Many people know how to valuate the fair price of a stock but does not know how to valuate the fair price of a bond. Here is how it works & why I think it is useless to the individual:

The fair price of a bond (or equivalent the yield-to-maturity) of a bond is often made in comparison to the opportunity cost of investing in the bond. The opportunity cost measured is often the risk-free rate. Unlike the risk-free investment, a bond is not risk free and is subjected default risk. If there is no default, by definition both the par and the promised couples are paid to the investor. If there is a default, only part of those would be return. The fraction of the amount that can be return to the investor upon default of the bond is called the recovery rate.

Before I proceed, it is important to understand a probability theory called the total probability rule for expected value which states that:

E(A) = P(B) x E(A | B) + P( B') x E (A | B')

Where E(A) refers to the average value for a random variable A. “B” is an event and “B'” is a complementary event

Let R be the return of a bond and F be the risk free rate. The prudent investor would ensure that his average return of the bond is at least equal to the risk free rate otherwise he is better of investing in the risk-free rate. Thus,

E( bond return ) = total return from risk free rate

E( bond return ) = 1 + F

but using the total probability rule for expected return,

E( bond return ) = P ( did not default ) x E( bond return | did not default ) + P ( default ) x E( bond return | default )

Let's put some numbers it. Presume we use a fixed deposit 1 year maturity yielding F = 1%. Let's assume the probability of the bond defaulting is P(default) = 0.03 and let R be the return of the bond and the recovery rate is $0.30 for each dollar of principal.

[1 - P(default) ] * (1 + R) + P (default) * RecoveryRate * (1 + R) = 1 + F

(1 + R) { ( 1 – P (default) + P(default)*RecoveryRate } = 1 + F

(1 + R) = (1 + F) / { ( 1 – P (default) + P(default)*RecoveryRate }

R = (1 + 0.01)/ { (1 – 0.03 + 0.03*0.30 } - 1 = 3.17%

Thus, bond yield to maturity should be at least 3.17%. Although the calculation is highly theoretical it can be both useful and useless. Consider a structured note which is structured to “look” like a bond. One of a popular structure note is to linked it with the credit risk of a few entities. The “first to default” of one of these entity would cause the note to default and having recovery rate of zero. Let's assume our structured note is linked to 6 entities “first to default”. If we assume the probability of default for each entity to be the same and that they are independent of each other:

P (structured note defaulting) = P(entity 1 default OR entity 2 default OR entity 3 default OR ... ) = 6 * P (entity 1 default)

If we assume the entity credit rating is the same as the above plain vanilla bond than P (structured note default ) = 6 * 0.03 = 0.18.

Thus, based on the new updated parameters for a structured note,

R = (1 + 0.01)/ { (1 – 0.18 + 0.18*0.0 } - 1 = 23%

In layman language it shows that a structured note that is “first to default” with 6 entities increases the required rate of return from 3.17% to 24%. While the parameters of the above is just for illustrations it shows that if an investors would to buy a “first to default” structured note, he should at least enjoy a return that is at least in ball park 5 times that of a plain vanilla bond of the same credit rating. Some people just think that if they can get 5 times that of a fixed deposit interest they will be happy. This is the wrong. They should demand at least 5 times that of an equivalent plain vanilla bond of the same credit rating of the underlying entity.

If we look at the structured note being sold to retail and high networth investors – you can see how much the institutions have earned. So say instead of selling a structured note that should yield 23%, they will sell say a 5% yield. The difference in yield goes into commissions, distribution cost and profit margins of everybody else other than the investors.

This being said, the calculation above is quite useless for the individual investors. Consider there are only two scenarios for a bond or a note. Either the security default or do not default. In the event which the bond default, the losses can be massive and after netting of legal fees and lawsuit, the likely recovery rate of a bond is as good as zero. In other words, the “unlikely” outcome of a bond – which is when it defaults – is an outcome that is totally unacceptable to the individual. I feel say that many retirees who may not be that rich have invested in plain vanilla corporate bond. They think that the “unlikely” event of default will not happen but in reality it can happen. If it does happen, it is highly likely that the recovery rate is low and therefore the investment losses can be near 100%. Many corporate bonds require at least $250,000 to buy. It is not say $1000 or $10,000. Thus a retiree with say an asset of $1,000,000 (this figure does not qualify the person to be “high networth”) would to invest $250,000 into a single bond than if the bond defaults – 25% of his networth will be gone into the drain. Obviously this is a total unacceptable outcome.

The calculation of the “fair value” of a bond is appropriate for a portfolio manager that holds hundreds of bonds. If a bond default, not much harm is done since there are so many other bonds in the portfolio. In fact, after receiving coupons from existing bonds the defaulted bond's losses can be easily offset.