Deriving the Duration of a Continuously Compounded Zero-Coupon Bond

Summary

The document derives the interest-rate duration of a zero-coupon bond under continuous compounding. Its price at time t for maturity T is represented by discounting the single payment at rate r. Duration is defined as the negative derivative of price with respect to the rate, divided by the bond’s price.

Differentiating the exponential price expression multiplies it by the negative time to maturity. Substituting that derivative into the duration definition cancels the price terms and leaves time to maturity. Thus, under these assumptions, the bond’s duration equals the remaining time until payment. The explanation is concise and gives the result algebraically; it does not cover coupon bonds, discrete compounding, changes in yield curves, or other risk measures, so the conclusion should be read within the stated zero-coupon, continuously compounded setup.

Key ideas

• With continuous compounding, a zero-coupon bond’s price is an exponential discount of its maturity payment.
• Differentiating its price with respect to the rate yields the price multiplied by negative time to maturity.
• Dividing the negative price sensitivity by price gives duration equal to remaining maturity.
• The derivation assumes a zero-coupon bond and a single continuously compounded rate.

Tags

# Calculate duration of zero coupon bond


# Calculate duration of zero coupon bond












I am currently studying interest rate risk management, and i can't seem to get the derivation right, and I would like to do all of the steps, to be sure that I understand what is going on.

Let Pz (t, T ) be the price of a zero coupon bond at time t with maturity T and continuously compounded interest rate r.

Duration = $-\frac{1}{P} \frac{d P}{d r}$

Let A and a be two constants and x be a variable. Let $F(x)=A \times e^{a x}$ be a function of x. Then, the first derivative of F with respect to x, denoted by $\frac{d F}{d x}$, is given by

Derivative of F(x) with respect to $x=\frac{d F}{d x}=A \times a \times e^{a x}=a \times F(x)$

The book shows (duration of zero coupon bond): $D_{z, T}=-\frac{1}{P_{z}(t, T)}\left[\frac{d P_{z}(t, T)}{d r}\right]$

$=-\frac{1}{P_{z}(t, T)} \times\left[-(T-t) \times P_{z}(t, T)\right]$

$=T-t$

Because I know the theory this makes total sense, but I cannot derive it. Does someone know how to do this?

## Answer by siou0107 (score 1)

https://quant.stackexchange.com/a/50711

$$P_z\left(r, t, T\right) = e^{-r\left(T - t\right)} \Rightarrow \partial_r P_z = -\left(T - t\right)P \Rightarrow D_{z, T} = T - t$$

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