Spero Consulting

Archive for November 13th, 2008

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.

This a small point, but the pedant in us isn’t above it.

Yesterday’s (Novem­ber 12) Wall Street Jour­nal con­tained an opin­ion col­umn, Is Now the Time to Buy Stocks? by John H. Cochrane, a finance prof at the Uni­ver­sity of Chicago. Rather than com­ment on his data-​mining exer­cise, we’d rather repeat his qual­i­fi­ca­tion that “His­tory is not a guar­an­tee – this time it could be dif­fer­ent.” This, of course, is the Prob­lem of Induc­tion or at least non­sta­tion­ar­ity and is some­thing we’ve writ­ten about in many posts.

In those posts, we’ve often men­tioned St. James and his admo­ni­tion about uncer­tainty, which appeared in his only Epis­tle. (Inter­est par­ties can read it on the Quotes page.)

But today we’re ignor­ing the larger issues to focus on a smaller one: not philo­soph­i­cal notions of mod­el­ing in prob­a­bil­ity and sta­tis­tics, but rather basic definitions.

It is extremely likely the Prof. Cochrane under­stands the dis­tinc­tion between the “mean” and the “median” of a pop­u­la­tion but either he or his edi­tor failed to clearly make that dis­tinc­tion in the fol­low­ing sen­tence: “We all like to think we’re smarter than aver­age, but at least half of us are deluded.”

The aver­age is the mean of the expected value of the dis­tri­b­u­tion, while the median is the point that sep­a­rates the top 50% from the bot­tom 50%. So, he uses the word “aver­age” in the first clause but implies the median in the sec­ond one.

We know that it is a small point, but if the mean exists, then it equals the median only when the den­sity or mass func­tion is sym­met­ric. (There are sym­met­ric den­si­ties that have medi­ans but no means.)

Prof. Cochrane likely has a bell curve – which is quite sym­met­ric – in mind when he wrote his sen­tence, but it is quite sloppy lan­guage and con­fus­ing to those who are new to the terms, and frankly, we hope for more than that from our edu­ca­tors. It is those sim­ple, innocu­ous, off-​hand state­ments that leave stu­dents con­fused and unsure of their grasp of the mate­r­ial, and we don’t like that.

Writ­ing the above reminded us of a con­ver­sa­tion we had with a dean some time ago when we were a young fac­ulty member.

The dean wanted the entire fac­ulty to be above-​average teach­ers. (Well, we doubt that he actu­ally cared about the qual­ity of teach­ing; instead, he wanted the entire fac­ulty to have above aver­age teach­ing rat­ings, which is an entirely dif­fer­ent thing. Note that even with a fac­ulty of excel­lent teach­ers – which should have been the true aim – as long as there was some vari­a­tion, some­one would have to have below-​average ratings.) 

We attempted to explain that out­side of Lake Wobe­gon it was quite impos­si­ble to make every­one above aver­age. In fact, the best that our dean could hope for in a fac­ulty of N profs was to N — 1 (strictly) above-​average teach­ers. Unless they were clones with iden­ti­cal and high rat­ings, he’d need one fac­ulty mem­ber with really, really bad rat­ings to make the rest pos­sess above aver­age ratings.

Being humor­less, he didn’t appre­ci­ate our advice, but then we didn’t expect him to – not even 50% of the time.

Anal­o­gously: The Gangs That Can’t Shoot Straight

Last week in The Under­state­ment of the Year! we wrote, “The prob­lem, dear reader, is that few senior man­agers (and almost no board members) understand the val­u­a­tion and risk mod­els used for securitizations…”

Today, there is an arti­cle in The Wall Street Jour­nal, Citi Direc­tors Mull Replac­ing Chair­man, that pro­vides addi­tional evi­dence to sup­port our claim.

To be frank, unless it is we, we don’t really care who Citi selects as a chair­man, and we doubt that you do, also.

We’re more inter­ested in the way that the article’s writ­ers describe board mem­ber Richard Par­sons as “one of the few Cit­i­group direc­tors with expe­ri­ence in finan­cial services.”

One of the largest finan­cial ser­vice firms in the world, and only a few direc­tors with (any type of) finan­cial ser­vice expe­ri­ence. How could they lose? we ask sar­cas­ti­cally. There is a mul­ti­tude of types of expe­ri­ence with finan­cial ser­vices firms; so, we’d argue that while such expe­ri­ence is nec­es­sary, it is by no means suf­fi­cient to under­stand and eval­u­ate com­pli­cated prod­ucts, hedges, strate­gies, and risks.

To be faced with such inex­pe­ri­ence, it must be the case that either senior man­agers are par­tic­u­larly poor judges of tal­ent or those inex­pe­ri­enced direc­tors were nom­i­nated specif­i­cally because they lacked expe­ri­ence or despite their lack of experience. 

The for­mer rea­son for pur­posely select­ing the inex­pe­ri­enced is clearly cyn­i­cal and involves senior man­age­ment attempt­ing to nom­i­nate mem­bers who are much more likely to be weak and unable to pro­vide the req­ui­site level of oversight.

The lat­ter rea­son may or may not be cyn­i­cal. For exam­ple, an unknowl­edge­able direc­tor may have been cho­sen because he or she is par­tic­u­larly savvy and a fast learner (not cyn­i­cal) or because he or she has a mem­ber­ship at Augusta or Oak­mont or some other exclu­sive golf club where senior man­agers might like to play (very cynical).

Now, we’re will­ing to stip­u­late that in many mar­ket and eco­nomic set­tings, it may not seem to mat­ter. In fact, it is pos­si­ble in the over­whelm­ing major­ity of the cases that it doesn’t seem to mat­ter, but that doesn’t mean that such nom­i­na­tions are indeed consequence-​free.

For such cases, we like the anal­ogy of a cop who is a par­tic­u­larly bad shot. That fact is almost never directly rel­e­vant as law enforce­ment offi­cers rarely draw their weapons and fire. So, it may seem that it doesn’t matter.

Unfor­tu­nately, the self-​aware offi­cer real­izes that he or she is a poor shot and acknowl­edges his or her inabil­ity to respond effec­tively to extreme sit­u­a­tions. This knowl­edge likely col­ors or influ­ences his or her behav­ior in all set­tings, includ­ing inci­dents where only a very small prob­a­bil­ity of esca­la­tion exists. 

Such behav­ior is usu­ally cor­rectly inter­preted by the other rel­e­vant par­ties as weak­ness. How­ever, in some cases the officer’s may over-​react or behave in an extremely risk-​averse man­ner due to his or her per­sonal inse­cu­rity. Regard­less, in both cases, the officer’s and society’s well-​being has been compromised.

It is much the same with gov­er­nance and risk man­age­ment within firms. Those direc­tors lack­ing ade­quate fire­power are unlikely to deter anti-​social behav­ior; thus, weak boards are more likely to induce exces­sive risk-​taking and increased odds of a dis­as­ter (although that real­iza­tion may not occur). Is that increased prob­a­bil­ity of dis­as­ter worth 18 holes at a world-​famous course? Don’t answer that!