# Interest Rate Risk of Bonds

Investors expect a fair rate of return from bonds, based on prevailing interest rates, term length of the bonds, and their credit rating. Since prevailing interest rates change continually, there is interest rate risk in holding bolds if the investor wants to sell the bonds before their maturity. For instance, if a bond, with a $1,000 par value, is issued with a nominal interest rate of 5% when bonds with similar risk and terms are also at 5%, then the bond can be sold for $1,000. But if interest rates rise to 6%, then the price of the bond will drop so that the bond's $50 interest payment per year will have a ** yield to maturity** (**YTM**) of 6%. Hence, there may be capital gains and losses associated with bonds if they are sold before maturity, so even with securities that are considered risk-free in terms of default, such as U.S. Treasuries, there is still interest rate risk.

Another way to look at bond prices and yields is to note that the price of a bond = the sum of the present values of the coupon payments and the principal.

Bond Value = Present Value of Coupon Payments + Present Value of Par Value

B | = | T ∑ t=1 | C_{t}(1+r) ^{t} | + | P (1+r) ^{T} |

B = current bond price = coupon paymentC = par value of bondP = time until paymentt T = issued term of bond r = interest rate per payment period |

When interest rates change, then the present value of those payments changes, also, causing the price of the bond to change with it. Note that since the interest rate factor is in the denominator, it is inversely related to the bond price.

Fixed

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The relationship between bond prices and prevailing interest rates is neither simple nor linear. How much bond prices rise or fall depends on the terms of the bonds, the current bond yield, and whether the bonds have embedded options, such as being callable or putable. Burton G. Malkiel has described most of the important general relationships between interest rates and bond prices.

- The most obvious relationship, easily seen in the graph below, is that when interest rates rise, then bond prices fall, increasing the YTM to the current market interest rate for bonds of equal term length and credit rating, and vice versa.
- An increase in a bond's yield to maturity results in a smaller bond price change than a decrease of equal magnitude. As you can see in the graph below, decreasing the yield by a certain amount increases the bond price more than increasing the yield by the same amount decreases the bond price.
- The higher the bond's yield, the less it changes in price per unit change in yield. In the graph below, high-yielding bonds would be in the right-lower part of the graph where the change in bond prices is less per unit change in yield. Thus, the following corollaries are true:
- Interest rate risk is inversely proportional to the current yield to maturity of a bond — the higher the yield to maturity, the less the price will change for a given change in interest rates. This makes sense because any change in interest rates will be a smaller percentage of a high YTM than for a smaller YTM.
- Interest rate risk is also inversely proportional to the coupon rate of the bond. The higher the coupon rate, the less the bond price changes for a given change in interest rates. Since the coupon payments are larger, the bondholder receives more money earlier, which increases the present value of the bond's future cash flow.

**Term Length**- The prices of long-term bonds changes by more than the prices of short-term bonds for the same change in interest rates. This results because the present value of the later coupon payments and the payment of principal is further into the future for long-term bonds, and, thus, is more affected by interest rates.
- The change in bond price per change in interest rate increases as the term of the bond increases, but at a proportionately lesser rate.

The interest rate risk of a bond portfolio, or other similar fixed-rate security, can only be assessed if the risk can be quantified. There are 2 methods of ascertaining interest rate risk: the full-valuation approach and the duration/convexity approach.

## Full-Valuation Approach (aka Scenario Analysis)

The price of a bond in regard to interest rates is the sum of its future cash flows discounted by the interest rate — in other words, the sum of the present value of those cash flows. Hence, one way to calculate interest rate risk is to calculate what actual bond prices would be after a change in interest rates — the **full-valuation approach**. A bond portfolio manager would typically calculate the bond prices for a number of interest rates. Since bond prices only decline if interest rates rise, the manager would mostly be interested in the value of the portfolio if the interest rate rises by specific increments, such as 100, 200, or 300 basis points. By calculating the value of the bond portfolio for each increment, the manager can determine the actual interest rate risk if the interest rates rise by the calculated amount. Because interest rate risk is determined for specific scenarios, the full-valuation approach is also known as **scenario analysis**.

### Example: Scenario Analysis

The table below lists bond prices and the corresponding price changes for bonds with a coupon rate of 5% for several different market interest rates and bonds of different terms.

Interest Rate | Term | Bond Price | Price Change | YTM |
---|---|---|---|---|

6.0% | 1 Year | $990.43 | -0.96% | 6.00% |

6.0% | 5 Years | $957.35 | -4.27% | 6.00% |

6.0% | 10 Years | $925.61 | -7.44% | 6.00% |

6.0% | 20 Years | $884.43 | -11.56% | 6.00% |

7.0% | 1 Year | $981.00 | -1.90% | 7.00% |

7.0% | 5 Years | $916.83 | -8.32% | 7.00% |

7.0% | 10 Years | $857.88 | -14.21% | 7.00% |

7.0% | 20 Years | $786.45 | -21.36% | 7.00% |

8.0% | 1 Year | $971.71 | -2.83% | 8.00% |

8.0% | 5 Years | $878.34 | -12.17% | 8.00% |

8.0% | 10 Years | $796.15 | -20.39% | 8.00% |

8.0% | 20 Years | $703.11 | -29.69% | 8.00% |

The full-valuation approach works well for option-free bonds, but the analysis becomes more complicated if the bonds have embedded options, such as being callable or putable. For instance, if the bonds are callable, then any price increases will be capped by the call price and the price increase of the bond will slow as the call price is approached. If the bond is putable, then decreases in the bond price will have a floor at the putable price, which is usually par value. If the bond's price falls below this, then the bondholder can sell the bond back to the issuer for the put price. Hence, the price decline slows as the put price is approached, then levels off at the put price.

**Floating rate securities** also have complications. The floating rate is usually a specified number of basis points above a benchmark, such as a U.S. Treasury. The rate is reset at specific time intervals, such as every month or every 6 months, and there is usually a cap on the interest rate of the security, which is the maximum amount of interest that can be earned. While the interest rate is below the cap, a given change in interest rates will result in larger price changes the more time that is left until the reset date. The market may also demand a greater basis point spread than is being offered by the security, resulting in lower prices. Finally, there is **cap risk**, where any increases in interest rate above the cap price will cause the bond price to decline just as with an ordinary bond.

Picking which scenarios to analyze depends on the investment objective of the manager, and possibly regulations. For instance, depositary institutions are often required to test a portfolio for a 1%, 2%, and 3% increase in interest rates. Highly leveraged portfolios, such as those managed by hedge funds, may test extreme scenarios — **stress testing** — since even small changes in interest rates can result in large losses in highly leveraged portfolios.

With a large bond portfolio, the full-valuation approach becomes computationally intensive, hence statistical models that can be performed more quickly and cover more scenarios have been developed to calculate interest rate risk.