ber, we’ve said absolutely noth­ing about prob­a­bil­i­ties in the real world that our exam­ple rep­re­sents. Risk neu­tral prob­a­bil­i­ties and default rates are derived from a set of assump­tions that per­mits (rel­a­tively) easy cal­cu­la­tion, but those prob­a­bil­i­ties and rates only work in our model, and they do not rep­re­sent real fre­quen­cies. For more on that, please see our other posts on the topic.

As we hope that you can see, CDS is iden­ti­cal to term life insur­ance – except mil­lions and mil­lions of sim­i­lar firms don’t die each year; so, there is lit­tle empir­i­cal evi­dence of var­i­ous fac­tors, includ­ing loss given default rates.

By the way, we’ve ignored counter-​party risk and a host of other com­pli­cat­ing assumptions.

As with many of our longer posts, we’ll likely edit this one in the near future.

Copy­right © 2008 Spero Consulting.

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Foot­notes:

Update: Decem­ber 12, 2008. While none of our analy­sis or cal­cu­la­tions was incor­rect, we did have a minor error in the penul­ti­mate para­graph. We should of said “first” not “last.” To make amends, here is a multi-​period prob­lem, Multi-​period Bond Price Implied Default Rates and CDS, but it won’t make sense with­out read­ing this one first. We also added a few para­graphs below, which should help explain the multi-​period case.

Fur­ther update: April 14, 2008. We also have a new, related post on default rates. It is Cal­cu­lat­ing Coun­ter­party Credit Reserves from April 8, 2009. Much of that post involves default rates, too.

We see that we’re get­ting a num­ber of hits from search engines for folks look­ing for infor­ma­tion about price-​implied default rates – pos­si­bly col­lege stu­dents with home­work assign­ments or peo­ple try­ing to under­stand the var­i­ous types of default rates they may encounter in their jobs or readings.

We have a num­ber of posts on risk-​neutral default rates, includ­ing Implied Risk Neu­tral Prob­a­bil­i­ties (of Default) , implied RISK NEUTRAL prob­a­bil­ity of default, redux, Risk Neu­tral Val­u­a­tion: There Are at Least Two Expected Val­ues, but we doubt if those set­tings are the ones that all guests want to see, espe­cially those look­ing for help on their home­work. (Of course, we think they are all worth read­ing.) So, as a pub­lic ser­vice, we offer an exam­ple of a sim­ple, one-​period bond prob­lem. (It is single-​period because it is gratis, after all.)

Sup­pose that a zero-coupon, risky bond with a face value of $1,000 matures in exactly one year. (Yeah, we said it was sim­ple.) We’ll ignore com­pound­ing issues and assume that the annual risk-​free rate is 5%. We’ll also assume that this risky bond’s yield-​to-​maturity is 8%.

Let’s cal­cu­late and dis­cuss a few things before we pro­vide addi­tional assumptions.

We’ll cal­cu­late the bond’s price that cor­re­sponds to an 8% yield, and we’ll cal­cu­late the bond’s price if it were risk­less; of course, by risk­less we mean free of default risk or credit risk, only. Our sim­ple one-​period model doesn’t really per­mit inter­est rate risk, which is a type of mar­ket risk.

The bond’s price with a 8% annual yield is: $1,000 ÷ (1 + 0.08) = $925.93.

Now, if the bond were risk-​free, its price would be $1,000 ÷ (1 + 0.05) = $952.38,

which is $26.45 higher. So, the price drops and the yield increases (over their risk-​free equiv­a­lents) because the owner(s) of the bond is forced to bear some type of credit risk or prob­a­bil­ity of loss.

That $26.45 will appear again later, but at this point we can’t say much more than it is the dif­fer­ence in the prices of a one-​period risk-​free bond and our one-​period risky bond.

The prob­lem with sim­ple cal­cu­la­tions – whether in one or mul­ti­ple peri­ods – is that they ignore all of the fac­tors that actu­ally affect and deter­mine prices. In other words, we’ve com­pletely ignored the mar­ket dynam­ics and fac­tors that would cause the price to be $925.93.

The market-​clearing price would depend upon sup­ply and demand con­sid­er­a­tions.¹ Those con­sid­er­a­tions would depend upon the pref­er­ences, beliefs, and endow­ments of actual and poten­tial sell­ers and buy­ers. In our sim­ple set­ting, the impor­tant pref­er­ences would be risk and time pref­er­ences, which could pos­si­bly be expressed as util­ity func­tions; beliefs would involve the prob­a­bil­ity of default as well as other prob­a­bil­i­ties asso­ci­ated with each agent’s wealth in other assets if they exist – i.e., their endowments.

So, we can think of the price of $925.93 as a “func­tion” of pref­er­ences, U(·); beliefs, f(·); and endow­ments, w.² Unfor­tu­nately, in real life, we don’t know those fac­tors; so, we’ll never be able to solve the actual prob­lem, but we can solve a sub­sti­tute problem.

All we know is that the price is $925.93, and it can be expressed as a yield-​to-​maturity – or a yield curve for multi-​period prob­lems – of (our assumed) 8%. So, the yield could be viewed as a func­tion of the price if you want, but they’re really deter­mined simultaneously.

As we’ve writ­ten many times before in related posts, because of sev­eral clever researchers in eco­nom­ics and finance, we can actu­ally do more than just dis­cuss the tau­tolo­gies of price and yield.

In cer­tain cases, we can assume that mar­ket par­tic­i­pants are risk-​neutral – that takes care of U(·) and makes the w irrel­e­vant – and we can assume a par­tic­u­lar form of a den­sity or dis­tri­b­u­tion func­tion of out­comes, f(·). Very impor­tantly, with those assump­tions, if we don’t know one of the para­me­ters of f(·) we can solve for it if we know every­thing else. That would be like solv­ing for the mis­named implied vol or implied default rate, which is what we will do here.³

Here’s the key to all risk-​neutral pric­ing: under cer­tain assump­tions, if agents are (assumed to be) risk-​neutral, then we can treat prices as equal to the expected value of the asset’s cash flows accord­ing to an asso­ci­ated den­sity func­tion. That’s the only time we can treat prices as expected cash flows, rather than expected util­i­ties, but depend­ing upon the level of the course, some profs are pretty bad at explain­ing that fact.⁴

So, there are three things to consider. First, if agents are risk neu­tral, we can assume that they care only about expected values.

Second, if agents are risk neu­tral, then they won’t pay a pre­mium for tak­ing risk like risk-​lovers would, nor will they need to be paid a pre­mium for tak­ing risk like risk-​averse agents would need to be paid.

Third, that means we can assume that risk neu­tral agents are sat­is­fied earn­ing the risk-​free rate. ⁵ So, given all of our words above, that means that risk neu­tral agents would value assets at the dis­counted value of the expected cash flows – dis­counted at the risk-​free rate.

So, as we showed above, if the bond were actu­ally risk-​free, then price would have been $952.38, but the price is $925.93. That means that mar­ket par­tic­i­pants must expect to receive less than the face value of $1,000 at least some per­cent­age of the time, and that per­cent­age is the prob­a­bil­ity of default.

Let’s see exactly how much less than $1,000, but first note that we could write the price of a risk-​free bond in a slightly expanded way. Risk-​free means 100% chance of get­ting $1,000; so,

Equa­tion A:

$952.38 = 100% × ($1,000 ÷ (1 + 0.05)) + 0% × (value given default ÷ (1 + 0.05))

We did noth­ing but add zero to our pre­vi­ous cal­cu­la­tion of a risk-​free bond.

Let’s make it risky. Let p rep­re­sent the prob­a­bil­ity of default, then for a risk-​neutral per­son, we could write that same line as:

price = (1 — p) × ($1,000 ÷ (1 + 0.05)) + p × (value given default ÷ (1 + 0.05))

Thus, with a price of $925.93, we could write:

$925.93 = (1 — p) × ($1,000 ÷ (1 + 0.05)) + p × (value given default ÷ (1 + 0.05))

There are two unknowns: the prob­a­bil­ity of default, p, and the value of the bond given default, which has to be less than $1,000. In fact, we could put a deter­mine a upper bound that is less than $1,000 if we wanted to do so. (How?)

Now, look at the last equation. Once we know or assume the value given default, we could find the prob­a­bil­ity of default, p, or vice versa.

Usu­ally, one assumes the value given default and solves for p. There’s not really a good rea­son for doing it other than that’s what just about every­one does. (Don’t let any­one attempt to fool you with some lame jus­ti­fi­ca­tion. It’s tra­di­tion, cus­tom, con­ven­tion. Regard­less of the word, it is arbitrary.)

So, let’s make-​up – er, we mean assume – a value given default. This is often given in terms of a loss given default, a loss given default rate, or a recov­ery rate, but they’re all equiv­a­lent as one can see in the fol­low­ing relationships.

value given default = $1,000 — loss given default

value given default = $1,000 — loss given default rate × $1,000 = $1,000 × (1 — loss given default rate)

value given default = $1,000 × (1 — loss given default rate) = $1,000 × recovery rate

The loss given default is often abbre­vi­ated LGD. Unfor­tu­nately, the loss given default rate is some­times abbre­vi­ated as LGD. Don’t let the bad nota­tion fool you. Now, where were we?

That’s right. Let’s sup­pose that the loss given default rate is 40%. That means the recov­ery rate is 60%, which is its com­ple­ment. Regard­less, of how that assump­tion is stated, that means that the value given default is $600. So, now we have another num­ber to put into our equation:

$925.93 = (1 — p) × ($1,000 ÷ (1 + 0.05)) + p × (600 ÷ (1 + 0.05))

or,

Equa­tion B:

$925.93 = (1 — p) × $952.38 + p × 571.43.

If we did the arith­metic cor­rectly, then solv­ing for p gives a prob­a­bil­ity of default of almost 7%: 6.94%. Clearly, all things equal, which means hold­ing every­thing else con­stant, as the loss given default increases, the prob­a­bil­ity of default decreases. One can make a graph of that rela­tion­ship as we did in Implied Default Prob­a­bil­i­ties and Risk Neu­tral Mod­els in June, 2008.

Now, under the assump­tion of risk-​neutral agents, the dif­fer­ence between the two bond prices of $26.45 can be express as the dif­fer­ence in the present value of their expected cash flows. The dif­fer­ence in the present val­ues of the expected cash flows in Equa­tions A and B is the present value of the expected loss. The loss given default is $400. The undis­counted expected loss is: 0.0694 × $400 = $27.76. The present value of the expected loss is – not sur­pris­ingly – $27.76 ÷ 1.05 = $26.45.

That’s not the most some­one would spend for insur­ance. That insur­ance pre­mium depends upon the person’s risk-​aversion.

