Duration

Bond Valuation and Analysis in R

Clifford Ang

Senior Vice President, Compass Lexecon

What is duration?

Estimated price change for a 100 basis point change in yield
- Two bonds with the same duration will have same estimated price change
A way to manage the risk of interest rate sensitive liabilities

$$D = \frac{P(\text{down})-P(\text{up})}{(2 \cdot P \cdot \Delta y)}$$

$$\frac{\Delta P}{P} = -D \cdot \Delta y$$

$$\Delta P = -D \cdot \Delta y \cdot P$$

$\dfrac{\Delta P}{P}$ = Percent change

$D$ = duration

$\Delta y$ = Change in yield

$\Delta P$ = Dollar change

$P$ = Current price

Example: $100 par value, 5% coupon rate, 10 years to maturity, initial yield = 4%, expected increase in yield = 1%

(p <- bondprc(100, .05, 10, .04))

108.1109

(p_down <- bondprc(100, .05, 10, .03))

117.0604

(p_up <- bondprc(100, .05, 10, .05))

(duration <- (p_down - p_up) / (2 * p * 0.01))

7.890234

(duration_pct_change <- - duration * 0.01)

-0.07890234

(duration_dollar_change <- duration_pct_change * p)

-8.530203

Yield vs. price

Bond Valuation and Analysis in R