Spero Consulting

Archive for December 2nd, 2008

In early Octo­ber, we wrote So Much for the ‘Hedge’ Part of Hedge Funds. In that post, we noted that many equity funds were suf­fer­ing like the rest of us.

A month ear­lier, way back on 9/​11, in Spec­u­la­tors, Hedge Funds and Lehman Broth­ers, we dis­agreed with a sub­ti­tle from an arti­cle in the pre­vi­ous day’s edi­tion of The Wall Street Jour­nal. The sub­ti­tle was: Big Cash Bal­ances Sig­nal Bull­ish Times Ahead.

We wrote, “We’re not so sure. It seems more likely that many funds are plan­ning for (or have been warned of) massive with­drawals and redemp­tions in the com­ing months. We’ll see if our con­jec­ture is cor­rect by January.”

It’s not Jan­u­ary – merely early Decem­ber – but ignor­ing increases in Trea­sury bond prices, there is no sign of bull­ish­ness any­where in the finan­cial mar­kets. And, hedge fund redemp­tions – or at least requests for redemp­tions – seem to con­tinue unabated.

Today The Wall Street Jour­nal reports in J.P. Morgan’s Once-​Hot High­bridge Goes Cold that investors have requested with­drawals of 36% of assets in its main multi-​strategy fund.

An arti­cle in yesterday’s edi­tion of The Wall Street Jour­nal, Tudor Halts Redemp­tions in Largest Fund, states that $40 bil­lion was pulled from hedge funds in Octo­ber, and with­drawal requests con­tinue – even if they are not being hon­ored. Pre­sum­ably, that’s why Mr. Jones halted redemp­tions in his fund despite receiv­ing requests for redemp­tions of about $1.4 bil­lion of the $10 bil­lion fund.

Accord­ing to that arti­cle, Mr. Jones’ fund is down only 5% year-​to-​date; the arti­cle was posted before the close of yesterday’s hor­ren­dous trad­ing day, but still, it seems that he has sub­stan­tially out­per­formed the mar­ket and other multi-​strategy funds; the main High­bridge fund is down 25%.

Mr. Jones’ prob­lem is one of the ironies of finan­cial life; (rel­a­tively) good per­form­ing invest­ments are sold to min­i­mize the recog­ni­tion of real­ized losses, and some of the real losers are kept.

As almost every­one knows, besides wealthy indi­vid­u­als, investors in hedge funds include insti­tu­tions – like endow­ments, foundations, and pen­sion plans – and cor­po­ra­tions, includ­ing finan­cial ser­vice cor­po­ra­tions. While indi­vid­u­als may attempt to with­drawal cash from all funds, espe­cially the worst per­form­ing ones, we could imag­ine pub­lic insti­tu­tions – finan­cial firms in par­tic­u­lar – attempt­ing to simul­ta­ne­ously liq­ui­date assets with­out incur­ring large real­ized losses. There­fore, redemp­tions are high at all per­for­mance funds. Unfor­tu­nately, it seems that many of funds can’t afford to liq­ui­date their posi­tions or draw down their cash bal­ances. That may be pru­dent, but it is not a good sign.

More­over, it seems that the large amounts of cash that were set aside in the sum­mer and Sep­tem­ber for redemp­tions are now insuf­fi­cient three months later in December.

That reduced flex­i­bil­ity doesn’t bode well for investors or the econ­omy if addi­tional neg­a­tive shocks affect the mar­kets, e.g., large-​scale ter­ror­ist attacks, nat­ural dis­as­ters like a huge Cal­i­forn­ian earth­quake, or sur­prise defaults of seem­ingly healthy firms. (They still exist don’t they?)

Yes­ter­day, we wrote Volatil­ity and Losses: No End in Sight. We hope to be proven wrong. In all earnest­ness, we do wish these fund man­agers good luck dur­ing the com­ing months. Oth­er­wise, their win­ter may be as bleak, dreary, cold, and bit­ing as late fall has been in the hills of West­ern Pennsylvania.

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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