Multi-​period prob­lems aren’t that much dif­fer­ent, but they require bonds of mul­ti­ple matu­ri­ties if one is attempt­ing to derive a credit curve, and one works for from the last first period for­ward solv­ing maturity-​by-​maturity. Oth­er­wise, one can find an “aver­age” annual mar­ginal prob­a­bil­ity of default. (We talk about a sim­i­lar issue in Good Col­umn, Bad Math.) So, in our multi-​period exam­ple, we’ll explain the price of a two-​year bond as the dif­fer­ence in present val­ues between a risky and risk-​free two-​year bond. Then we’ll say much much of that can be attrib­uted to the first period and then the sec­ond period.

Note: WE“VE SAID ABSOLUTELY NOTHING ABOUT THE REAL PROBABILITY OF DEFAULT! If all of the agents are risk-​averse, then the unknown real prob­a­bil­ity of default will be less than the risk-​neutral rate, but that’s not too help­ful, is it? Some of our older posts do illus­trate this idea.

Good luck with the assignment.

Copy­right ©2008 Spero Consulting.

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Foot­notes:

But You’ll Never Know the One

We also have a newer post, Price Implied Default Rates, that pro­vides an exam­ple more like a risky bond, and this one: Multi-​period Bond Price Implied Default Rates and CDS. And we’ll have more related posts soon.

We’ve noticed that our few posts on risk neu­tral prob­a­bil­i­ties and implied default prob­a­bil­i­ties have been among our most pop­u­lar con­tent for read­ers through­out the world. (And it is cool to write “through­out the world.”)

So, we’ve finally start­ing com­pos­ing a longer essay to cover con­tin­u­ous den­sity func­tions, but an ear­lier post this morn­ing – Novem­ber 13–about means and medi­ans reminded us of a source of con­fu­sion regard­ing risk neu­tral pric­ing and val­u­a­tion meth­ods, and that is the fact that there are (at least) two dif­fer­ent means to consider.

In fact, there are (at least) two dif­fer­ent dis­tri­b­u­tions to con­sider: the real one, which can never been known, and the assumed one, which per­mits cal­cu­la­tions to be made based upon mar­ket prices (and many assump­tions). Actually, there may be far more than two, but we can illus­trate our point with only two.

What Makes a Mar­ket: Mar­kets per­mit indi­vid­u­als with dif­fer­ent pref­er­ences; beliefs about future uncer­tain­ties; endowments;and plan­ning hori­zons to exchange resources and claims mutu­ally max­i­mize (some mea­sure of) each person’s prospec­tive sat­is­fac­tion (accord­ing to their indi­vid­ual pref­er­ences or tastes).

Those beliefs about future uncer­tain­ties can be thought of as sub­jec­tive (or per­sonal) prob­a­bil­i­ties of events (or com­bi­na­tions of events) that affect the indi­vid­ual or the world.

Gen­er­ally, those sub­jec­tive prob­a­bil­i­ties are rep­re­sented as dis­tri­b­u­tion func­tions (and com­bi­na­tions of events as joint dis­tri­b­u­tion functions).

As one could well imag­ine, know­ing those beliefs or uncer­tain­ties or sub­jec­tive prob­a­bil­i­ties or dis­tri­b­u­tion func­tions along with know­ing the mar­ket par­tic­i­pants’ pref­er­ences would be quite use­ful for pre­dict­ing future prices. Unfor­tu­nately, there’s no eco­nomic way to do so. (In fact, we’d argue that in the real world, there is no way to do so – partly because many indi­vid­u­als can’t clearly spec­ify pref­er­ences or beliefs and partly because there’s not enough time to do it.)

Regard­less – and we’ll write in the sin­gu­lar – the real dis­tri­b­u­tion func­tion and pref­er­ences are unknown.¹

What we do know – or, more pre­cisely, what we can model – is that the observed price of a good or secu­rity is some “func­tion” of pref­er­ences, plan­ning hori­zon, endow­ments, and the “true” dis­tri­b­u­tion of out­comes. (We put “func­tion” in scare quotes because we’re using that term quite loosely and not in a strict, math­e­mat­i­cal sense.)

So, if we observe a price of, say, a claim against a stream of cash flows, we know that it is the result of com­bin­ing those fac­tors men­tioned above for both actual or would-​be mar­ket par­tic­i­pants: think sup­ply and demand curves. 

We also know (from Jensen’s Inequal­ity) that if all of the par­tic­i­pants are risk-​averse, then price will be less than the distribution’s expected value–although we don’t know that “true” expected value of the cash flows; so, we don’t know the dif­fer­ence between the two.

Now, that expected value of the cash flows is one of the two expected val­ues ref­er­enced in the post’s title. The “real” expected cash flow from a stock or bond or option or other finan­cial claim. Again, it is some­thing that we can­not observe in the real world. (Of course, for cer­tain dis­tri­b­u­tions, expected val­ues do not exist, but that is another topic, and our goal here is pro­vide a bit of intuition.)

The other expected value and – more gen­er­ally, the other dis­tri­b­u­tion – is known but is not “real” so-​to-​speak. That dis­tri­b­u­tion func­tion is an assump­tion, which can be con­sid­ered a fig­ment of the analyst’s imag­i­na­tion. (It is very, very sad to know how many prac­ti­tion­ers con­fuse that assump­tion with the real world, but we shan’t attempt to ruin the hopes and dreams of any­one today. Plus, we’ve writ­ten about it in other posts.)

A Brief Account­ing: So, there is a real, but unknow­able dis­tri­b­u­tion func­tion, and imag­i­nary, but know­able one. (Real dis­tri­b­u­tion func­tions are only truly known for games of chance like dice or the lot­tery.) Since we can’t know it, we must assume one to go any further.

Unfor­tu­nately, assum­ing a dis­tri­b­u­tion func­tion isn’t very use­ful with­out also know­ing somthing about pref­er­ences – in terms of, say, a util­ity func­tion. If we did have both a dis­tri­b­u­tion func­tion and util­ity func­tions, then with addi­tional assump­tions about mar­ket mech­a­nisms and hori­zons and endow­ments, we could cal­cu­late expected utilities, which would allow us to cal­cu­late mar­ket prices.

So, what to do? Through a few clever appli­ca­tions involv­ing the math­e­mat­i­cal notion of change-​of-​measure and eco­nomic notion of no arbi­trage (via cost­less repli­ca­tion of a posi­tion), researchers showed that one could assume that, say, investors were risk-​neutral and go from there. (Tech­ni­cally, as we under­stand it, one could use square-​root util­ity pric­ing if they wanted to, but it would just com­pli­cate mat­ters, and risk neu­tral pref­er­ences are so, so, nice and linear.)

So, if investors were assumed to be risk neu­tral, then they’d only care about expected cash flows, and one could then assume that those risk-​neutral investors val­ued expected cash on a util-​for-​dollar basis. (Tech­ni­cally, risk neu­tral­ity means lin­ear pref­er­ences but not nec­es­sar­ily util-​for-​dollar pref­er­ences; they could be mul­ti­ples or fractions.)

Now with assumed pref­er­ences and a dis­tri­b­u­tion function, the mean or expected value of the assumed dis­tri­b­u­tion could be set equal to the observed price, and one could then work with that preference-​distribution com­bi­na­tion rather than the true unknown ones. (Note that we were a bit loose with the first clause of last sentence. Technically, it involves mov­ing from today’s price to a future price and then dis­count­ing back­wards to get a present value. In con­tin­u­ous time mod­els, this shows as mul­ti­ply­ing by both e^rt and e^–rt, respec­tively, but it is obscurred in the usual slide-​rule pre­sen­ta­tion of Black-​Scholes.)

So, the selec­tion of a dis­tri­b­u­tion func­tion – which hope­fully rep­re­sents some­thing that we’ve inferred about the true but unknow­able one – and the assump­tion of risk neu­tral­ity allows us to treat prices as expected cash flows, which both per­mits and sim­pli­fies cal­cu­la­tions. How­ever, as any prac­ti­tioner can tell you, that doesn’t mean that the cal­cu­la­tions are simple.

So, set­ting the price equal to the mean of the assumed dis­tri­b­u­tion func­tion is the sec­ond expected value ref­er­enced in the title.² And that is okay IF (and that’s a big IF) the claim against cash flows can be repli­cated or hedged with other instru­ments. (And that’s hedged, not nedged or sledged.)

Finally, and briefly, as we noted back on June 22, when a para­me­ter value of the assumed dis­tri­b­u­tion is unknown, it can often be inferred or found if enough other infor­ma­tion is avail­able. Unfor­tu­nately, these inferred para­me­ters are often called “implied” as in implied volatil­i­ties. They’re implied by the assump­tion of the par­tic­u­lar dis­tri­b­u­tion func­tion and by the assump­tion that mar­ket par­tic­i­pants are risk neu­tral, but one needs to make infer­ences to find them.

We hope this helps those strug­gling with the con­cepts, espe­cially those in math-​finance pro­grams who are hin­dered by a weak back­ground in eco­nom­ics. If it is not, send a note and let us know why or ask a ques­tion of us. It is likely that we’ll con­tinue to edit this post.

Update: we have newer posts on the topic, too, includ­ing Risk Neu­tral Val­u­a­tion: There Are at Least Two Expected Val­ues, that describes the dif­fer­ence between real and risk neu­tral dis­tri­b­u­tions. We also have: Price Implied Default Rates that pro­vides an exam­ple more like a risky bond, and a multi-​period exam­ple: Multi-​period Bond Price Implied Default Rates and CDS.

The Wall Street Jour­nal has an arti­cle about Iceland’s finan­cial prob­lems in today’s paper: After­shocks Felt From Ice­land. It turns out that the coun­try has more prob­lems than being a small, cold island in the mid­dle of the North Atlantic.

Any way, we’re not writ­ing about its cli­mate, espe­cially since West­ern PA’s is prob­a­bly worse and we have no beaches and few tall blonds. No, we’re writ­ing about the graph in the arti­cle and the blurb that states, “Trad­ing in the credit default swap mar­ket puts the prob­a­bil­ity of a default by Ice­land on its debt at a lit­tle over 50%.”

As pre­sented, that state­ment is highly mis­lead­ing and non­sense, and the pur­pose of this post is to explain why.

We’ve writ­ten about Implied RISK NEUTRAL prob­a­bil­i­ties of default a few times. In the aptly titled Implied Risk Neu­tral Prob­a­bil­i­ties (of Default) we pro­vided an exam­ple that illus­trated the dif­fer­ence between the actual prob­a­bil­ity of default, which is never known in the real world, and the model–implied prob­a­bil­ity of default, which could be cal­cu­lated from ANY model–regard­less of its valid­ity–that per­mits at least two out­comes, e.g., survival and fail­ure of the entity. Such a model may or may not assume risk neu­tral­ity, but risk neu­tral­ity makes the cal­cu­la­tion simpler.

Regard­less of whether Ice­land goes bank­rupt or not, we pro­vide sev­eral exam­ples that dis­tin­guish the risk-​neutral, implied default rate from the true default rate.

