###### Cash Flows of Fixed-income Securities

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The modified duration is a measure of the percentage price change of a bond given a change in its yield-to-maturity. On the other hand, the money duration of a bond is a measure of the price change in units of the currency a bond is denominated. In the United States, the money duration is commonly called “dollar duration.”

The money duration (MoneyDur) is calculated as the annual modified duration times the bond’s full price (PV^{Full}), including accrued interest.

$$ MoneyDur = \text{Annual ModDur} × PV^{Full}$$

and

$$ ΔPV^{Full} ≈ \text{-MoneyDur} × ΔYield$$

Assume that an investor holds a $5 million (par value) in a 4.5% bond maturing on March 31, 2015. The bond is priced at 97.250 per 100 of par value to yield 5.250% on a semi-annual basis for settlement on 30 June 2014. The total market value including accrued interest is $4,980,000 or 4.980 per 100 par value. The bond’s annual Macaulay duration is 2.500.

The money duration is equal to the annual modified duration times the full price per 100 of par value:

$$ \text{MoneyDur} = \text{Annual ModDur} × PV^{Full}$$

$$=\frac{MacDur}{(1+y)} × PV^{Full}=\frac{2.5}{1+\frac{0.0525}{2}}×$97.25=$236.90$$

Note: periodicity = 2; hence the reason for the \(1+\frac{0.0525}{2}\).

This means that for a 1% in yield, the bond price will lose $239.90.

Another version of the money duration is the price value of a basis point (PVBP) for the bond. The PVBP estimates the change in full price given a 1 bp change in the yield-to-maturity.

$$PVBP=\frac{(PV_- )-(PV_+)}{2}$$

PV_{– }and PV_{+} represent the bond prices calculated after decreasing and increasing the yield-to-maturity by 1 bp. The PVBP is also called the “PV01”, standing for the “price or present value of 01”, where “01” means 1bp. In the United States, it is commonly called the “DV01” (Dollar value).

A bond with exactly five years remaining until maturity offers a 4% coupon rate with annual coupons. The bond, with a yield-to-maturity of 6%, is priced at 91.575272 per 100 of par value. Estimate the price value of a basis point for the bond.

Lowering the yield-to-maturity by one basis point to 5.99% results in a bond price of 91.615115:

$$PV_-=\frac{4}{1.0599^1} +\frac{4}{1.0599^2} +\frac{4}{1.0599^3} +\frac{4}{1.0599^4} +\frac{100+4}{1.0599^5} =91.615115$$

Increasing the yield-to-maturity by one basis point to 6.01% results in a bond price of 91.535451:

$$PV_+=\frac{4}{1.0601^1} +\frac{4}{1.0601^2} +\frac{4}{1.0601^3} +\frac{4}{1.0601^4} +\frac{100+4}{1.0601^5} =91.535451$$

$$PVBP=\frac{(PV_- )-(PV_+)}{2}=\frac{91.615115-91.535451}{2}=0.039832$$

Alternatively, the PVBP can be derived using modified duration:

$$Approx. \quad modified \quad duration = \frac{PV_–PV_+}{2×Δyield×PV_0 }$$

$$=\frac{91.615115-91.535451}{2×0.0001×91.575272}=4.349646$$

$$PVBP = 4.349646 × 91.575272 × 0.0001=0.039832$$

QuestionA bond has a money duration of 4. By how much will the full price of this bond change after a 0.25% change in yield-to-maturity?

- $0.01
- $1
- $4

SolutionThe correct answer is

A.\(ΔPV^{Full} ≈ \quad -MoneyDur × ΔYield\)

\(≈ \quad -4 × 0.25\% = -0.01\)