## Wednesday, December 28, 2016

### Bond Basics: Part 2

In Part 1 of this series I described how a bond is basically a loan, or more precisely, the contract defining the terms of a loan, where you are the lender and a company or government entity is the borrower. I explained that the terms of this loan contract, or bond, include the principal amount, referred to as the face value, an interest rate, referred to as the coupon rate, a payment schedule for the coupon payments, typically every six months, and a due date for the final coupon payment and repayment of principal, referred to as the maturity date.

Toward the end of Part 1 I introduced the concept of yield to maturity (YTM), often simply referred to as yield. A bond's yield incorporates both the coupon rate and the change in bond price between the day you buy the bond and the day the bond matures. A bond's yield provides a reasonable measure of the rate of return you can expect for a bond held to maturity. I explained that the market price of a bond may be different than the face value of the bond, that bond yield is inversely related to bond price, and said that I would explain all of this with an example in Part 2. Read on for the explanation.

As explained in Part 1, although bonds typically are sold in increments of \$1,000, bond prices are quoted as a percent of face value (also referred to as par value, or simply par). So in the examples that follow, when I say you pay 100 for a bond, or that a bond's price is 100, it means that the bonds price is 100% of face value, or \$1,000 per bond. Similarly, a bond price of 99 means that the bond's price is 99% of par, or \$990 per bond, and a price of 101 means that the bond's price is 101% of par, or \$1,010 per bond.

Say you buy a bond today at par value, with a coupon rate of 1%, that matures in one year. At issuance this bond has a price of 100, and it also will have a price of 100 at maturity; i.e., you will get back the same amount of principal that you paid for the bond. So for every \$100 of bond value that you buy, you will receive \$1 in interest, in the form of coupon payments (1% of \$100 equals \$1), and at maturity you also will receive the principal amount of \$100. At issuance the yield (YTM) of the bond also is 1%; since you paid 100 for the bond, and the price at maturity also is 100, the coupon rate of 1% comprises the entire rate of return for the bond--there is no price change to factor into the yield.

We can generalize the information in the previous paragraph, and say that for a bond priced at par (or simply a par bond), the yield equals the coupon rate. However, a bond priced above or below par will have a yield that is lower or higher than the coupon rate. Here's how this works.

Say that the day after you buy the 1-year par bond with the 1% coupon rate, the coupon rate on new 1-year bonds bought at face value increases to 2%. The buyer of that bond will earn \$2 in coupon payments for every \$100 of bond value, so the yield also is 2%. Since investors can now earn 2% in one year by buying this new bond, no one would pay you 100 for your bond with a coupon payment of only 1%. They will pay you an amount that equalizes the rate of return, or yield, for the two bonds.

Your bond pays only 1% in coupon payments, but an investor will only pay you a price for the bond that will give them a 2% return over the one year until maturity. Since they need an extra 1% in return to give them the same 2% return they can get on the new bond with the 2% coupon rate, the price of your bond will fall by about 1% to about 99.

Now someone who buys your bond for about 99 will earn 1% in coupon payments and 1% in price appreciation (since the bond will mature at 100), giving them the same 2% total return at the end of one year. The yield of both bonds is 2%; the new bond pays the 2% return through the 2% in coupon payments, and your bond pays the 2% return through 1% in coupon payments and 1% in price appreciation.

Here we see that when the 1-year-bond yield increased, the existing 1-year-bond price decreased. The price of the existing bond decreased by an amount that caused the yield to increase to the new market rate of 2%, so that buyers of either the new bond or the existing bond would earn the same rate of return.

We can use similar reasoning to work out the approximate price increase for a one percentage point yield decrease in a 1-year bond, from 1% to 0%. If the one-year yield decreased to 0% the day after you bought your bond with the 1% coupon rate, your bond price would increase to about 101. Your bond would earn 1% in coupon payments and lose 1% in price depreciation, earning the same 0% as the buyer of the new bond with a coupon rate of 0%. The price of your 1% coupon bond increased so that the yield is 0%, the same as the 0% yield of the new bond with the 0% coupon rate.

This brings us full circle to the observation I shared in Part 1, which was that a National Financial Capability Study found that only 28% of American adults understand the relationship between interest rates and bond prices. Here is the question as it was asked in the study:

If interest rates rise, what will typically happen to bond prices? Rise, fall, stay the same, or is there no relationship?

Hopefully you are now in the 28% of American adults that can correctly answer, "Fall". Here is the explanation provided in the online quiz that asks the questions asked in the study:

When interest rates rise, bond prices fall. And when interest rates fall, bond prices rise. This is because as interest rates go up, newer bonds come to market paying higher interest yields than older bonds already in the hands of investors, making the older bonds worth less.

The example in this post went through some simple bond math that explains this. However, the question and answer in the study and online quiz are not very precise, and don't really accurately describe the relationship between bond prices and bond yields. The reason is that there are many different interest rates in our economy, and the only interest rate relevant to the price of a particular bond is the yield of bonds that are similar to the bond in question. I'll discuss this in more detail in Part 3 of this series.

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