In our ear­lier post, Implied Default Prob­a­bil­i­ties and Risk Neu­tral Mod­els, we com­mented on a sim­i­lar graph in another WSJ arti­cle from last June, and men­tioned many of the fac­tors that would be involved in such a cal­cu­la­tion. Unfor­tu­nately, we recently and acci­den­tally deleted a very nice com­ment about that post, which expanded the analy­sis to include coun­ter­party credit risk: the risk that the pur­chaser of a CDS con­tract would not get paid (the insur­ance pro­ceeds) in case of bank­ruptcy because the insurer or CDS writer was also insol­vent – kind of like AIG.

In this post, we’ll pro­vide another numer­i­cal exam­ple with a dif­fer­ent assumed, risk-​averse, util­ity func­tion for the insur­ance buyer.

We’ll again assume a sin­gle period, but we will not use the 50% prob­a­bil­ity of bank­ruptcy that we did in the ear­lier post; it would be too confusing. In fact, the 50% prob­a­bil­ity of default men­tioned in the arti­cle is likely the cumu­la­tive prob­a­bil­ity of default over the five years. It may or may not be based on equal mar­ginal prob­a­bil­i­ties of default for each of the five years, regard­less the annual mar­ginal prob­a­bil­ity of default is not 10%; the 50% men­tioned for five years was not found by mul­ti­ply­ing five years times 10%.

Read­ers inter­ested in an exam­ple of a discrete-​time, multi-​period sur­vival prob­lem that illus­trates these issues should see Good Col­umn, Bad Math. Read­ers inter­ested in a calculation-​intensive, similarly-​structured, discrete-​time prob­lem, should see our research paper on moral haz­ard: Dead­lines as Man­age­ment Con­trol Devices, which is based upon our dis­ser­ta­tion. In that paper, the game ends with suc­cess, rather than fail­ure, but the out­come tree is very similar.

So, will pro­vide a cou­ple exam­ples sim­i­lar to our square root prob­lem in August.

Case 1: Assume that the per­son has nat­ural log­a­rith­mic util­ity, which is strictly con­cave fun­tion and makes him risk-​averse. We’ll also assume that the per­son has an ini­tial endow­ment of $75.858, which we choose for con­ve­nience as you’ll see below. We’ll ignore time-​value-​money cal­cu­la­tions and inter­est rates today; they’re inessential.

Assume that a firm will be worth $100 if it sur­vives and $10 if if fails. That makes the loss given default (LGD) $90, and the loss given default rate $90/$100 equal to 90%. In the real world, e don’t know the loss given default until a default occurs, the firm’s assets are liq­ui­dated, and the resid­ual cash is paid to the debtholders. LGD rate is always assumed in CDS and other sim­i­lar cal­cu­la­tions and, from our experience, seems to be con­sid­ered much less than implied default rates.

Assume that the actual prob­a­bil­ity of default is 12%, i.e., the prob­a­bil­ity of get­ting $10 from the invest­ment is 12%. REMEMBER, two items that we never know in real life are the mar­ket par­tic­i­pants pref­er­ences – expressed here as a ln(·) util­ity func­tion – and the actual prob­a­bil­ity of default, 12%. It is cru­cial never to for­get this ignorance.

Also, we gen­er­ally don’t know the person’s entire endow­ment, specif­i­cally his other wealth inde­pen­dent of the gam­ble. In this first case, we clev­erly chose the person’s endow­ment so that his other wealth, not tied up in this par­tic­u­lar invest­ment, is zero. (You’ll that fact below.)

We’ll do what we need to do to cal­cu­late the risk-​neutral prob­a­bil­ity of default and then later we’ll change a few assump­tions to see how those changes affect the answer.

First, we’ll cal­cu­late the person’s expected util­ity with the invest­ment. Now, with log­a­rith­mic util­ity it is:

10% × ln($10) + 90% × $ln($100) = 4.375 utils.

Now, to get the same 4.375 utils of sat­is­fac­tion from a cer­tain gam­ble (involv­ing no risk), the per­son should be will­ing to spend up to:

e^4.375 = $75.858.

So, that $75.858 is his cer­tainty equiv­a­lent, or the most he would pay for the uncer­tain invest­ment. (That’s why we clev­erly set his ini­tial wealth at the same $75.858, so there would be no money left-​over after the invest­ment.) With the same hand-​waving (about mar­ket inter­ac­tions) that we per­formed in August, we’ll sup­pose that the $75.858 is also the price, i.e., com­pe­ti­tion among similarly-​preferenced and endowed buy­ers drive the price to the break-​even point; tech­ni­cally, it is an indif­fer­ence point but only pedan­tics like our­selves care.

Now, a risk neu­tral per­son could–but need not – be mod­eled as car­ing only about expected cash flows on a dollar-​for-​dollar basis; so, for a risk-​neutral per­son, we could set his util­ity equal to dol­lar val­ues and expected dol­lar val­ues. In other words, he would value $10, $75.858, and $100 as 10 utils, 75.858 utils, and 100 utils, respec­tively. (We wrote “but need not” above, because we could add a con­stant and mul­ti­ply by a pos­i­tive num­ber with­out chang­ing the essence of the analysis.)

Remem­ber, in the real world, we don’t know the 12% or the actual mar­ket participant’s pref­er­ences, which we assumed to be log­a­rith­mic here, or his start­ing wealth, BUT if we assumed that he was risk neu­tral in our dollar-​for-​dollar way, then we solve for the cor­re­spond­ing prob­a­bil­ity of default, i.e., find p such that:

p × 10 + (1 — p) × 100 = 75.858.

Rear­rang­ing and solv­ing for p, we get the risk neutral-​implied prob­a­bil­ity of default, p, equals about 26.83% (ver­sus the real prob­a­bil­ity of default of 12%, which, again, we never know in real life).

So, the WSJ writer or edi­tor is call­ing that 26.83% the prob­a­bil­ity of default, when it is, in fact, the implied prob­a­bil­ity of default assum­ing that mar­ket par­tic­i­pants were risk-​neutral. (Here, our “model” is so sim­ple as to be innocu­ous, but in more robust set­tings – with more details – that’s not the case.)

That risk-​neutrality, which pro­vides lin­ear­ity of pref­er­ences, is what allows the ana­lyst to view the price and set it equal to the expected value of the cash flows in the pos­si­ble out­comes, e.g., sur­vive or fail, for a pos­si­ble prob­a­bil­ity, p. In real life, ana­lysts would use dif­fer­ent dis­tri­b­u­tions to cal­cu­late an implied prob­a­bil­ity of default based upon their spe­cific model in much the same way that they would cal­cu­late a model-​implied volatil­ity when using Black-​Scholes or a vari­ant. (Pro­vide mar­ket vari­ables or guesses about those vari­ables, pro­vide a model, and solve for the last remain­ing unknown. Notice that there are quite a lot of assump­tions in such a process.)

(By the way, for those with a lit­tle knowl­edge of sto­chas­tic processes, set­ting the price equal to the expected value (under risk neu­tral val­u­a­tion) is why the phrase Mar­tin­gale Method is used. That’s what a Mar­tin­gale is: a process where the value today is equal to the expected value in the future, and it doesn’t really change if we add inter­est rates and discounting.)

Now please note, unlike in real-​life, in this exam­ple, we know that the true prob­a­bil­ity of default is 12%. To an out­side observer, with­out our infor­ma­tion to con­struct the cal­cu­la­tions, there is no clear rela­tion­ship between the 12% and the 26.83%. In other words, know­ing only the 26.83% says noth­ing about the true prob­a­bil­ity of default, and that is the error that the jour­nal­ist makes in today’s article.

Because the 50% for Ice­land is such a large num­ber, the graph and the blurb seem almost designed to insight hys­te­ria; how­ever, actual – albeit unknown rate – could be sub­stan­tially lower.

We’re sure that many WSJ read­ers along with the article’s writer mis­in­ter­pret that num­ber. We were and con­tinue to be amazed (and shocked) at the num­ber of folks who work or trade in the area who do not under­stand it. Thus, we view this post as a pub­lic service.

It is about 5:00 EDT, and proof­read the post like we promised. We’ll add to this post this later today with more exam­ples; so, please check back for updates that show why the price could drop and the implied RISK NEUTRAL prob­a­bil­ity of default could rise despite the TRUE prob­a­bil­ity remain­ing at 12%. (Note: the true default rate has lit­tle or noth­ing to do with the his­toric default rate. We’ve writ­ten a lot about that notion, too. See our essay on uncer­tainty man­age­ment for that discussion.)

Case 2: let’s keep every­thing the same, but make the per­son “more” risk-​averse. In micro­eco­nom­ics, that has a par­tic­u­lar, tech­ni­cal mean­ing hav­ing to do with the con­cav­ity (the curved­ness) of the util­ity func­tion, but here we’ll avoid the issue by reusing the nat­ural log­a­rth­mic func­tion recur­sively, i.e., our util­ity func­tion is now ln(ln(·)).

In such a prob­lem, the addi­tional con­cav­ity reduces the cer­tainty equiv­a­lent of the gam­ble, and pos­si­bly the price. We’ll wave our hands again as a way to stay on course, and assume that the price falls to the new cer­tainty equiv­a­lent. To make it work, with­out try­ing to hard, we’ll arbi­trary assume that as soon as the per­son pur­chases the firm, his pref­er­ences, via util­ity func­tion, (and risk aver­sion) changes to the double-​log thing, ie.,

12% × ln(ln($10)) + 88% × ln(ln($100)) = 1.444 utils.

For the changed per­son to get the same 1.444 utils of sat­is­fac­tion for sure, he’d be will­ing to sell it for:

e^exp(1.444) = $69.243.

(As his risk aver­sion increases, the value of a gam­bles decreases.) Now, in the real world, a decrease in a poten­tial seller’s reser­va­tion price doesn’t nec­es­sar­ily change the mar­ket price, but we’ll assume that it does. So, imme­di­ately, the price is $69.243. We can now find the revised risk-​neutral probabilities:

p × 10 + (1 — p) × 100 = 69.243.

Solv­ing for p yields a new, risk-​neutral, implied prob­a­bil­ity of default of 34.175%. So, a change in risk pref­er­ences will change the implied prob­a­bil­ity of default. You may call it the mar­ket implied prob­a­bil­ity of default, but it is really the implied prob­a­bil­ity of default using the mar­ket price and assum­ing that buy­ers are risk-​neutral, but that gets kind of long. The real prob­a­bil­ity of default is still 12%.

Case 3: Now, let’s go back to our first case, where we used the nat­ural log, ln, only once, not twice. Let’s assume that right after the pur­chase, the new owner dis­cov­ers that the loss given default is really $99 dol­lars, not the $90 that (it was assumed that) the mar­ket knows.

In that case, the new expected util­ity is 0 + .88 × ln(100) or 4.145 utils. Tak­ing the inverse gives e^4.053 = 57.544.

