# Investment Banking: How Bond Prices are Affected by Changes in Interest Rates

All bonds are sensitive to changes in the general level of market interest rates, but the sensitivity varies from bond to bond in investment banking. The interest rate sensitivity of a bond varies inversely with the coupon rate and directly with its term to maturity. Analysts have developed a measure of interest rate sensitivity called *duration* that takes into account both the coupon rate and term to maturity factors.

## The coupon rate effect

Bonds with higher coupon rates are less sensitive to changes in interest rates than bonds with lower coupon rates. For example, you can contrast the interest rate sensitivity of a five-year zero-coupon bond with an initial yield to maturity of 5 percent, with that of a five-year 5 percent coupon bond with an initial yield to maturity of 5 percent (assuming that the bond pays interest annually).

The five-year zero-coupon bond selling to yield 5 percent to maturity will be priced at

The price of the five-year, 5 percent coupon bond selling to yield 5 percent will be $1,000 because the yield to maturity and the coupon rate are identical.

Now, assume that yields in the marketplace rise by 100 basis points due to general economic uncertainty and that the bonds both sell at a yield to maturity of 6 percent. The new price of the zero-coupon bond will be

With this increase in interest rates of 100 basis points, the price of the zero-coupon bond fell by 4.63 percent.

If the yield to maturity rises to 6 percent, the price of the 5 percent coupon bond is

With this increase in interest rates of 100 basis points, the price of the 5 percent coupon bond fell by 4.21 percent. Thus, the lower coupon bond is more sensitive to changes in interest rates than the higher coupon bond. If interest rates rise, you would rather be holding higher coupon bonds than lower coupon bonds.

## The term to maturity effect

Bonds with longer terms to maturity are more sensitive to changes in interest rates than bonds with shorter terms to maturity. To illustrate this point, contrast the price change of similar zero-coupon bonds, one with 5 years to maturity and one with 30 years to maturity, when yields go from 5 percent to 6 percent.

The market price of a five-year zero-coupon bond falls from $783.53 to $747.26, a decrease in price of 4.63 percent when the yield to maturity on the bond rises from 5 percent to 6 percent.

Contrast that with the change in price on a 30-year zero-coupon bond when the yield to maturity on the bond rises from 5 percent to 6 percent. At a yield of 5 percent, the price of the bond is

At a yield of 6 percent, the price of the bond is

When yields rise from 5 percent to 6 percent, the price of the 30-year zero-coupon bond falls by 24.75 percent. Suffice it to say, you'd rather be holding shorter-term bonds than longer-term bonds if interest rates rise.

## Duration

Price volatility of a bond varies inversely with its coupon rate and directly with its term to maturity. *Duration* is a combined measure of interest rate sensitivity that takes into account both of these properties.

Duration is one tool investment bankers use to determine how risky a bond investment is. It's defined as the time-weighted term to maturity of a bond in which the cash flows are weighted according to when they're received in a present value sense.

You compute the duration for a bond as a measurement of years. You calculate duration simply by finding the present value of each cash flow as a percentage of the price of the bond and multiplying that value by the year in which the cash flow is received. The sum of those values is the time-weighted term to maturity of the bond and represents the duration of the bond.

The duration for this bond is a ten-year, 5 percent coupon bond with a 7 percent yield to maturity (assuming annual interest payments). This bond has a duration of 7.94 years. It will show interest rate sensitivity that is identical to a zero-coupon bond with 7.94 years to maturity.

Year | Cash Flow | Present Value of Cash Flow at 7% | Present Value as a Percent of Price | Year x Present Value as a Percent of Price |
---|---|---|---|---|

1 | $50 | $46.73 | 0.0544 | 0.0544 |

2 | $50 | $43.67 | 0.0508 | 0.1016 |

3 | $50 | $40.81 | 0.0475 | 0.1425 |

4 | $50 | $38.14 | 0.0444 | 0.1776 |

5 | $50 | $35.65 | 0.0415 | 0.2075 |

6 | $50 | $33.32 | 0.0388 | 0.2328 |

7 | $50 | $31.14 | 0.0362 | 0.2534 |

8 | $50 | $29.10 | 0.0339 | 0.2712 |

9 | $50 | $27.20 | 0.0316 | 0.2844 |

10 | $1050 | $533.77 | 0.6210 | 6.2100 |

Total |
$859.53 |
7.9354 |

Duration has many useful properties. One is that you can compute the duration of a fixed-income portfolio simply by computing a weighted average (weighted by value) of the bonds in the portfolio. In that way, an investor can determine the interest rate sensitivity of his fixed-income portfolio and can estimate how much the portfolio will change in value given a change in market interest rates.

Portfolio managers often adjust the duration of their portfolios by buying and selling bonds in anticipation of interest rate changes. For example, if you thought interest rates were going to decline (and bond values would rise), you would want to lengthen the duration of your portfolio. Conversely, if you thought interest rates were going to rise, you would want to shorten the duration of your portfolio.