Bond Prices and Present Value

DIY Investor previously discussed the concept of present value. Present value puts a monetary value on income to be received in the future. As a simple example, $200 to be received 5 years from now would have a present value of $149.45, if the discount rate (i.e. yield) is 6%. This is calculated as follows:

                                                               $149.45 = 200/(1.06)^5. 

Flipping this around reveals that, if you invest $149.45 today at 6% compounded annually, you will  have $200 five years from now.

This is the key to bond prices, since bonds are merely a stream of payments to be received in the future. The general price for a bond is calculated by adding up the present value of the payments to be received and can be expressed with the following scary-looking formula:

P = C/(1 + yld.) + C/(1+yld.)^2 +...+C/(1+yld.)^n + PRIN/(1+yld.)^n

C is the coupon payment, yld. is the all important yield-to-maturity, and PRIN is the principal amount.

Understanding this formula is key to understanding how bonds work. To put it into specific terms, consider the price of a 3-year maturity with a coupon of 6% with market yields for similar bonds at 6%. Assume a principal amount of $100. The price can be written as :

P = 6/(1.06) + 6/(1.06)^2 + 6/(1.06)^3 + 100/(1.06)^3

P = 5.66 + 5.34 + 5.04 + 83.96 = 100

This just shows that when market rates are equal to the yield on the bond, the price is $100, or par.

But suppose yields rise. Suppose the yield on similar bonds in the market are 7%. Redo the calculations:

P = 6/(1.07) + 6/1.07)^2 + 6(1.07)^3 + 100(1.07)^3

P = 5.61 + 5.24 + 4.90 + 81.63 = 97.38

Notice how the present value of each term is now lower. Playing with the present value formula by changing yields and maturities etc. will enable you to understand everything you need to know about bonds. The simple demonstration here shows the inverse relationship between bond prices and yields that confuses so many investors. Studying the formula further reveals why longer maturity bonds are more volatile in price than shorter maturity bonds. If you don't believe DIY Investor, do the same exercise above for a 5-year maturity. Think about the impact of more terms. Think about what happens to the present value of $100 to be received 5 years from now compared to the 3-year period above.

You can even go further and think about the percentage drop in bond price calculated above for the assumed 100 basis point increase in market yields. This gets you even more sophisticated in that it takes you to the concept of duration, which is merely an extension of maturity. Now, when you go on Morningstar or similar sites and see the duration of various bond funds, you hopefully understand how the underlying implied volatility is figured.

Bottom Line

All investors, including the most aggressive, should have a portion of their assets in fixed income or bonds. The key to understanding how this portion of their assets works is understanding the basic principle of present value. DIY Investor believes that this requires, for most people, rolling up the sleeves, getting out the calculator, and playing around with the numbers.