measure the change in bond prices with respect to changes in yield. This is the default if the basis is omitted. Modified Duration, a variation on Macaulays duration is modified duration, which is a price sensitivity measure that is the percentage derivative (rate of change) of price with respect to yield. However, changes in perception of the risk of default may also change bond prices, blunting or augmenting what duration would predict. Because the formula is complicated and time consuming, most investors use calculators to do the math. Treasuries generally have lower coupon rates and current yields than corporate bonds of similar maturities because of the difference in default risk. As a general rule, for every 1 change in interest rates (increase or decrease a bond s price will change approximately 1 in the opposite direction, for every year of duration. In cell B20 we entered the following formula for Bond 1: mduration(B2,B3,B5,B6,B9,B10) We then copied this formula to C20 to find the modified duration of Bond.
Securities with the same duration have the same interest rate risk exposure. Convexity is the rate that the duration changes along the price-yield curve, and, thus, is the 1st derivative to the equation for the duration and the 2nd derivative to the equation for the price-yield function. Hence: Bond Price Change Formula Bond Price Change Duration Yield Change Convexity Adjustment Important Note! The convexity can actually have several values depending on the convexity adjustment formula used. Using duration, you can estimate how much a bond s price is likely to rise or fall if interest rates change (the bond s price sensitivity and it can be thought of as a measurement of interest rate risk.
Using the modified duration, we can approximate the percentage change in a bonds price for a change in yield: Change in Price -DMod YTM In Figure 2, we replicated this formula in cell B22, this time assuming that the yield increased from.0.1. Recall, too, that the coupon rate is the interest rate on the bond that is established at the time the bond is issued. Morgan North American High Yield Research, December 2008,. Note, however, that this convexity approximation formula must be used with this convexity adjustment formula, then added to the duration adjustment:. So investors need to consider both yield and duration.
Not to be confused with maturity which is how long a fixed-income investment lasts bond duration measures how long it takes, in years, for an investor to be repaid the bond s price by the bond s total cash flows. A modified version of the Macaulay model that accounts for changing interest rates. The formula for the duration of a coupon bond is the following: Duration Formula for Coupon Bond Coupon Bond Duration 1 y y (1 y) T (c y) c (1 y)T 1 y y yield to maturity c coupon interest rate in decimal form. This is because duration is a tangent line to the price-yield curve at the calculated point, and the difference between the duration tangent line and the price-yield curve increases as the yield moves farther away in either direction from the point of tangency. Beyond improving the predicted percentage change in bond prices, convexity is also useful for comparing bonds of equal durations and yields. Below are the durations and yields for various fixed income asset classes.1. A Visual Representation As we mentioned earlier, modified duration only works for small changes in the yield since it is a linear approximation to a nonlinear function. Note that the price-yield curve is convex, and that the modified duration is the slope of the tangent line to a particular market yield, and that the discrepancy between the price-yield curve and the modified duration increases with greater changes in the interest rate. If we have two bonds that are identical with the exception on their coupon rates, the bond with the higher coupon rate will pay back its original costs faster than the bond with a lower yield. Convexity As stated, using modified duration to calculate the change in price for a given change in yield only works for small changes in yield. Figure 1 from the Fourth Quarter 2012 Spreadsheet Corner article illustrated the relationship between market interest rates and bond prices. The price/yield curve is just that: curved.
Conversely, if interest rates fall, the value of bonds will increase. Looking back through history when we have seen rates rise, we have certainly not seen weak returns in the high yield market. One such measure is known as duration. 2 actual/360 3 actual/365 4 European 30/360.