Now, IF every­one knows that the two states are {$1, $100}, then the risk-​neutral prob­a­bil­ity of default satisfies:

p × 1 + (1 — p) × 100 = 57.544,

and equals 42.9%. Remem­ber the actual prob­a­bil­ity of default is still 12%, but the low out­come is par­tic­u­larly low for a log util­ity func­tion. So, the implied, risk-​neutral prob­a­bil­ity of default is more than 3.5 times the true prob­a­bil­ity of default.

Case 4: let’s take Case 3, and assume that the buyer knows that the loss given default has increased from $90 to $99, but a trader or ana­lyst at another firm has not observed that change but has observed the new price of $57.544. In that case, the ana­lyst very likely keep the same LGD assump­tion and solve for a new implied prob­a­bil­ity of default of (using the erro­neous, but assumed $10, rather than the cor­rect $1:

p × 10 + (1 — p) × 100 = 57.544.

In that case, solv­ing for p gives an model-​implied, under the assump­tion of risk-​neutrality prob­a­bil­ity of default of 47.2%. Of course, once again, the real prob­a­bil­ity of default is 12%.

The dif­fer­ence between the 47.2% and $42.9% implied default rates is solely attrib­uted to the (incor­rect) assump­tion about the loss given default. In our expe­ri­ence, the LGD is the least-​challenged, least-​investigated assump­tion used to price CDS and related prod­ucts. In real-​life, it would be extremely com­mon to main­tain that assump­tion in the face of falling prices.

We’ll prob­a­bly refine this post in the com­ing days, but our four sim­ple cases should be suf­fi­cient to cast deep sus­pi­cion on Iceland’s reported prob­a­bil­ity of default, when it is really a model-​implied, default rate under the assump­tion of risk neu­tral­ity. Remem­ber in all of our cases, the real prob­a­bil­ity of default is 12%. The mod­els used to cal­cu­late that rates involve more vari­ables and more cal­cu­la­tions, but apply no more knowl­edge than do our sim­ple exam­ples here.

If you have any ques­tions or com­ments, please write.

Copy­right © 2008 Spero Consulting.

We read in Monday’s (Sep­tem­ber 15) Wall Street Jour­nal that the Fed­eral Reserve nudged Mer­rill Lynch towards a merger: Cri­sis on Wall Street as Lehman Tot­ters, Mer­rill Seeks Buyer, AIG Hunts for Cash.

We think it is a mis­take, and we’re not cer­tain of the Fed’s goal when its pro­pose such arrange­ments. Pre­sum­ably, such a nudge is ratio­nal­ized on the basis of “sta­bi­liz­ing” the finan­cial sys­tem, but we’re not so sure that such a ratio­nal­iza­tion is a jus­ti­fi­ca­tion. It seems like a knee-​jerk reac­tion for the sake of “tem­po­rary sta­bil­ity.” We believe that it has the poten­tial to lead to wider swings – mean­ing greater losses – in the future.

We’re sure that with all that is hap­pen­ing in the finan­cial mar­kets, such a merger seems con­ve­nient and, in some sense, reduces by one the num­ber of prob­lems that the reg­u­la­tors must cur­rently con­front. A quick scan of the papers reveals com­ments like “sav­ing a rel­a­tively healthy patient,” “one less firm to worry about,” etc. But given the small num­ber of such firms, we don’t believe that expe­di­ence jus­ti­fies the long-​term increase in sys­temic risk to the econ­omy. Unless, of course, the “patient” is in much worse health than we have been led to believe.

While most are focus­ing on the short-​term we shall empha­size the long hori­zon. We do this pri­mar­ily because it is our nature and because we do not view the self-​inflicted prob­lems on Wall Street to be an indi­ca­tion of the gen­eral health of the econ­omy. As we have argued many times dur­ing the past sev­eral months, it seems that the losses are par­tic­u­larly con­cen­trated this time. (We argue that it has to do with poorly-​aligned incen­tives, which led to “exces­sive” risk-​taking within the indus­try and pos­si­bly a wis­en­ing up” of some of its cus­tomers.) While there are and will con­tinue to be indi­rect effects that harm the econ­omy, e.g., overly-​tight credit stan­dards and high risk pre­mia, etc, those fric­tions seem sustainable.[1.In fact, it is pos­si­ble that ille­gal immi­grants have hit the hard­est by the reduc­tion in home sales and build­ing, and their incomes are rarely reported to the gov­ern­ment.] (Note also that inter­est rates are low because there is plenty of avail­able capital.)

Our argu­ment is against the merger is based upon a clas­sic centralization/​decentralization argu­ment com­bined with a diver­si­fi­ca­tion argu­ment. How­ever, in the inter­est of brevity, we don’t expand on the centralization/​decentralization argu­ment that much.

For all the bravado and despite some firms insist­ing that they are “a breed apart,” (moo) there is a sub­stan­tial amount of herd behav­ior in the finan­cial ser­vices indus­try, par­tic­u­larly among the large, most heavily-​regulated firms. We attribute much of that mim­icry to the profit-​motive but also to gov­ern­men­tal reg­u­la­tion, where the reg­u­lated insists to the reg­u­la­tor that all is well because it is just like its peers (in terms of ratios, con­cen­tra­tions, etc). (Those out­side of the indus­try would likely be amazed the num­ber of sense­less, nearly content-​free, peer-​group stud­ies per­formed.) Despite those regulation-​induced sim­i­lar­i­ties, there is still some ben­e­fit to decen­tral­iza­tion and diver­si­fi­ca­tion; not all firms have fallen on dif­fi­cult times and those in trou­ble are in trou­ble to vary­ing degrees.
We won’t pro­vide a detailed sta­tis­ti­cal model, but we have in mind a typ­i­cal sum-​of-​normal-​variables-​type of argu­ment. Think of 100 small, equally-​sized, inde­pen­dent firms each mak­ing their own invest­ment deci­sions and earn­ing normally-​distributed returns. 

Now think of that num­ber being reduced due to, say, acqui­si­tions. There are many ways to imple­ment such a sequence, but we’ll take the sim­plest one: one firm begins buy­ing another and another and another, etc. At each point, it suc­cess­fully inte­grates the new sub­sidiary. That means the acquirer begins mak­ing all of the invest­ment deci­sions for the new subsidiary. 

Now in real life, this isn’t quite true. As firms merge they’re not always able to merge sys­tems, etc; so, cer­tain port­fo­lios may remain sep­a­rate and sep­a­rately directed. Even­tu­ally, though, one would expect that asset allo­ca­tion deci­sions would be shifted to a cen­tral author­ity; so, our argu­ment would seem to hold in the long-​run.

After the first acqui­si­tion, the acquirer has twice the mar­ket power as any­one else, and its con­cen­trated returns will be weighed twice as heav­ily as any other firm’s. As it gets larger and larger, its idio­syn­cratic is no longer idio­syn­cratic; it becomes sys­temic. At the extreme, when it is the only remain­ing firm, its return is the mar­ket return. So, the idio­syn­cratic is the sys­temic. (It is like increas­ing the covari­ance among the orig­i­nal 100 firms.)
Now if the gov­ern­ment wants finan­cial sys­tem secu­rity, is hav­ing one mega-​firm the best way to achieve it? We doubt it. Espe­cially, if by sta­bil­ity, one means a VaR-​type argu­ment of not los­ing more than some amount some per­cent­age of the time – if such tail events can even be cal­cu­lated. (We assume the returns were nor­mally; so, they could be.)

With­out per­form­ing any cal­cu­la­tions to deter­mine the effi­cient fron­tier, it seems that the remain­ing firm would have to be some type of “super-​investor” for the expected gains (from higher aver­age returns and lower costs) to out­weigh the poten­tial for mas­sive cor­re­lated losses, and we’ve seen lit­tle evi­dence of super-​investors. We’d pre­fer a set­ting with less effi­cient, stum­bling invest­ment com­mit­tees – with pos­si­bly more local­ized knowl­edge – mak­ing rel­a­tively small mis­takes and hav­ing those mis­take cancel-​out to a set­ting with a few large megafirms mak­ing – pos­si­bly – fewer mis­takes but larger and more cor­re­lated ones.

We’d like to say more, but we are pow­er­less to do so today. Look for more as power returns.

No one can pre­dict the future; see our St. James’ quote on our quotes page. It sum­ma­rizes our outlook. 

At the end of the day, it is still indi­vid­ual tastes, pref­er­ences, and desires that matter. At invest­ment com­mit­tee meet­ings, the argu­ment is not always won by the most knowl­edge­able or most thought­ful or even the luck­i­est. Given a grown-up’s under­stand­ing of human nature and luck, does such con­cen­tra­tion seem wise?

We view this con­cen­tra­tion of decision-​making to be prob­lem­atic. Not for an sin­is­ter rea­son. As we men­tioned, it could sim­ply be a mat­ter of (bad) luck. We joked in April when UBS tried to pin $37 bil­lion in losses on one per­son. As firms grow larger and larger, such an out­come would not be a joke.

The Good: In today’s (Sep­tem­ber 3) The Wall Street Jour­nal, Hol­man Jenk­ins has a nice Busi­ness World col­umn enti­tled, The Infla­tion Hur­ri­cane.

The print version’s blurb per­fectly sum­ma­rizes the essay: “Does the fed­eral gov­ern­ment have to be respon­si­ble for every­thing?” By now, through con­stant media rein­force­ment, we know that every­thing that goes wrong is Bush’s fault, but Mr. Jenk­ins seem to mean why do we tax­pay­ers — at, say, 1,200 feet above sea level — need to sub­si­dize folks who choose to live below sea level? 

Ignore the poor folks, who may be help­less, for whom we have a bit of compassion. Instead focus of the wealthy with vaca­tion homes or con­dos in flood– or storm-​prone coastal areas. Exactly, why should we sub­si­dize their lifestyles? Yeah, we couldn’t come up with a good rea­son, either. Please read the col­umn for your­self. How­ever, before doing so, note that Mr. Jenk­ins does make a few math errors.

The Bad: Mr. Jenk­ins writes very well and we often agree with his point-​of-​view, but his knowl­edge of prob­a­bil­ity and math seems to no bet­ter than that of other jour­nal­ists. (That wasn’t intended as a compliment.) In the essay, he writes about the prob­a­bil­i­ties of a “100-​year flood.” Now, ignore the sta­tis­ti­cal issues like sam­ple size and sta­tion­ar­ity, which affect what con­sti­tutes a “100-​year flood,” and focus solely on his numer­i­cal example. 

He puts the prob­a­bil­ity of a 100-​year flood at one-​percent per year. So, far, so good, but then he puts the prob­a­bil­ity of a 100-​year flood in ten years to be 10%, and 30 years to be 30%. Extrap­o­lat­ing, that would mean that the prob­a­bil­ity of a flood in 130 years is 130%, and that doesn’t make sense.

Mr. Jenkins’ mistake is that he is adding the prob­a­bil­i­ties as if they were inde­pen­dent events, i.e., 1% + 1% + 1%…up to 10 years or 30 years, but he is mis­in­ter­pret­ing what the one per­cent rep­re­sents. The one per­cent is the con­di­tional prob­a­bil­ity of a flood in any given year.

