Spero Consulting

Archive for October 9th, 2008

We’re not sure whether to start this post on a sar­cas­tic tone or not. The sar­cas­tic start is “it’s too bad the short-​selling ban expired, could the dear reader imag­ine how far stocks would have fallen with­out it?” But, the mar­ket did tank again today.

Instead, we’ll note that when such a ban is arbi­trar­ily imposed dur­ing a seem­ingly neg­a­tive event, most par­tic­i­pants infer/​conclude that the event is worse or much worse than they would have oth­er­wise thought or were led to believe – pos­si­bly worse than any­one ever imagined. (By arbi­trar­ily, we mean spur-​of-​the-​moment and not due to a mechan­i­cal, pre­com­mit­ment to impose such a ban when, say, a mar­ket vari­able hits a pre-​established barrier.)

Such infer­ences tend to make investors and other indi­vid­u­als ner­vous, pos­si­bly pan­icky, and much more likely to sell their hold­ings thereby neg­a­tively affect­ing prices and bring­ing about a result oppo­site of the ini­tial desire affect.

Now, such a con­clu­sion seems obvi­ous even to some­one as unso­phis­ti­cated as we; so, that makes it seem that gov­ern­ment offi­cial are so extremely ner­vous, pan­icky, and out-​of-​their-​element that they are at or pass the point of sense­less­ness. They remind us of middle-​schoolers play­ing bas­ket­ball, as the ten­sion increases, they tend to stop breath­ing when they have the ball. The lack of oxy­gen does not enhance judg­ment, and bad things – turnovers and missed shots – tend to accu­mu­late (and exac­er­bate that nervousness_​. It almost looks like they’re play­ing with plas­tic bags over their heads or play­ing under­wa­ter with­out SCUBA: must…get…rid…of…ball…to…restore…air…flow…gasp…gasp.

When other firms, out­side of the ini­tial ban, try to be cov­ered by it, too, investors tend to get sus­pi­cious of them, too. (It is like being sub­poe­naed as a wit­ness and ask­ing for immu­nity.) The nat­ural response by both inter­ested and indif­fer­ent par­ties is that the firm is hid­ing some­thing, and they tend not to hide good news.

As we have stated repeat­edly in other posts dur­ing the past sev­eral weeks, we think that both elected and unelected fed­eral offi­cials have per­formed mis­er­ably dur­ing the recent cri­sis, and, in fact, have made it worse through their conduct. See almost any­thing that we have writ­ten in the past few weeks, includ­ing, Planes, Trains, and Auto­mo­biles and Banks and Farms and States…, Shame on Them!, Prin­ci­ples Lost and More, SOX’s Roles in the Finan­cial Cri­sis of ‘08, and >Out of Their Ele­ments to list just a few.

The Fed’s recent deci­sion to lend directly to firms (via the com­mer­cial paper mar­kets) and the SEC’s unwill­ing­ness to extend the ban are the only hope­ful, sen­si­ble actions that we’ve observed are fear­ful lead­ers take in quite some time.

the rest of the world complains.

On Tues­day, we posted The Impor­tance of the Rule of Law, which describes how Russia’s prob­lems are unique from the West’s and, as we see it, are the result of self-​destructive poli­cies and desires.

In that entry, with­out pro­vid­ing addi­tional links to arti­cles, we insin­u­ated that The Wall Street Jour­nal’s reporters seemed to be agree­ing with Russ­ian lead­ers that, of course, Rus­sia was an inter­na­tional vic­tim of the credit cri­sis in the USA. 

In our mind, the only link­age – while large – is the decline in energy prices as the US econ­omy adjusts – by using less – to the higher level of oil and gas prices.

It is nice to see the The Wall Street Jour­nal edi­to­r­ial page crit­i­ciz­ing the Russ­ian Pres­i­dent for his attack on the U.S in its essay, Dmitry’s Dia­tribe.

Don’t expect other coun­tries to come to our defense when the US is crit­i­cized, how­ever. The recent losses in global equity mar­kets show that the rest of the world still heav­ily depends upon the USA. Those coun­tries, big and small, fast or slow-​growing, suf­fer at least pro­por­tion­ally when some­thing neg­a­tive hap­pens in the U.S., and many remain crit­i­cally depen­dent upon the US.

It seems that if we don’t buy their stuff, no one does, either, espe­cially their own citizens.

Don’t expect grat­i­tude for keep­ing the world afloat. Instead expect con­tin­ued resent­ment for that depen­dence upon us; for­eign lead­ers are kind of like teenagers in that respect.

So, while the teenagers of the world will crit­i­cize us, the inse­cure politi­cians in the US will try to be more like them, when all-​the-​while, they want to be more like us. (Kind of like Danny and Sandy in Grease.)

Aside: say, doesn’t adopt­ing inter­na­tional account­ing stan­dards seem like a great idea right about now? We hear that stan­dards elim­i­nate greed, stu­pid­ity, inef­fec­tive gov­ern­ment poli­cies, and risk con­cen­tra­tions. If we’d fol­lowed them, we could be as well of as the rest of the world right now. (And, had we fol­lowed them sev­eral years ago, Con­gress would never, ever have pres­sured Fan­nie Mae and Fred­die Mac to do stu­pid things, but who knew?)

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.

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