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.

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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.

A bond portfolio manager would typically calculate the bond prices for a number of interest rates. But if interest rates rise to 6%, then the price of the bond is going to drop so that the bond’s $50 interest payment per year will have a yield to maturity (YTM) of 6%.

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. 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. 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. For instance, if the bonds are callable, then any price increases are going to be capped by the call price and the price increase of the bond will slow as the call price is approached. 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.

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.

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. Hence, the price decline slows as the put price is approached, then levels off at the put price.Floating rate securities also have complications. An increase in a bond’s yield to maturity results in a smaller bond price change than a decrease of equal magnitude. The floating rate is usually a specified number of basis points above a benchmark, such as a U.S. 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. 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. 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. 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. 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 market may also demand a greater basis point spread than is being offered by the security, resulting in lower prices.

The higher the coupon rate, the less the bond price changes for a given change in interest rates. 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. 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 LengthThe prices of long-term bonds changes by more than the prices of short-term bonds for the same change in interest rates. 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.

When interest rates drop low enough, the issuer will call the bond, and issue new bonds at the lower interest rate.

However, at lower prices and higher yields, the callable bond has price-yield characteristics similar to the option-free bond. Similarly, a putable bond will not drop below the put price, which is usually par value, since the bondholder can sell the bond back to the issuer for the put price. At higher prices and lower yields, the putable bond has a price-yield relationship that is similar to the option-free bond.The interest rate risk of a bond portfolio, or other similar fixed-rate security, can only be assessed if the risk can be quantified.

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