Note that if the flood occurs in, say, two years, there are no more tri­als and the exper­i­ment ends. Mr. Jenk­ins is not inter­ested in the prob­a­bil­ity of hav­ing, say, three “100-​year” floods in the first, say, 50 years. He is inter­ested in only one flood — the next one — and it seems that he is mis­tak­ing the prob­a­bil­ity of the flood occur­ring in year T with the con­di­tional prob­a­bil­ity of the flood occur­ring in year T, given that it didn’t occur in any period from one to T — 1.

The prob­a­bil­ity of the flood occur­ring in the first year is indeed 1%. The prob­a­bil­ity of it occur­ring in the sec­ond year is the prob­a­bil­ity that it didn’t occur in the first year, ( 1 — 1%), multiplied by the con­di­tional prob­a­bil­ity of it occur­ring in the sec­ond year, 1%. That means that the prob­a­bil­ity of the flood occur­ring in the sec­ond year is: (1 — 1%) ·1% = 0.99% = 0.0099. So, the prob­a­bil­ity of a flood within the first two years is the sum: 1% + 0.99% = 1.99%. That isn’t much dif­fer­ent than the 2% that Mr Jenk­ins would have cal­cu­lated, but the dif­fer­ence grows through time as we mul­ti­ply the con­di­tional prob­a­bil­ity of a flood in period T, which is one per­cent (1%), by the prob­a­bil­ity of no flood in the pre­vi­ous T — 1 peri­ods, which is 0.99^{(T — 1)}, which of course gets smaller as T gets bigger.

So, the prob­a­bil­ity of the flood occur­ring in year ten is: (1 — 1%)⁹ · 1% = 0.94%. Sum­ming those prob­a­bil­i­ties for each of the ten years gives 9.6% chance that it would have flooded in the first ten years.. Still not too far from Mr. Jenk­ins’ 10%, but clearly not going in the right direc­tion. Here is one way to write the sum of 9.6%:

= 1% + (1 — 1%) ·(1% + (1 — 1%) · (1% + (1 — 1%) · (1% + (1 — 1%) · (1% + (1 — 1%) · (1% + (1 — 1%) · (1% + (1 — 1%) · (1% + (1 — 1%) · (1% + (1 — 1%) · 1%)))))))))¹

Or, if we fac­tor cor­rectly, we have p(T), the cumu­la­tive prob­a­bil­ity of a flood by period T, we have:

p(T) = 9.6% = 1% ·(1 + 0.99¹ + 0.99² + 0.99³ + 0.99⁴ + 0.99⁵ + 0.99⁶ + 0.99⁷+ 0.99⁸ + 0.99⁹).

For those who care, we could write it sym­bol­i­cally with p tak­ing the role of 1%, and p(T) rep­re­sent­ing the cumu­la­tive prob­a­bil­ity of a flood by year T.

By thirty years, Mr. Jenk­ins sum is off by about 4%, i.e., 30% ver­sus the cor­rect 26.03%. At 100 years, he is off by almost 37%. Pre­sum­ably, he would have 100%, when in fact it is 63.40%. It takes about 460 years to get a 99% per­cent chance that a flood would have occurred by that time; very dif­fer­ent than 99 years that he would say.

So, while we enjoy read­ing Mr. Jenk­ins’ columns, but the next time there is math involved, we do hope that he asks some­one. We’ll do it for free and con­fi­den­tially if he asks.

Finally, note that our cumu­la­tive prob­a­bil­ity of a flood, p(T), is exactly the same as the cumu­la­tive default if, say, New Orleans were a firm and p were the con­stant, con­di­tional prob­a­bil­ity of default (in a dis­crete time setting).

An Illus­tra­tion with Only One Sim­ple Equation.

Update: we have sev­eral newer, related posts: Implied RISK NEUTRAL prob­a­bil­ity of default, redux, expands the analy­sis pre­sented here by using a dif­fer­ent util­ity func­tion and con­sid­er­ing a cou­ple of dif­fer­ent sit­u­a­tions;Risk Neu­tral Val­u­a­tion: There Are at Least Two Expected Val­ues describes a com­mon source of con­fu­sion: the var­i­ous dis­tri­b­u­tions involved; and Price Implied Default Rates pro­vides an exam­ple that is more like a risky bond.

We get a decent num­ber of hits on the lit­tle that we have actu­ally posted about risk neu­tral prob­a­bil­i­ties. We take that as a sign that many folks have ques­tions about them, and that most extant expla­na­tions are poor. We could write an entire post — make that an essay — on why that is the case, but we will defer that to another day.

While we would like to com­plete our planned essays on com­mon risk man­age­ment con­cepts and cal­cu­la­tions, we doubt that it will be pos­si­ble in the near future. (Those read­ers who are think­ing, “no rest for the wicked” should be ashamed of them­selves, and those read­ers think­ing we are too busy to accept new clients should be doubly-​ashamed of them­selves. We shall set up a Pay­Pal account so that they can buy indulgences.)

So, to fill the gap between what we have writ­ten and what we will write, we’ve pre­pared a numer­i­cal exam­ple to illus­trate what risk-​neutral prob­a­bil­i­ties are and how they dif­fer from real­ity. To mas­ter the topic, one needs to know a decent amount from a vari­ety of top­ics in math, prob­a­bil­ity, and eco­nom­ics. Our numer­i­cal exam­ple won’t pro­vide mas­tery, but will pro­vide enough knowl­edge to be dan­ger­ous (and annoy­ing) to oth­ers, and that in itself is an accomplishment.

Before that, though, we note that we once had a pro­fes­sor cor­rect our use of Eng­lish, par­tic­u­larly our word choice. (Now, the reader who thinks, “only one prof?” should also be ashamed of him­self or her­self.) We were cor­rected because we said that we were sur­prised by someone’s behav­ior. He replied that there was no rea­son to be sur­prised—it was per­fectly pre­dictable — but one should be shocked by it. We find the wide­spread mis­in­ter­pre­ta­tion of risk neu­tral prob­a­bil­i­ties to be another exam­ple of a unsur­pris­ing, though shock­ing phenomena.

In a sim­i­lar vein, our hero, Nas­sim Nicholas Taleb, and his colleague, Dan Gold­berg, of Deci­sion Research Lab­o­ra­tory, show how volatil­i­ties (stan­dard deviations) are also mis­in­ter­preted by a sur­pris­ing num­ber of finan­cial pro­fes­sion­als: We Don’t Quite Know What We are Talk­ing About When We Talk About Volatil­ity. Shock­ing, but no longer surprising.

Again, we’ll have more to say about the topic of risk neu­tral prob­a­bil­i­ties, par­tic­u­larly the assump­tions and applic­a­bil­ity of such analy­sis, but our empha­sis here is to pro­vide just enough of a frame­work so that our sim­ple cal­cu­la­tions make sense. So, we begin as always, at the beginning…

Read the rest of this entry »

Or when is a “hedge” not a hedge? —when it is a nedge or a sledge or a wild*** guess, of course.

To para­phrase St. Fran­cis de Sales, the road to hell is paved with good inten­tions because exe­cu­tion mat­ters! (Else­where he scolds per­fec­tion­ism, too, and argues for a bal­ance: do not be rash, do not over-​analyze. Real­ize that it is not a sin to be imper­fect, but it is a sin to do wrong.)

Back on May 21, we posted On N’edges and Sl’edges and Bil­lions Lost in ref­er­ence to a WSJ arti­cle, “Trou­ble Hid in the Hedges.” That arti­cle cited the likely con­tin­u­a­tion of losses at large invest­ment banks to due inef­fec­tive hedges, par­tic­u­larly due to the firms using var­i­ous CMBX indices to hedge CMBS (com­mer­cial mortgage-​backed secu­rity) invest­ments. Some of those losses have now been rec­og­nized: “Lehman Talks a Rosy Talk.” A few weeks ago on June 10, we posted They’re Los­ing $Bil­lions, But Doesn’t She Have Nice Clothes and Shoes? in which we again men­tioned nedges and sledges and alluded to losses from inef­fec­tive hedges.

We define nedges as near hedges and sledges as some­what–like hedges. (Pre­sum­ably, if we were younger and hip­per, we could have thought of iHedges for inef­fec­tive hedges, but we would pre­fer that the word “hedge” not appear in it’s entirety with­out scare quotes or ital­ics, and we are not clever enough to come up with iHedge.)

We invented those terms to indi­cate that while some sta­tis­ti­cal rela­tion­ship might exist between the two items — in the com­mer­cial mort­gage case, a credit index and bonds — only a fool would believe that buy­ing one and sell­ing the other (in what­ever “optimal” proportion) would elim­i­nate the risk of loss. Unless one is buy­ing and sell­ing the same item at the same moment in time, then the com­plete elim­i­na­tion of value or return risk is not pos­si­ble either in the short run or in the long run. Moreover, it is worth noth­ing that elim­i­nat­ing mar­ket risk usu­ally comes at the cost of addi­tional counter-​party risk. That exchange of one risk for another might remind the reader of a vari­a­tion of the arcade game, whack-​a-​mole, with the pre­sumed goal to have smaller and smaller rodents pop-​up after each hit or trans­ac­tion. (Of course, one small mole can still do tremen­dous dam­age to a well-​manicured front lawn.)

In the long run, if the same trans­ac­tion could be repeated ad infini­tum, aver­age losses would likely be reduced if the rel­e­vant, return prob­a­bil­ity dis­tri­b­u­tion func­tion were well-​behaved. Unfortunately, in the real world we can never be quite sure of that fact. We can, in theory, think of nice prob­a­bil­ity func­tions that arise from, say, repeated coin flips (where one gains a dol­lar for heads or loses a dol­lar for tails on each flip). At the limit when the num­ber of flips approaches infinity, extreme losses would be rare, but avoid­ing any loss is not assured. In such an exper­i­ment, break-​even might be expected, but even there it is not guaranteed.

In the short-​run, no such asymp­totic rule comes into play, and like the CMBS exam­ple, there is the chance — in some cases quite a high chance — of los­ing on both legs of the trade.

We believe that exclud­ing fraud, the will­ful igno­rance of first prin­ci­ples is the rea­son behind many huge trad­ing losses. Such losses, while often attrib­uted to bad luck, are not usu­ally due to the lack of advanced “knowledge” or “sophis­ti­ca­tion.” So, it doesn’t take a PhD. It takes an PhD or MBA for­get­ting, ignor­ing, or never inter­nal­iz­ing the basics, espe­cially when some level of pseudo-​sophistication is com­bined with a healthy dose of hubris, pos­si­bly due to the mis­spec­i­fi­ca­tion of past good for­tune. We′ll have other posts on this issue in the near future because we find it quite annoy­ing and extremely dangerous.

