How do I find the present value of a bond by discounting each cash flow at a different rate?

Question

Suppose I have a $100 par value bond that pays $6 in coupons semi-anually (i.e. $3 every 6 months). The bond matures in 2 years. There will be a total of 4 cash flows, so the present value calculation is:

PV = ($3/[1 + d1]) + ($3/[1 + d2]) + ($3/[1 + d3]) + ([$3 + $100]/[1 + d4])

d1, d2, d3, and d4 represent the discounting rate for the individual cash flows at 6 months, 12 months, 18 months, and 24 months respectively. Each cash flow has its own discounting rate because their durations are different (e.g. d3 should be higher than d1, because cash flow received in 6 months has a greater present value than cash flow received in 18 months).

Now the problem is: what values do I plug in for d1, d2, d3, d4 in order to calculate the present value? At the moment, all I know is that d1 < d2 < d3 < d4, but I do not know their exact numerical value. How do people find these values for valuing bonds?

Answer 1

Typically you would use a discount curve that represents roughly the same risk (probability of default) as the bond in question. This is not a trivial task by any means, since the time until each cashflow will not line up exactly with other bond (meaning you'd have to find bonds that mature at each of those times). Even for government bonds, where there are many fixed tenors, it's virtually impossible to find a bond that matures in exactly 8 years, for example.

In academia, the discount rates will usually be given to you, and probably as a single rate until you get into more advanced fixed income classes.

In reality, analysts will either use their own method of calculating these curves (bootstrapping is a relatively simple algorithm) that vary in complexity and accuracy.

Also, you can not always assume that d1 < d2 < d3 < d4. You may have been given that, but it is not always the case in reality. It is not uncommon to have short-term interest rates that are larger that long-term interest rates.

Answer 2

You don't.

There are two major approaches when comparing investments. NPV analysis and IRR analysis.

NPV uses your formula, with known coupon rate, maturity value, and yield curve of corresponding credit rating to arrive at a PV, then compare the PV against the quoted Price today.

IRR uses your formula, with known coupon rate, maturity vlue, Price today, then solve for a single discount rate (1st year i, 2nd year (1+i)^2-1, 3rd year (1+i)^3-1, etc), then compare against the average yield of corresponding credit rating and maturity.

It does not make sense to use such formula when both quoted Price today and average yield are not known.