Mastering Modified Duration and Money Duration | CFA Level I Fixed Income
Today, we’ll explore modified duration and its related measure – the money duration. These measures help us better understand a bond’s interest rate risk.
Modified Duration: Estimating Bond Price Changes
Modified duration provides an approximate percentage change in a bond’s price for a 1% change in yield to maturity. It is calculated as the Macaulay duration divided by one plus the bond’s yield to maturity.
EXAMPLE
Let’s calculate the modified duration for a 3-year bond with a Macaulay duration of 2.792 and a yield-to-maturity of 5%:
The modified duration of 2.659 suggests that if the bond’s yield increases by 1%, the bond’s price would drop by approximately 2.659%, and vice versa.
Approximate Percentage Change in Bond Price = -Modified Duration * Change in Yield to Maturity
Calculating Modified Duration for Semi-Annual Coupon Bonds
For bonds with semi-annual coupons, we need to adjust our calculations. First, calculate the Macaulay duration and then divide it by one plus half the yield-to-maturity. To convert the modified duration to an annual basis, divide the semi-annual modified duration by 2.
Money Duration: Absolute Price Change in Currency Units
Money duration, also known as dollar duration, is the absolute price change in currency units given a 1% change in the bond’s yield-to-maturity. It can be expressed based on the full price of a bond position or per 100 of bond par value.
Based on Full Price of Bond Position
Money duration = ModDur x Full price of bond position
Per 100 of Par Value
Money duration = ModDur x Full price of bond per 100 of par
EXAMPLE
Let’s say Peter has a position of 300 units of the bond mentioned earlier, which has a price of $948.02 and an annual modified duration of 1.765:
Money Duration for Full Price of Bond Position: $501,976
Money Duration per 100 of Par Value: $167.33
If the bond’s yield-to-maturity increases by 1%, Peter’s total position would decrease by approximately $5,020.
Price Value of a Basis Point (PVBP)
Price Value of a Basis Point (PVBP) estimates the change in a bond’s full price for a 1 basis point change in yield. Instead of using the modified duration for calculation, we calculate the decrease in price when the yield increases by one basis point and the increase in price when its yield decreases by one basis point. The estimated price change is the difference divided by 2.
PVBP = (V– – V+)/2
EXAMPLE
Let’s calculate the PVBP for Peter’s bond at 12% yield with 2 years to maturity:
Calculate the value of the bond if the yield-to-maturity falls by 1 basis point: V– = 94.81907
Calculate the value of the bond if the yield-to-maturity rises by 1 basis point: V+ = 94.78561
Calculate the PVBP: (V– – V+) / 2 = 0.01673
If the yield-to-maturity rises by 1% (100 basis points), the total change in Peter’s position would decrease by approximately $5,019. This figure is quite close to what we calculated earlier using the money duration.
Modified duration and money duration are powerful tools for estimating the change in a bond’s price due to changes in interest rates. By mastering these concepts, you can better manage your bond investments and understand the risks associated with interest rate fluctuations. In our next lesson, we’ll explore how to correct errors in these measures using convexity adjustments.
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