In our exam­ple below, we use sim­u­lated val­ues rather than real return data so that we can con­struct it just the way we want it. We pro­vide the exam­ple as an EXCEL spread­sheet to also show the mechan­ics of a sim­ple sim­u­la­tion — as well as make our point about slopes, lin­ear cor­re­la­tions, nedges, and sledges. More­over, it would be dif­fi­cult to pro­vide an exam­ple using CMBX and CMBS rela­tion­ships because the data are so lack­ing. In fact, many such instru­ments were not marked on a daily basis until this past win­ter. Fur­ther­more, note that mark­ing val­ues on a daily basis is quite dif­fer­ent than wit­ness­ing daily trans­ac­tions and observ­ing new daily prices. Deep and liq­uid com­pet­i­tive mar­ket prices do not exist for many of these secu­ri­ties, bonds, and instru­ments; so, often it is mark-​to-​untested-​quote, rather than mark-​to-​market transaction.

Because our exam­ple is an intro­duc­tory level sta­tis­tics prob­lem, some aca­d­e­mics might con­sider it to be a straw man, i.e., a weak oppo­nent designed by us to be eas­ily defeated (by us). Fear not; we take pride in our clev­er­ness and humil­ity, but we try hard not to be devi­ous to fool oth­ers or to be so dull that we fool ourselves.

Instead, we would argue that aca­d­e­mics that would make that crit­i­cism are truly aca­d­e­mic and haven’t spent much time in firms or other orga­ni­za­tions, where it is pos­si­ble to over­hear com­ments like, “We don’t have time to think about a strat­egy. We have to do something!” Or, the equally bewildering, “We have to do some­thing, or it will look like we don’t know what we are doing!” As it turns out, with­out a sub­stan­tial degree of luck or divine inter­ven­tion, such deci­sions rarely pay-​off. As a girls bas­ket­ball coach, such com­ments remind us of closely-​contested, mid­dle school games, when the panic sets into the lead­ing team, team­mates begin to play “hot potato” with the ball (“Ouch, it burns. I don’t want it. You take it.”) and the girls for­get to breathe in their desire not to make a mis­take. (Don’t worry arbiters of sex­ism. We’ve seen it hap­pen with boys and men, too, but just we haven’t expe­ri­enced it while coach­ing them.)

In this file which we have pro­tected, we (1) gen­er­ate three cor­re­lated ran­dom vari­ables, and (2) regress two of those vari­ables, n and sl, against the other one, x. For illus­tra­tive pur­poses, we do this in a very sim­ple fash­ion to show that by design the betas, b, or line slopes should be the same due to the com­mon covari­ance that x shares with n and sl. (In a sim­u­la­tion, they may not be exactly equal.) With x as the regres­sor or inde­pen­dent vari­able, the slopes, b_n, and b_sl, are equal to cov(x, n) ÷ var(x) and cov(x, sl) ÷ var(x), respec­tively.

So, for exam­ple, when traders or risk man­agers don’t have time to think because they MUST ACT, they might select either n or sl as a suit­able hedge for x. Focus­ing only on the expected, least-​square min­i­miz­ing rela­tion­ship, they would be indif­fer­ent between the two. Moreover, such meth­ods and think­ing might allow the trader or man­ager to under­es­ti­mate or ignore the scale of the poten­tial risks that they may be inflict­ing upon the firm via their “hedg­ing activ­i­ties,” and this is where the road leads to hell. For example, if the firm owns x and uses n or sl to hedge by short-​selling either of those instru­ments, then low val­ues of x com­bined with high val­ues of n or sl would be par­tic­u­larly dam­ag­ing, and the chance and mag­ni­tude of the loss is related to the total vari­ance of n or sl—not just the covari­ance. With suit­able lever­age, it is pos­si­ble to lose big, with a nedge like n, whereas with sl it is pos­si­ble to gen­er­ate large, two-​legged loses with­out leverage.

This can be seen in the fol­low­ing graph where we have cho­sen many of the para­me­ter val­ues so that the two depen­dent vari­ables can be eas­ily distinguished. Notice, also, that as promised, both n and sl have the same slope with respect to x, and in theory, either one could be used to “hedge” x, as both would have the same expected ben­e­fit. It is just that there is a dif­fer­ence between expec­ta­tion and realization.

Now, some may argue that they are more sophis­ti­cated than our por­trayal and would not employ such a sim­plis­tic tech­nique unless no other alter­na­tives existed. How­ever, given its per­va­sive use, we find that dif­fi­cult to believe that it is a method of last resort. More impor­tantly, while there are dif­fer­ent and more com­plex hedg­ing strate­gies, unless the mar­ket risk is elim­i­nated via for­ward pur­chases or sales, all prob­a­bil­ity and statistics-​based strate­gies suf­fer from the same under­ly­ing prob­lem that we illus­trate here — there is resid­ual ran­dom­ness and the pos­si­bil­ity to lose on both legs.

With more com­pli­cated hedg­ing strate­gies it might be more dif­fi­cult to see this prob­lem and while some addi­tional vari­a­tion might be reduced, some still remains. Thus, the old adage of los­ing sight of the for­est for the trees seems par­tic­u­larly rel­e­vant here. We also note the many folks spend much time and energy ana­lyz­ing ret­ro­spec­tive rela­tion­ships to deter­mine such hedg­ing strategies, and, of course, such rela­tion­ships need not be per­sis­tent — the oft-​mentioned prob­lem of induction. Dynamic, unsta­ble rela­tion­ships will inval­i­date his­tor­i­cal analy­ses as will sta­ble rela­tion­ships with rare events and small sam­ple histories. CMBX — CMBS series have extremely small sam­ples. So, bas­ing hedg­ing strate­gies and posi­tions on his­tor­i­cal analy­ses poses addi­tional risk due to mis­spec­i­fi­ca­tion. That is why the recent and rel­a­tively long period of low volatil­ity in many mar­kets was so dam­ag­ing to those who for­got or ignored this fact — per our point of ignor­ing first principles.

In addi­tion, while CMBX offers pro­tec­tion on a bas­ket of CMBS, if either (1) your firm holds secu­ri­ties not in the CMBX bas­ket or (2) your firm holds only some secu­ri­ties in the bas­ket, your firm is likely to have sledges rather than hedges when using an index to try to off-​set a position.

Finally, a slightly more tech­ni­cal crit­i­cism is based on our obser­va­tion that some ana­lysts seem to for­get that risk-​neutral pric­ing meth­ods don′t actu­ally elim­i­nate risk; they just, in some sense, ignore it for cer­tain purposes. (Out-of-sight is often out-​of-​mind for the har­ried and/​or unwit­ting.) We won′t dwell too much on this issue here or any­where else until we pub­lish more ref­er­ence mate­r­ial, includ­ing a sim­ple expla­na­tion of risk neu­tral val­u­a­tion. How­ever, please do note that some folks do tend to believe that only expec­ta­tions mat­ter, and often these same folks tend to also for­get or repress the social and behav­ioral ele­ment of trad­ing, espe­cially if they haven’t shown a pre­vi­ous inter­est in human nature in their careers or education. Such obser­va­tions make us won­der whether their hir­ing man­agers are cyn­i­cal or naively-​ignorant? See this post, Caveat Emp­tor, for a related complaint.

P.S. As we men­tioned, the EXCEL sim­u­la­tion file is pro­tected; so, other than click­ing the link to our web site or press­ing the func­tion key, F9, to gen­er­ate a new batch of ran­dom num­bers, there is lit­tle that one can do. Inter­ested par­ties should con­tact us directly for a non-​protected version.

Update: If this topic is of inter­est, please see our more recent posts. Many pro­vide bet­ter numer­i­cal illus­tra­tions of risk neu­tral prob­a­bil­i­ties: Implied Risk Neu­tral Prob­a­bil­i­ties (of Default) from August; implied RISK NEUTRAL prob­a­bil­ity of default, redux from Octo­ber 9; Risk Neu­tral Val­u­a­tion: There Are at Least Two Expected Val­ues from Novem­ber 13; and Price Implied Default Rates from Decem­ber 2. We also have a new multi-​period exam­ple: Multi-​period Bond Price Implied Default Rates and CDS from mid-​December. All of our posts are designed to illus­trate and com­mu­ni­cate the basic notions so that the reader has the req­ui­site foun­da­tion to con­sider addi­tional details per their spe­cific interest.

This weekend′s WSJ has an arti­cle enti­tled, “Bond Insur­ers Inflict Fur­ther Pain on the Mar­ket.” While the arti­cle is inter­est­ing for sev­eral rea­sons, in this post we focus on one tiny aspect of it, which will still take a long time to explain. In the arti­cle, there is a graph labeled “Default Fears,” and it includes the sub­ti­tle, “The annual cost to buy pro­tec­tion against default on $10 mil­lion of MBIA and Ambac debt for five years.” Accord­ing to the graph, such pro­tec­tion costs about $2.0 mil­lion per year for each firm. The pur­pose of this post is to describe the mean­ing of the $2.0 mil­lion, includ­ing var­i­ous interpretations.

To sim­plify mat­ters, we assume that the pro­tec­tion, i.e., the insur­ance con­tract, lasts only one year. In many ways, such a pol­icy is almost iden­ti­cal to term life insur­ance. (For those inter­ested in more advanced topics, we will try to add a post that ana­lyzes a multi-​year, term con­tract, but we want to start with the sim­plest case.)

The insur­ance pre­mium will depend upon the (1) pref­er­ences, (2) beliefs, and (3) endow­ments of the buy­ers and sell­ers. In the near future, look for a rather detailed essay on these top­ics, but for now think of (1) pref­er­ences as the parties′ likes and dis­likes, includ­ing their lev­els of risk aver­sion; think of (2) beliefs as fore­casts or spec­i­fi­ca­tions of what might hap­pen, which hope­fully can be expressed as prob­a­bil­ity dis­tri­b­u­tions; and think of (3) endow­ments as the par­ties′ cur­rent and future resources.

We men­tion these fac­tors because ceteris paribus, which means every­thing else equal, we know that (a) the more risk averse the per­son, the more they are will­ing to pay for a given amount of pro­tec­tion, and (b) the greater the belief in the like­li­hood of some­thing bad hap­pen­ing — in this case default — the more they are will­ing to pay for a given amount of pro­tec­tion. If we could express these pref­er­ences and beliefs as math­e­mat­i­cal func­tions, and if the func­tions pos­sessed cer­tain char­ac­ter­is­tics, we could then deter­mine the exact price that the par­tic­i­pants would be will­ing to pay or receive.

Unfor­tu­nately, in real life, it is impos­si­ble to have this knowl­edge. One of the great things about a free econ­omy is that this infor­ma­tion isn’t needed to trans­act and trade. (Within the past few years, researchers have won Nobel prizes deter­min­ing just how much (or how lit­tle) infor­ma­tion is needed for such a math­e­mat­i­cal ver­son of the econ­omy to func­tion properly.)

