The money duration of a bond is a measure of the price change in units of the currency in which the bond is denominated, given a change in annual yield to maturity.
Money Duration = AnnModDur x PVFULL
ΔPVFULL≈ -MoneyDur x Δyield
Consider a bond with a par value of $100 million. The current yield to maturity (YTM) is 5% and the full price is $102 per $100 par value. The annual modified duration of this bond is 3. the money duration can be calculated as the annual modified duration (3) multiplied by the full price ($102 million): 3 x $102 million = $306 million. If the YTM rises by 1% (100 bps) from 5% to 6% the decrease in value will be approximately $306 million x 1% = $3.06 million. If the YTM rises by 0.1% (10 bps), the decrease in value will be $306 million x 0.1% = $0.306 million.
An important measure which is related to money duration is the price value of a basis point (PVBP). The PVBP is an estimate of the change in the full price given a 1 bp change in the yield-to-maturity. The formal equation is given below.
where PV_ and PV+ are full prices calculated by decreasing and increasing the YTM by 1 basis point.
A quick way of calculating the price value of a basis point is to take the money duration and multiply by 0.0001. For example, if the money duration of a portfolio is $200,000 the price value of a basis point is $200,000 x 0.0001 = $20. (1 bp = 0.01% = 0.0001)
Example 13: Calculating money duration of a bond
A life insurance company holds a USD 1 million (par value) position in a bond that has a modified duration of 6.38. The full price of the bond is 102.32 per 100 of face value.
Example 14: Calculating PVBP for a bond
Consider a $100, five-year bond that pays coupons at a rate of 10% semi-annually. The YTM is 10% and it is priced at par. The modified duration of the bond is 3.81. Calculate the PVBP for the bond.
Money duration = $100 x 3.81 = $381.00
PVBP = $381 x 0.0001 = $0.0381
The graph below shows the relationship between bond price and YTM. It shows the convexity for a traditional fixed-rate bond.
Source: CFA Program Curriculum, Understanding Fixed-Income Risk and Return
Interpretation of the diagram:
Here we need to factor in the convexity. The percentage change in the bond’s full price with convexity-adjustment is given by the following equation:
Change in the price of a full bond:
Expression in first braces: duration adjustment
Expression in second braces: convexity adjustment
Approximate convexity can be calculated using this formula:
PV_ and PV+ = new full price when YTM is decreased and increased by the same amount
PV0 = original full price
The change in the full price of the bond in units of currency given a change in YTM can be calculated using this formula:
Convexity is good
The following exhibit shows the price-yield curves for two bonds with the same YTM, price, and modified duration, and why greater convexity is good for an investor.
Interpretation of the diagram:
The relationship between various bond parameters with convexity is the same as with duration.
For a fixed-rate bond,
For bonds whose cash flows were unpredictable, we used effective duration as a measure of interest rate risk. Similarly, we use effective convexity to measure the change in price for a change in benchmark yield curve for securities with uncertain cash flows. The effective convexity of a bond is a curve convexity statistic that measures the secondary effect of a change in a benchmark yield curve. It is used for bonds with embedded options.
Effective Convexity =
Here is a summary of some important points related to bonds with embedded options:
Example 15: Calculating the full price and convexity-adjusted percentage price change of a bond
A German bank holds a large position in a 6.50% annual coupon payment corporate bond that matures on 4 April 2029. The bond’s yield to maturity is 6.74% for settlement on 27 June 2014, stated as an effective annual rate. That settlement date is 83 days into the 360-day year using the 30/360 method of counting days.
There are 15 years from the beginning of the current period on 4 April 2014 to maturity on 4 April 2029.
1. The full price of the bond is 99.2592 per 100 of par value.
FV = 100, I/Y = 6.74, PMT = 6.50, N = 15, CPT PV; PV = -97.777.
2. PV+ = 99.1689
FV = 100, PMT = 6.5, I/Y = 6.75, N = 15, CPT PV; PV = -97.687.
PV_ = 99.3497.
FV = 100, I/Y = 6.73, PMT = 6.5, N = 15, CPT PV; PV = -97.869.
The approximate modified duration is 9.1075.
The approximate convexity is 201.493.
3. The convexity-adjusted percentage price drop resulting from a 100 bp increase in the yield-to-maturity is estimated to be -8.1% (-9.1075 + 1.00746).<
Modified duration alone estimates the percentage drop to be 9.1075%. The convexity adjustment adds 100.746 bps (0.5 × 201.493 × .012 = 1.00746%).
4. The new full price if the yield-to-maturity goes from 6.74% to 7.74% on that settlement date is 90.7623. The actual percentage change in the bond price is -8.5603%. The convexity-adjusted estimate is -8.1%.
Example 16: Calculating the approximate modified duration and approximate convexity
The investment manager for a US defined-benefit pension scheme is considering two bonds about to be issued by a large life insurance company. The first is a 25-year, 5% semiannual coupon payment bond. The second is a 75-year, 5% semiannual coupon payment bond. Both bonds are expected to trade at par value at issuance.
Calculate the approximate modified duration and approximate convexity for each bond using a 5 bp increase and decrease in the annual yield-to-maturity.
In the calculations, the yield per semiannual period goes up by 2.5 bps to 2.525% and down by 2.5 bps to 2.475%.
The 25-year bond has an approximate modified duration of 14.18.
PV+ : FV = 100, I/Y = 2.525, PMT = 2.5, N = 50, CPT PV, PV = -99.2945.
PV– : FV = 100, I/Y = 2.475, PMT = 2.5, N = 50, CPT PV, PV = -100.7126.
Similarly, the 75-year bond has an approximate modified duration of 19.51 and an approximate convexity of 708.