What is the Bond Price Formula?

Source: Capital Pixel

A basic concept in investing that baffles many investors is how bond prices work.  Even the idea that bond prices go up and down and the fact that the investor doesn't need to hold bonds to maturity seems mysterious to some.  This is not good for the simple reason that investors have been piling into bonds and bonds funds over the past couple of years and will be in for an unpleasant surprise when interest rates rise.

To get a complete understanding, it is sometimes worthwhile to go to the basic math--as I'll show here.

How Bonds Work 
 
A bond is a promise to pay a stream of income over time.  To know the stream of income, you need to know the principal amount, the coupon rate, and the time to maturity.  Suppose, then, that we buy $10,000 principal of a 3-year maturity bond that has a coupon of 6%.  The stream of income, then, is:

Year 1   $600
Year 2   $600
Year 3   $600 + $10,000

There are some real world intricacies to know.  First, bonds typically pay interest twice/year.  Thus the above example would have interest payments of $300 every 6 months until the bond matures. Secondly, if you buy a bond in the secondary market, you'll pay accrued interest to the holder of the bond equal to interest earned since the last interest payment.  Thirdly, bonds are typically sold in $1,000 units but priced in terms of $100 principal.

Here's the key to understanding bond prices:  $600 received one year from now is not equivalent to $600 received today.  After all, if I had $600 today, I could earn interest on it for one year and have a greater amount at the end of the year.  Suppose, then, that I could in fact earn 6%.  Then, if I had $566.03 today, I would have $600 a year from now if I earned 6%.

$566.03 is said to be the present value of $600 to be received a year from now if the discount rate is 6%!

The next question, formulated in terms of our new jargon, is:  what is the present value of $600 to be recieved 2 years from now if the discount rate is 6%?  Again, we think in terms of how much we would have to invest today at 6% compounded to get $600 2 years from now.  That amount is 600/(1.06)^2 = 534.

It is important that you check this out and stick with it until you have a thorough understanding.  Think this through.  If we take $534 and earn 6% for the 1st year we will have $ 566.04 (534 * 1.06).  If we then earn 6% on the $566.04 the second year (note that we are earning interest on interest, i.e. the compounding effect), we will have $600 at the end of the 2nd year.

If you have the hang of this, you see that the present value of $600 to be received 3 years from now is $503.77 and the present value of $10,000 to be received in 3 years is $8,396.2.

Thus, we have the following present values:

Year 1   $566.03
Year 2   $534
Year 3   $503.77 + $8396.20

If you add these up, you get $10,000.  Divide by $100 and you see the price of the bond is $100, i.e. par.

Now, here comes the $64,000 question:  what would happen to the price of the bond if the yield was 5%, and what would it be if the yield was 7%?

For year 1 at 5%, the present value would be $600/1.05 = $571.03.  Complete the calculation and find how much higher the price of the bond would be!  (Answer:  price = $102.72!)

For year 1 at 7%, the present value would be $600/1.07 = $560.75.  Again, complete the calculation and find out how much lower the price would be.

Play around with this until you are completely comfortable with the idea that bond prices and yields move in opposite directions.

General Formula

All of this can easily be generalized in terms of the following formula:

P = cpn./(1 + yld.) + cpn./(1 + yld.)^2 + ... + cpn./(1 + yld.)^n + prin./(1 + yld.)^n

where cpn. = coupon payment ($600 in example above),
yld. = discount yield,
n = years to maturity.

Study this formula and you can see that longer maturity bonds vary more in price as yields change because they have many more terms and the principal gets impacted more.

You can also understand the impact if the probability of default increases.  If instead of getting $1,000  at maturity, assume you only get $.80 on the dollar!