In the real world, one can assume that most peo­ple are risk-​averse much of the time, but as we men­tioned and reit­er­ate, no one knows these preferences. In addition, no one except God knows the true prob­a­bil­ity dis­tri­b­u­tion func­tion of future events. All we do know is that through com­bi­na­tions of beliefs, pref­er­ences, and endow­ments pos­sessed by the poten­tial and actual buy­ers and the poten­tial and actual sell­ers, we see a price, which on Fri­day was a pre­mium of $2.0 mil­lion for $10 mil­lion of credit pro­tec­tion for one year.

Can any­thing fur­ther be said? Thanks to some very clever researchers, yes, we can say more as long as we under­stand the hypo­thet­i­cal nature of such statements.

Risk Neu­tral­ity: These finan­cial econ­o­mists real­ized that there was no hope dis­cov­er­ing actual (risk-averse) preferences or the true way that ran­dom­ness occurs (the actual prob­a­bil­ity func­tions), but these researchers were smart enough to real­ize that cer­tain prob­lems could be solved in an eas­ier man­ner if they assumed that indi­vid­u­als involved in eco­nomic activ­i­ties were risk neu­tral, rather than risk averse, and then pro­ceeded with their analy­ses under that assumption.

Assumed Dis­tri­b­u­tions & Implied Val­ues: The assump­tion of risk neu­tral­ity allows observed prices to be viewed as the expected val­ues of pos­si­ble cash flows. That is gen­er­ally not true in the real world, but it is okay in a hypo­thet­i­cal, risk neu­tral world.

So, once pref­er­ences were assumed to be risk neu­tral, and once the researcher fur­ther assumed a spe­cific form for the prob­a­bil­ity dis­tri­b­u­tion func­tion, e.g., nor­mal or log-​normal or expo­nen­tial or what­ever, he could set the price equal to the expected value and solve for any unknown prob­a­bil­ity dis­tri­b­u­tion para­me­ters, like the mean or the vari­ance of a nor­mal dis­tri­b­u­tion. Those implied val­ues could then be used for other pur­poses, i.e., valuing other things that behave in a sim­i­lar way.

If the reader is famil­iar with options, this is the same idea behind implied volatil­i­ties, which is short-​hand for the implied stan­dard devi­a­tion of the assumed dis­tri­b­u­tion under the assump­tion that mar­ket par­tic­i­pants are risk neu­tral. (Of course, it is actu­ally inferred, but we′ll ignore that fact.)

Unfor­tu­nately, there are many mar­ket par­tic­i­pants who use implied volatil­i­ties and won­der why they don’t equate to actual, his­tor­i­cal volatil­i­ties. Ignor­ing the obvi­ous prospective-​retrospective dif­fer­ences, it is in some sense like won­der­ing why ten miles isn’t equal to ten kilo­me­ters or won­der­ing why a black-​and-​white pho­to­graph doesn’t cap­ture all aspects of a col­or­ful set­ting. We will have more to say about this below, but now we get back to the issue at hand.

Assumed Expected Loss: If we observe a pre­mium of $2.0 mil­lion on $10 mil­lion of debt for a one-​year pol­icy, and if we assume that par­tic­i­pants are risk-​neutral, then in a com­pet­i­tive mar­ket (another big assump­tion that we will ignore) we can then con­clude that the (risk neu­tral) expected loss from default is $2.0 million.

In fact, if real buy­ers of pro­tec­tion are isk-​averse, then the pre­mium of $2.0 mil­lion implies that they expect the loss to be less than $2.0 million. This has to do with the fact that being risk-​averse, the buy­ers don′t like to take chances; so, they will pay a (risk) pre­mium above the actual expected loss to sta­bi­lize their wealth, i.e., to avoid the risk of los­ing greater amounts.

When all goes right, this is how insur­ance com­pa­nies make money. Buy­ers are will­ing to pay more than their expected losses to avoid larger, cat­a­strophic losses. On aver­age, and when losses aren′t too cor­re­lated, the insur­ance com­pany prof­its by the sum of the dif­fer­ences, which is the sum of the risk premiums.

Prob­a­bil­ity of Default: For­tu­nately, in cases of default (in dis­crete time), we have very sim­ple prob­a­bil­ity dis­tri­b­u­tions — either the firm defaults, or it does not; so, we don′t have to choose a dis­tri­b­u­tion func­tion, A sim­ple binary one works. We will let p rep­re­sent the prob­a­bil­ity of default, and (1 — p) is the prob­a­bil­ity of sur­vival or no default.

Face Value & Loss Given Default: We assume that the $10 mil­lion men­tioned above is the face value of the debt, which we will label as F. That means that at the bond′s matu­rity, if the firm still exists, it will have to repay $10 million.

In the article, the author isn′t very clear whether the $10 mil­lion is the loss given default or the face value, but such pro­tec­tion con­tracts are usu­ally spec­i­fied on the notional or face value of the debt; so, that is what we will assume.

As it turns outs, except in cases of com­plete fraud, cred­i­tors usu­ally recover some por­tion of loan or bond value. Banks usu­ally col­lect or recover more than bond-​holders because they are bet­ter orga­nized for such events; it’s always ad hoc for a par­tic­u­lar coali­tion of bond­hold­ers. Regard­less, to esti­mate or infer a prob­a­bil­ity of default, we need to assume a recov­ery rate, or its com­ple­ment, a loss given default rate. We will use L to rep­re­sent the loss given default rate.

Taken together, the prob­a­bil­ity of loss, p, mul­ti­plied by the loss given default rate, L, mul­ti­plied by the face value, F, is equal to the expected loss assum­ing risk neu­tral­ity, which under that assump­tion is equal to the price. (We’re ignor­ing coun­ter­party credit risk in this example.)

price = expected loss under risk neu­tral­ity = p · L · F + (1 — p) · 0 · F = p · L · F

With­out a default there is no loss; so, the sec­ond adden­dum is equal to zero.

Thus, hypo­thet­i­cally at least, the pre­mium is equal to the prod­uct of the risk-​neutral prob­a­bil­ity of default, the loss given default rate, and the face value. We observe the pre­mium of $2.0 mil­lion and the face value of $10 mil­lion, So, we have 2 = p · L · 10 or p · L = 0.20 = 20%. With two vari­ables and one equa­tion, if we assume one, we can solve for the other.

Gen­er­ally, folks assume a loss given default rate, L, and solve for the risk-​neutral, implied prob­a­bil­ity of default, p. We are pedan­tic and also believe that proper nota­tion informs and reminds; so, we will write this implied (or solved for) prob­a­bil­ity of default as p(L). Sup­pose the loss given default rate is 40%, then it is easy to see that in our case, the risk-​neutral prob­a­bil­ity of default would then be 50%. The graph below shows the implied prob­a­bil­ity of default for each assumed loss given default rate.

Now, there are a cou­ple of points worth mak­ing. The first regards the par­tic­u­lar sit­u­a­tion described in the article.

Gen­er­ally, the loss given default rate is less than 100% (of face value). How could it not be? Because MBIA and Ambac guar­an­tee the debt of oth­ers, it is easy to imag­ine that some orga­ni­za­tions could lose more than the face value of their MBIA and Ambac bonds if either firm defaults, and so it might seem that the loss rate in greater than 100%. (But that’s because the notion­als of other, guar­an­teed items might not be included.)

Usu­ally, firms will assume that the other expo­sure can be con­verted into equiv­a­lent notional value of bonds rather than set a higher loss rate, and the con­ver­sion requires a rather sub­stan­tial assump­tion. That is one of the rea­sons that say, $1 bil­lion worth of pro­tec­tion might be writ­ten on $100 mil­lion bond issue.

The arti­cle cited above (and many other arti­cles last win­ter, spring, and sum­mer) have described the losses that large banks have suf­fered on their inven­to­ries of munic­i­pal bonds and other secu­ri­ties due to the decreased cred­it­wor­thi­ness of the insurers.

In fact, the arti­cle men­tioned that those banks may face another $10 bil­lion of losses due to the recent rat­ings down­grades of the insurers.

Many read­ers will recall that dur­ing this past win­ter, a con­sor­tium of banks attempted to sta­bi­lize the cap­i­tal posi­tions of the insur­ers in hope of pre­vent­ing fur­ther write­downs. Of course, the cir­cu­lar­ity of some of these trans­ac­tions should remind one of a snake eat­ing it′s own tail, i.e., we will invest in them so that they (who are now us) guar­an­tee that we lose noth­ing if bad things hap­pen. Not that dif­fer­ent than writ­ing life insur­ance on one′s own life…if I die, come to the house so that I can pay you…

The sec­ond point is more gen­eral. As we men­tioned above, many peo­ple expect to see risk-​neutral prob­a­bil­i­ties and volatil­i­ties in real life. That is kind of like using a map to nav­i­gate and then believ­ing the world is flat because the map is use­ful within some rel­e­vant range or being sur­prised the states aren’t actu­ally red, green, blue, or yel­low per the map.

Aside: With our own ears, we′ve heard man­agers argue that their sub­or­di­nates were wrong because the latter′s his­tor­i­cal, default rates didn′t equal the former’s risk-​neutral, implied default prob­a­bil­i­ties. Your apple doesn′t look like our orange. Your orange must be wrong! Where did you get it? Do you know what you are doing? (The fact that there need not be any rela­tion­ship between his­tor­i­cal and prospec­tive rates is a sep­a­rate issue.) As we’ll explain, there dif­fer­ent notions mea­sured in dif­fer­ent ways.

Con­sider that the more risk-​averse the buyer, the more he is will­ing to pay for insur­ance; so, the higher the implied, risk-​neutral prob­a­bil­ity of default, and thus the greater the dif­fer­ence between a fixed “true” or buyer-​believed default rate and the implied default rate based on the pre­mium. (Again, this assumes all else equal and some notion of know­ing the true rate and hav­ing a clear notion of more risk averse, etc.)

So, why would any­one think that the two notions (actual and risk neu­tral rates) should be equal — that would hap­pen only if risk neu­tral folks set prices in the real world, and as it turns out that because they are NEUTRAL or indif­fer­ent towards risk, they tend not to be the most promis­ing insur­ance customers.

We will likely add to this post as we refine this last crit­i­cism and recall for­got­ten events where the method­ol­ogy was abused and mis­un­der­stood. Also, look for sev­eral risk-​related and valuation-​related essays in the com­ing weeks. After we pub­lish those essays, we will have a plat­form from which to crit­i­cize sim­i­lar shoddy think­ing (but in more com­pli­cated sett­tings.) At some point in the future, we will likely turn this post into an essay, and list it under the fal­lac­ies page.

Today’s Heard on the Street com­men­tary in The Wall Street Jour­nal, “Trou­ble Hid in the Hedges,” reminded us that we have been mean­ing to write about nedges and sledges for some time. The reg­u­lar reader may have noticed that we are rather pedan­tic, par­tic­u­larly about the mean­ing of words. In finan­cial risk-​related activities, three words are fre­quently mis­used, and two of them are almost never used prop­erly. Those three words are: (1) mar­ket, (2) arbi­trage, and (3) hedge. Per the title, we will pro­pose a few alter­na­tives to “hedge.” In another post we will have more to say about the overuse of the word “mar­ket,” par­tic­u­larly when it is used as a label for a hap­haz­ard sequence of sporadic exchanges.

