Bond Pricing and Perpetuities

Michael Sack Elmaleh , CPA, CVA

There are many types of bonds, issued by many types of government and private companies. In this chapter I am interested in discussing bonds that can be easily bought and sold in securities markets, otherwise known as publicly traded bonds. Time value formulas come into play in determining the fluctuating prices of these bonds.

Bond prices vary because interest rates vary. Variation in interest rates on specific bonds are primarily a function of five factors:

* The risk associated with the issuer of the bonds.

* The term of the bond (how long the bond must be held to maturity).

* The general or prevailing level of interest rates set by the Federal Reserve Bank (The Fed).

* The taxability of interest earned on the bonds.

* Fear of inflation.

Certain general principles usually hold. In theory any bond issuer can default on their bonds. But certain issuers like the U.S. government are virtually certain not to, and certain financially shaky private companies are much more likely to. In between are the bonds of state and local governments and highly solvent profitable corporations. Here the general rule is that the greater the risk of default, the higher the interest rate.

Another general, but not absolute rule, is that the longer the term to maturity the greater the interest rate. This factor is related to the other factors. Other things being equal the longer the maturity the greater the probability of default. Also, the longer the maturity the greater the chances that inflation will reduce the buying power of the principal that will be repaid when the bond is redeemed.

Fear of inflation directly impacts interest rates. The greater the potential for inflation, the higher the interest rate required on the bonds. This factor is directly related to the Fed’s policy actions in keeping inflation at bay. If the Fed increases or decreases the money supply to stimulate or cool off the economy general interest rates will be directly impacted. When inflation is less of a concern the Fed will keep interest rates low, when inflation risk is deemed to be too high interest rates will be increased.

Finally, interest payments on certain government bonds are tax favored compared to bonds issued by private companies. Certain municipal and state government bonds are exempt from federal taxation. Treasury bond interest is not subject to state income taxes. These tax-exempt bonds have lower interest rates than non-tax-exempt bonds issued by private corporations.

I should also note that certain corporate bonds have special features that can impact the level of interest payments. Some bonds are convertible to equity. Some bonds can be redeemed or called by the issuer before the stated maturity date. These features usually increase interest rates.

It is generally true that when bonds are initially issued and sold the face value of the bonds and the scheduled interest payments accurately reflects all the relevant factors I just described. If this is the case, then the theoretically correct bond price is given by this now familiar present value formula:

The fixed periodic payment is also called a coupon. Now because the coupon amount is fixed, we can eliminate the summation operator and utilize this variant of the above formula:

Now for most bonds issued by governments and large publicly traded corporations there is a secondary trading market. This means that the initial bond holders can sell their bonds to someone else. Even though the resale will take place after the original issuance of the bond the theoretical price of the bond will stay the same as long as the interest rate required by the new buyer is the same as the original interest rate on the bond.

But very frequently prevailing interest rates shift, and the buyer of the bond will want this change in interest rates to be reflected in the price paid. In this situation the periodic coupon payments and redemption amount are fixed. What is needed is an adjustment to the purchase price of the bond to reflect the change in prevailing interest rates. In order to determine the adjusted bond price, we simply plug in the new desired interest rate yield into the formula.

Example. Suppose we have a bond with a face amount and redemption value of $10,000 with a 5% stated interest rate and five annual $500 coupon payments. The table below shows that using the present value of the individual coupon payments plus the repayment of the principal at the end of year five equals a bond price of $10,000. If the prevailing interest rates stay at exactly 5% the bond will continue to be priced at $10,000. When the selling price equals the face amount the bond is said to be sold at par value.

Now suppose that soon after the bond is issued the prevailing interest rates rise to 6%. Here is the adjusted bond price.

Effectively for the buyer to receive the new desired yield of 6%, given the fixed coupon payment amount and redemption amount, the bond price must be changed to $9,579.

Now let’s see what happens when the prevailing interest rates decrease say to 4% soon after the original bond is issued at Par.

Now the buyer must pay $10,445 for the same bond to achieve the 4% prevailing rate of interest.

Publicly traded bond prices are listed as a percentage of par. So, in the example when the prevailing interest rose to 6% this bond would be listed at 95.8 while the listing would be 104.5 when the prevailing interest rate dropped to 4%.

In addition to coupon bonds corporations also offer what is known as “zero coupon “bonds. These bonds make no periodic payments of interest but agree to pay a face amount at some future date. Because these instruments pay no periodic interest, they are sold at a discount from the face amount. The discount is determined by the prevailing rate of interest plus usually a little extra to compensate the bond holder for deferring the collection of interest.

Example. If the prevailing interest rate is 4% a company may offer a 5% rate of return by selling the bond at a discount. Let’s say the face amount to be collected five years from now is $10,000. Assuming an annual compounding period of a year, this zero-coupon bond would be sold at the following amount:

Finally, there is a special type of bond offered by certain governments and private corporations called perpetuities. These bonds are sold at a set face amount at set interest rate to be paid at set intervals. But there is a twist. The twist is that the issuer has no legal obligation to ever redeem the bond. The bond has no maturity date. You heard me right the principal is never repaid. Who would buy such an instrument? And how much should someone pay for a bond?

Well, if you trusted the solvency of the perpetuity bond issuer and felt that the interest return was sufficiently attractive you might be willing to invest in this kind of instrument. The attractiveness of these bonds would be enhanced if there was a secondary market in which they could be sold.

And how much should someone pay for a bond? The standard formulas for determining bond pricing relies on knowing a set maturity amount at a set maturity date. In the absence of these elements the proposed value of a perpetuity is given by this formula: 

As an example, assume that a perpetuity offers a $500 annual payment, and the prevailing annual interest rate is 5%. By the formula this perpetuity should be valued at $10,000.

But there is something very interesting about this pricing approach. Theoretically the perpetuity price should equal the sum of the present values of all future interest payments. So just for laughs and giggles let’s create a spreadsheet of the future interest payments and let’s see what happens to the present value of each future annual payment and the cumulative present value of future cash flows as we go deeper and deeper into the future.

Since the payments go on forever let’s content ourselves with just the first 200 years of payments. This first table is a truncated view of the first fifty years of payments.

You can see that the present value of all the payments for the first 50 years only begin to approach the $10,000 price at $9,128. So, let’s continue. In truncated form this table below shows that only at around year 200 does the cumulative present value get to almost the $10,000 suggested formula price.

Also note what is happening to the present value of each additional $500 payment. At year 197 each additional $500 year payment is only adding about 3 cents more to the cumulative present value and this increment is actually decreasing. But due to rounding the table shows only about 3 cent increases for years 197 through 200.

So, what we are really seeing in the perpetuity pricing formula is a limit value. As we get further and further into the future, we get closer and closer to the $10,000 but we never quite get there. We will see this perpetuity formula again in another context.

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