Merton Model, and risk capital

[itsyourz]

hi,

in 06 practice exam part II, no.82

82. Consider a risky zero-coupon bond maturing in one year. At that time the issuer owes
USD 100 million. The issuer has no other debt and the bond can be priced using Merton's model.
The bond is the only asset of a bank. Which of the following statements is correct?

a. The amount of risk capital required for this bond by the bank necessarily increases if
the volatility of the assets of the issuer increases

b. The amount of risk capital required for this bond exhibits a hump shape - it first
increases with asset volatility and then falls

c. The shape of the relation between the amount of risk capital and asset volatility
cannot be determined without knowing the bank's RAROC hurdle rate

d. The shape of the relation between the amount of risk capital and asset volatility
cannot be determined without knowing the confidence level at which the bank's
credit-VaR is calculated

ANSWER: B

A risky bond can be decomposed into a risk-free bond and a put option. The price of
the bond equals the price of the default free bond minus the put option premium. As
the asset volatility increases, the put premium will increase and the price of the risky
bond will fall. Thus, a bond issued by a firm with extremely high asset volatility will be
almost worthless, so that it requires little capital.
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i think the explanation at bottom does not match the answer b.
could you help me to understand?

 

[David Harper CFA FRM]

Hi suk,

I agree, and it's a lame question.
Per the Merton model, the price of debt is falling with an increase in asset volatility
(i.e., for a given asset value, higher asset value implies higher call option, where the call option is equity. And, therefore, lower value to the debt).

So, probably the question means: higher asset volatility = lower debt value = lower risk capital.

But the problem with that logic is that it mistakenly treats risk capital as linear function of debt value.
If we think about Basel (standardized or IRB) and, say, the exposure = $100.

Then, initially maybe:
risk capital = 8% * $100 * 100% RW (for argument's sake)

Now assume value of debt goes down (i.e., higher asset volatility). This missing logic piece is that the capital charge will go up due to the higher risk, so maybe something like:

risk capital = 8% * $80 (i.e., lower debt value due to higher asset volatility) * 150% (i.e., offsetting higher risk weight)

In brief, higher asset value will lower the price of debt but an higher capital charge for higher risk will work in the other direction. Perhaps the first outweighs the second but i can't see why that is necessarily true.

So, i don't think the question can be answered (perhaps b refers to the hump in vega, but if so, that is not thought thru, so i don't get b either....)

David