## Thursday, September 23, 2010

### Yield to Maturity (YTM) & Interest Rate Risk

This post introduces YTM, and explains why rising interest rates result in falling bond prices.

##### Bond Price and YTM/Interest Rates

The above graph summarizes the relationship between the two; the post helps explain why this is so.

Calculating a Bond's Yield to Maturity
In the fundamentals of investment valuation post, we introduced the concept of present value, and arrived at
a fair price for a risk-free bond; we did this by summing the present value of the cash flows associated with the bond, using the prevailing interest rate. This is a typical way of evaluating the attractiveness of a bond, or any other investment. In the table below, given a 5% interest rate, the sum of the present values of the cash flows is \$1,000 -- a fair price for this bond. (click to expand)

Another approach to evaluating attractiveness is to calculate the bond's yield to maturity (YTM). The YTM is the interest rate that makes the present value of the cash inflows equal to the purchase price (the cash outflows), assuming the buyer holds the bond until maturity. To calculate the YTM we start with the cash flows and the price, and we calculate the interest rate. In this example, if the investor pays \$1,000, then the interest rate, his yield to maturity, will be 5%. Note: In the cash flow column the cash outflow (the purchase price) is negative to distinguish it from the positive cash inflows.

### Introduction to Interest Rate Risk

So far, we've seen that a \$1,000 5% bond purchased for \$1,000 has a YTM of 5%. Sounds fair enough, so let's assume you decide to buy this risk-free bond. Let's further assume that immediately after you buy the bond interest rates rise to 6%. How much would someone else now be willing to pay you for your brand new bond? With interest rates at 6%, new \$1,000 bonds will pay (\$1000 x 6%=) \$60/year in interest. If \$1,000 can buy them \$60 a year in interest, why would anyone give you \$1,000 for a bond that only pays \$50/year in interest; they wouldn't. What's your bond worth in a 6% world?

In the table above, we see that if someone pays \$1,000 for this bond, his yield to maturity will still be 5% -- less than the going rate of 6%. We also see that, using a 6% interest rate, the present value of the cash flows associated with your bond is only \$957.88. If a buyer pays \$957.88 for your bond, his YTM will be 6% -- the going rate. Click here to see the Excel spreadsheet and the underlying calculations. (If you have any problems accessing the spreadsheet, see this link.) If you change the purchase price from \$1,000 to \$957.88 in the spreadsheet, the new yield to maturity will be 6%.

If you continue to input purchase prices, the spreadsheet will calculate each of the associated YTM values. The lead graph shows the results of those calculations for purchase prices between \$500 and \$1500. For any given price on the horizontal axis, the graph shows you the associated yield to maturity on the vertical axis. From the graph it is clear that the higher the price you pay, the lower your YTM.

### Bond Performance: Why Rising/Increasing Interest Rates = Falling Bond Prices (& Falling Interest Rates = Rising Bond Prices)

The graph also works in the other direction.  Given any interest rate on the vertical axis, the graph shows you the appropriate price for this 5-year bond on the horizontal axis. Looking at the graph from this perspective, it is clear that the higher the interest rate, the lower the associated price. Similarly, the lower the interest rate, the higher the associated price. For example, if the interest rate were to fall to 4%, the value of the bond would increase to \$1044.52.

Interest Rate Risk is Unavoidable
You cannot avoid the possibility that interest rates will increase after you buy a bond. If rates increase your bond will be worth less on the open market than you paid for it. If you sell at that point, you will realize a loss on your sale; at best it is likely that you will be disappointed that you did not wait to make your purchase. However, you can avoid taking a loss on the bond by simply not selling it. In that case in a very real sense the rate change will have zero impact on your original investment. Your agreement was for 5%; your yield to maturity is still 5%. If you hold the investment until it matures you will earn the 5% return that you originally planned on.

The next post in this series, Bond Duration, discusses additional steps you can take to reduce the impact of interest rate changes.

Note: YTM is just a special case of the more general concept of internal rate of return (IRR). The YTM calculation assumes one cash outflow, the purchase of the bond, followed by cash inflows at a constant frequency (e.g., quarterly). The more general IRR calculation allows multiple cash outflows and inflows interspersed in any sequence. (If you link to the spreadsheet, you'll see the calculation done both ways.)

### Related Materials

See Wikipedia Yield to Maturity for additional discussion of this topic.
Fundamentals of Investment Valuation: introduction to present value
Bond Basics: relationship between interest, principal, interest rate and time.
100 Years of Treasury Interest Rate History: since 1900.
10-Year T-Note Real Returns discusses modifications to the above YTM calculation to determine inflation-adjusted returns.
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