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Archive for December 12th, 2008

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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