FinToolBox - Calculating YTM: Trial by Error

Calculating YTM: Trial by Error

In the previous section, we learnt how to calculate the PV of the bond, given we know the YTM (alongside with other inputs ). However, in real time, it is almost always the opposite: traders see the present value of bond from the market first, and they figure out what is the YTM.

The technique to calculate YTM is called "trial by error". Once you know what you are doing, it gets pretty simple. Let's walk it through via an example below.

Learning by Example: Trial by Error

Avocado, Inc. just issued a 10-year annual coupon bond. The face value of the bond is $1000, and its lucrative coupon rate is 6%. If the price of the bond is currently marked at $1480 by the market (they are crazy about Avo's next generation of Beehive technology!), what should be the closest YTM of this bond?

A. 8%

B. 3%

C. 6%

D. 1%

First of all, the idea of trial by error is as it sounds: you try all the options until you find the best one. What does that supposed to mean? Well, remember our equation for the PV of bond is:

For problem setting, we already know that:

PV = 1480

c = coupon rate ✕ Par Value = 6% ✕ 1000 = 60 (here the bond coupon is paid annually)

T = 10

FV = 1000

The only thing we don't know is the r, the real interest rate per period. Here since the coupon is paid on annual basis, r is thus the real interest rate per year, which is the APR in this case(APR = r ✕ number of payments per year = r ✕ 1 = r). Since we already know that YTM is APR, so r is the YTM itself.

So we have all the variables known in the equation above, except for r (= YTM in this case). What should we do?

The answer is simple, we try all the different r's in the right hand side of the equation, until the right hand side equals the left hand side of the equation, which is 1470 here.

Lucky for us, since this is a multiple choice problem, all we need to do is to try four options. Below is the solution for this problem step-by-step.

You might already feel exhausted trying the four options above. What about the real time problem where we don't have preset options? Well, that's how the mathematicians make a living.

What I can assure you is, Trial by Error is the exact method that either the computer or the financial calculator does. The only difference is they can do it much faster than us and they apply some "smart" algorithms to eliminate effort of trying.

Don't be daunted though. As a financial practitioner, you can always rely on a financial calculator or other tools to compute the YTM. Nobody will ask you to do it with hand (except your financial professor, who wants to make sure you understand what is in the black box. And it feels good, right?)

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