Understand Finance - Finance Tutorials and Finance How To's

Finding the value of a bond is pretty easy once you understand that bonds are just
time value of money problems in disguise. Naturally, you’ll want to have a firm grasp on
time value of money–if you don’t, we have plenty of tutorials to
help you build that foundation. With that disclaimer, let’s take a look at how bonds
work.

Why would anybody invest in a bond? Bonds will pay the investor a coupon
payment every year (or twice a year), and also par value (usually $1,000) at
maturity. Those coupon payments are an annuity, with a lump-sum par value at the end.
Sounds like a time value of money problem already! To show you a little more, let’s look
at a simple example.

Sample Problem
UnderstandFinance has outstanding bonds on the market with a par value of $1,000 and
mature in 5 years. The coupon rate is 9% and coupons are paid semiannually. If investors
require an 11% rate of return on these bonds, what should the price of the bond be?

Solution
We know the bonds mature in 5 years, but since the bonds make coupon payments
semiannually, there are 10 compounding periods. They tell us that the coupon rate is 9%.
This means that 9% of $1,000, or $90, will be paid each year in coupon payments. Remember
though, these bonds make semiannual payments! If they pay $90 a year, they must pay $45
every six months. Investors require 11% on these bonds, so that will be what we plug into
our financial calculator as I. Finally, we know we’ll get par value of $1,000 back at the
maturity date. This is our future value! Let’s see what this bond looks like on a
timeline.

So what you should notice here is that our coupon payment of $45 every period is just
a simple annuity. The $1,000 is a lump sum payment at the end and can be treated as a
future value. So, here’s what we should plug into our financial calulator:

Texas Instruments BAII Plus

Step 1. Clear the calculator: Step 2. Semiannual compounding: Step 3. Set N = 10 Step 4. Set I/Y = 11% Step 5. Set PMT = $45 Step 6. Set FV = $1,000 Step 7. Compute PV

Hewlett-Packard 10BII

Step 1. Clear the calculator: Step 2. Semiannual compounding: Step 3. Set N = 10 Step 4. Set I/YR = 11% Step 5. Set PMT = $45 Step 6. Set FV = $1,000 Step 7. Compute PV

So when we solve for PV, we find out that this bond is worth $924.62.
This problem illustrates a great concept. When investors require a rate higher than the
coupon rate, the bond will sell at a discount. In this case, investors require
11% but the bond only pays 9%. So we see that the bond sold for less than the $1,000 par
value. When investors require less than the coupon rate, the bond will sell for a
premium, or more than the $1,000 par value. What happens if the investors’
required rate equals the coupon rate? You guessed it! The bond will sell at par. This is
the inverse relationship between interest rates and bond price. When interest rates rise,
bond prices fall. When interest rates fall, bond prices rise. This is an important
concept so take some time to get cozy with it.

Here’s how I like to think about the inverse relationship. Let’s say you own a bond
with a 10% coupon rate. So, your bond pays $100 per year in coupon payments. If bonds of
similar risk are on the market with an 8% coupon rate, they’re only paying $80 a year. So
the bond you hold is more valuable, correct? If I want to buy your bond, you’re going to
charge a premium for it since it has a higher stream of coupon payments attached
to it.

Practice Problem 1
Big Apple Camera Shop has bonds on the market with a par value of $1,000 and mature in 15
years. The bonds have a 7% coupon rate and coupons are paid semiannually. If investors
require 6% for bonds of similar risk, what is the present value of these bonds?

Solution
Right away, we know that if investors require 6% but the bond pays 7%, these bonds will
sell for a premium. In fact, you should have gotten an answer of
$1,098. The coupon rate is 7% of $1,000 or $70 per year. However, since
the bond pays coupons semiannually, divide by 2 to get a PMT=$35. Also, make sure you put
N=30!

Practice Problem 2
Blue Jeans Coporation issued bonds 20 year bonds five years ago with a face value of
$1,000 and a coupon rate of 12%. If investors require 10.5% for bonds of this risk level,
what is a fair price for a Blue Jeans bond?

Solution
This one is a little tricky? If you got $1,112.08, then you’ve
definitely got the idea! If not, let’s see what could have gone wrong. These are 20 year
bonds, but there’s only 15 years left. So, we know N must be 30. Whenever you’re dealing
with bond valuation, you’re only going to be concerned with the time that is left. The
next issue is that they didn’t tell you the bonds were semiannual. A safe bet is to
assume that if the bond pays a coupon, it is a semiannual payment.