9.17 What is Duration? Price versus Reinvestment Risks
Initial (Old) Definitions:
- Duration is the weighted average life of the present value of the bond’s cash flows.
- Duration is the point which causes the fulcrum to be in balance.
- (Some analysts think of) duration as the “effective” maturity of the bond.
New Definition:
Duration represents the holding period at which a bond’s reinvestment and price risks will exactly offset one another. (The mathematical proof of this is coming up.)
Remember:
A change in YTM affects Price. This is what is meant by “Price Risk.” If YTM changes, both price and reinvestment rate will change. Price and reinvestment risks are inversely related. In the forthcoming analysis, we will assume that the new and changed REIN = YTM.
| if REIN = YTM, then RCY = YTM | and | if REIN ≠ YTM, then RCY ≠ YTM |
Some (New) Duration Characteristics:
- If rates go up during the life of a bond, the investor will benefit from a higher reinvestment rate, but be hurt in terms of price risk.
- If interest rates go down, the bond’s terminal value will be less.
- If reinvestment rates go up (down), the terminal value will be greater (less). The investor, accordingly, may have more or less than their future cash needs.
- At the duration point, we will find that RCY = YTM — even if the yield changes (as we assume, “instantaneously”), assuming the bond will be sold at that time. At other than the duration point, price and reinvestment risks will not offset one another, as will be seen below.
- If Price and Reinvestment risks were not offset, the RCY ≠ YTM. The Terminal Value of the bond, including the reinvestment of the coupons, is in the numerator of the RCY formula, while the Price, representing Price Risk is in the denominator.
Summary:
Duration balances the difference between the accumulated – or reinvested – value of the coupon payments on the one hand, and the assumed sale-value of the bond on the other hand. If rates go up, the initial impact is to lower the bond’s value, but this will be offset by a higher reinvestment rate. Let’s now see the precise mathematics.