Dur­ing our doc­toral stud­ies, we learned that “arbi­trage” meant “risk­less prof­its.” Usu­ally such (an eco­nomic) rent accrued through the simul­ta­ne­ous pur­chase and sale of the same item with two dif­fer­ent par­ties, pos­si­bly in two dif­fer­ent locations. If such oppor­tu­ni­ties do exist, they are usu­ally ephemeral. As we lament the pass­ing of our youth, we think their tran­si­tory nature is well-​captured by the refrain of Bruce Springsteen’s song, Glory Days: “…Glory days in the wink of a young girl’s eye…” (Although we pre­fer “blink” to “wink” as it seems more tragic.)

In real life, offsetting trans­ac­tions are usu­ally sequen­tial rather than simul­ta­ne­ous, and the time dif­fer­ence can be sub­stan­tial. In these cases, indi­vid­u­als often claim that their posi­tions are “hedged.” Now, when we learned the def­i­n­i­tion of “arbi­trage,” we also learned that “hedged” meant that the off­set was com­plete. Either via an imme­di­ate off­set­ting trans­ac­tion, or through a sequence of cost­less future ones, the arbi­trageur was immu­nized against the pos­si­bil­ity of loss.

Now, there are, in fact, ways to elim­i­nate risk and be fully-​hedged, but for many traders, these tac­tics are expen­sive rel­a­tive to the level of expected prof­its. (Makes one won­der about the effi­ciency of that desk’s risk-​return propo­si­tion.) Instead, approximate hedges are often used. These approx­i­ma­tions may reflect the direc­tion of the lin­ear rela­tion­ship, i.e., the cor­re­la­tion, between instru­ments but they ignore the strength of the rela­tion­ship. Two depen­dent vari­ables may have the same regres­sion coef­fi­cient or β (or slope) with respect to an inde­pen­dent vari­able, but the con­fi­dence inter­vals around the regres­sion line can be sub­stan­tially dif­fer­ent. These dif­fer­ences impose risk, which may be very costly. As today’s col­umn explains, sev­eral firms have recently lost bil­lions of dol­lars on bad hedges. (Time per­mit­ting, we’ll update the post with a graph to rein­force the fact that regres­sion lines are prob­a­bilis­tic in nature.)

We refer to these approx­i­mate hedges as “n’edges,” which is the con­trac­tion of “near hedges,” or more gen­er­ally as “sl’edges,” which is the con­trac­tion of “some­what like hedges.” While use of the for­mer term seems to give the risk mit­i­ga­tor the ben­e­fit of doubt, we hope that the lat­ter term con­jures the image of the use of a large, blunt and, pos­si­bly, inap­pro­pri­ate instru­ment. We believe our hyper­bole is jus­ti­fied given the rel­a­tive ease in which firms and indi­vid­u­als claim risk immu­nity due to such sledges. For more on the tech­ni­cal aspects of repli­ca­tion, we point the inter­ested reader to many of Taleb’s papers deal­ing with repli­ca­tion strate­gies at www​.Fooled​ByRan​dom​ness​.com. We will shortly post an essay and a data set to illus­trate a com­mon ana­lyt­i­cal mis­take when per­fom­ing sta­tis­ti­cal analy­sis. That essay also allows us to jus­tify our motto: Cog­i­ta­tio Ante Com­pu­ta­tion.

There is a decent arti­cle in today’s The Wall Street Jour­nal about econ pro­fes­sors at Prince­ton research­ing finan­cial mar­ket bub­bles. We write “decent” because there is much with which to both agree and dis­agree. In fact, one could write a book on the topic, and sev­eral have. Today, we offer only a short post. Our goal is to relate the the­ory to the over-​valuation of equity invest­ments, par­tic­u­larly where the float is very low. We have in mind par­tially corporate-owned, partially publicly-​owned sub­sidiaries where the unre­stricted pub­lic owns a tiny frac­tion of the shares outstanding.

Besides the WSJ arti­cle we — again — refer the inter­ested reader to Rick Bookstaber’s excel­lent book, A Demon of Our Own Design, espe­cially his dis­cus­sion of the burst­ing of the dot-​com bub­ble on about page 170. One will find many sim­i­lar­i­ties between his descrip­tion and exam­ples in the WSJ arti­cle. We empha­size a cru­cial sim­i­lar­ity by quot­ing from today’s arti­cle, “In mar­kets with lots of dis­agree­ment about val­ues, the opti­mists are bet­ter able to dom­i­nate when there are fewer shares avail­able.” The idea is that when there are few shares avail­able for pur­chase, say, when one or two large share­hold­ers hold the vast major­ity of shares — or as Book­staber dis­cusses when many out­stand­ing shares are restricted — the opti­mists on the demand curve set the price.

The impli­ca­tion for the val­u­a­tion of such sub­sidiaries should be clear, and makes us won­der about an antonym for syn­ergy? ¹ Our point: the mar­ket value of 100% of the sub­sidiary is likely to less than the sum of its parts. To be more explicit, sup­pose the float and daily trad­ing vol­ume are the same and equal 1% of the out­stand­ing. It is unlikely that the 99% restricted or non-​tradeable piece is worth 99 times that one percent’s mar­ket value. (Try plac­ing that order in the market.)

Depend­ing upon who owns the 99% and in what frac­tions, it is unlikely that the account­ing treat­ment would be mark-​to-​market, but if it were, our impli­ca­tion adds an addi­tional layer of com­plex­ity to any such mark­ing. More impor­tantly, it makes one won­der whether CEOs and CFOs should act with greater care when they report the mar­ket value of nearly-​wholly-​owned sub­sidiaries by extrap­o­lat­ing the price of very thinly-​traded shares. 

Okay, we can’t resist mak­ing three other comments.

1. We sus­pect that Prof Wei Xiong is attempt­ing to use non-​technical lan­guage and also to be hyper­bolic when he is quoted as saying, “The two most impor­tant char­ac­ter­is­tics of a bubble…People pay a crazy price and peo­ple trade like crazy.” How­ever, it is worth not­ing that almost any non­par­tic­i­pant could make those same state­ments about any trad­ing activ­ity and most other trans­ac­tions, as well. With­out restat­ing the assump­tions behind it, think of the poten­tial buy­ers and sell­ers at the top points of a typ­i­cal X formed by the inter­sec­tion of supply-​demand curves on an inverse demand curve. More­over, dear reader, think of all the things that one doesn’t buy and sell daily. It is just crazy. The actual trades are at the mar­gins of usu­ally large pop­u­la­tions of buy­ers and sell­ers. Such activ­ity doesn’t seem normal.
2. The arti­cle also states that “Mr. Brun­ner­meier and Stanford’s Ste­fan Nagel found that hedge funds on the whole “skill­fully antic­i­pated price peaks” in indi­vid­ual tech stock…” We haven’t read their work, but we encour­age inter­ested par­ties to: (1) read Mr. Bookstaber’s book, (2) read about sur­vivor­ship bias, and (3) contemplate whether such find­ings will hold if the past ten months were studied.
3. Like Alan Greenspan, we thought tech­nol­ogy stocks were unrea­son­ably val­ued in the mid-​90s, too. We didn’t make a speech about it in 1996 or coin a phrase like “irra­tional exuberance,” and for at least three years, the mar­ket seemed to ignore both of us. How could any­one rea­son­ably expect that a 70-​year-​old-​man, or even a younger and much hand­somer man, could fully grasp all of the tech­no­log­i­cal and finan­cial impli­ca­tions of every­thing that was hap­pen­ing then (or now)? Were we cor­rect and like Cas­san­dra in 1996, or we were the just lucky in 1999? We know some­one who in the sum­mer of 2003 claimed the 10-​year Trea­sury rates were going above 5%, pos­si­bly to 6% — 8% and stay­ing there for a long time. If and when that happens, we are sure they will take credit for their pre­dic­tion. While there have been ups and downs, the rate (~4%) is the same as at the time of the pre­dic­tion. Back then, “every­body knows that rates are going up.”

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 Footnotes:

We recently met an indi­vid­ual who is work­ing towards a PhD in math­e­mat­i­cal finance. He had a mas­ters degree in some­thing tech­ni­cal, too, and if we recall cor­rectly, has been in the PhD pro­gram for four years.

He wanted us to read his paper regard­ing the volatil­ity of volatil­ity. We didn’t have the time, but a quick skim revealed that it was suit­ably loaded with math and graphs — quite tech­ni­cal. ¹

When indi­vid­u­als dis­cuss volatil­ity, they usu­ally speak of one of two kinds: (1) his­tor­i­cal or real­ized vol or (2) implied vol. His­tor­i­cal or real­ized vol is the mea­sured ran­dom­ness in a past sequence of obser­va­tions of a par­tic­u­lar variable.

Implied vol is an esti­mate of future ran­dom­ness, and it is called “implied” because it is found by solv­ing a model, which will have any num­ber of assump­tions and restric­tions. For exam­ple, imag­ine a pric­ing model (a func­tion) of, say, input five vari­ables. ² Now, imag­ine that one can observe the actual price and four of the five input vari­ables. Then, under suit­able con­di­tions, one can solve for the fifth, pos­si­bly unknown, variable that is implied by the func­tion or model.

If one is using the Black-​Scholes option pric­ing model or one of its vari­ants, then (usually) the price of the option is known, and four of the five input vari­ables are known or can be inde­pen­dently esti­mated: (1) the exer­cise value of the under­ly­ing vari­able, (2) the cur­rent value of the under­ly­ing vari­able, (3) the time until expi­ra­tion, and (4) the risk-​free rate. Using the appro­pri­ate algo­rithm, one can then find the implied vol that sets the func­tion equal to the observed option price.

All of that just to say that when folks are con­cerned about the vol-​of-​vol, it almost always involves options, and when peo­ple are con­cerned about options or instru­ments with embed­ded options, they almost always use Black-​Scholes or some vari­ant. So, Black-​Scholes and vol-​of-​vol are kind of like bread and peanut butter. A lot things can be eaten with bread (Black-​Scholes), but peanut but­ter (vol-​of-​vol) is used almost exclu­sively with bread (B-​S). Of course, we know about peanut but­ter pie and crack­ers and cel­ery, but it is a decent anal­ogy, and we are always will­ing to hear of bet­ter ones.

So, what does all of this have to do with our new acquain­tance? We asked him to describe the Black-​Scholes model, which is like the bread of any vol sand­wich. His reply: when he had left his coun­try of birth — about six years ago — they did not have equity options; so, he didn’t really know it well enough to explain it to oth­ers. Not the answer that we sought. Morals: (1) make cer­tain that you have the right per­son ask­ing the right ques­tions, and (2) thought before cal­cu­la­tion is the pre­ferred order for the two.

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Foot­notes: