Modified duration is a measure of bonds’ volatility. It is related to the approximate percentage change in price for a given change in the yield or one percent parallel shift in the yield curve or the term structure of interest rates, assuming the term structure is flat due to the use of the yield to maturity as the discount rate. The fundamental principle that bond prices move in the opposite direction of the change in interest rates implies that there is an inverse relationship between modified duration and the approximate percentage in price for a given yield change.
The diagram shows the convex relationship between bond prices and interest rates. A tangent line has been drawn against the price function. The tangent’s gradient reflects the bond’s price change due to interest rate changes. Diagram: Duration as approximation for bond price changes3 Suppose that yield rises by dy, the price of the bond P0 falls to P1 along the function P(YtM), describing the inverse relationship between bond price and interest rates. Modified Duration is the slope of the tangent divided by the dirty price of the bond.
The restriction of this illustration is that it is only an approximation of the resultant change in the bond’s price (dP). The price:yield function is not linear but convex as is shown above. The impact of the Convexity increases in the event of large changes in the interest rate level. Nonetheless, the Modified Duration provides sufficiently adequate results for smaller interest rate level changes (100 basis points). The ways in which modified duration behaves depend on the bond, the coupon payments and its maturity.
“The higher the coupon on a bond the lower the modified duration. The reason is that the higher the coupon the larger the proportion of total cash flows received as coupons prior to redemption and thus the larger the proportion of cash flows received early compared to an equivalent lower coupon bond. Duration shortens as time to maturity shortens. Between coupon dates the duration shortens with time. However, immediately after coupon payment, duration increases slightly as the effect of an imminent cashflow is removed from total weighted average time to receipt of the cash flows”.
In addition, duration changes as bond price changes and is positively related to the price of a bond and negatively related to the yield. If interest rates (yields) fall, a parallel shift in the yield curve There is a consistency between the properties of bond price volatility and the properties of modified duration. With all other factors constant, the longer the maturity the greater the price volatility. Thus, modified duration can be interpreted as the approximate percentage change in price for a 100-basis-point change in yield.
As illustrated and explained above using the diagram, price yield curve of a bond with no embedded options is non-linear (convex), yet the first derivative of price with respect to yield is the tangent of the price yield curve. Thus, modified duration is only a reasonable approximation of the interest rate sensitivity for small changes in yield and it can only be applicable to measuring price risk over very small, and instantaneous changes in yield. Furthermore, we use yield to maturity as discount rate. Consequently, all future cashflows are discounted at the same rate.
This assumes that the term structure of interest rates is flat and only shifts in parallel, which is contrary to empirical evidence provided by researchers who have developed models that use the observed spot rates as the discount rates and allow for non-parallel shifts in the term structure. When there are large movements in the required yield, modified duration is not adequate to approximate the price reaction. Duration will overestimate the price change when the required yield rises, thereby underestimating the new price. When the required yield falls, duration will underestimate the price change and thereby underestimate the new price.
Relying on duration as the sole measure of the price volatility of bonds denominated in a single currency may mislead investors. First, deriving price volatility using modified duration assumes that all cash flows for the bonds are discounted at the same discount rate. The assumption of flat term structure and parallel shifts imposes a limitation of applying duration when the assumption does not hold, and the yield curve does not shift in a parallel fashion. This is extremely important when we use a portfolio’s duration to quantify the responsiveness of a portfolio’s value to a change in interest rates.
5 Duration of a portfolio is determined by a linear combination of the present value weighted durations of the individual bonds in the portfolio. Interest rate risk is included in the portfolio as the unexpected fluctuations in the bond price that affects all bonds in the portfolio. The duration of each bond in the portfolio is weighted by its percentage within the portfolio. If a portfolio has bonds, denominated in more than one currency, with different maturities, the duration measure may not provide a good estimate for unequal changes in interest rates of different maturities.
The justification to that lies in the assumption of perfect correlation between exchange rates around the world being far from rational. This limitation can be dealt with using rate duration. Rate duration is the approximate change in the value of a portfolio or bond to a change in the interest rate of a particular maturity assuming that the interest rate for all other maturities is held constant. Therefore, interest rates are allowed to change individually and disproportionately causing maturities to vary by different numbers of basis points. ).
