11.2 Duration · M11 Macro

Motivation

Rates move every day. A desk needs one number that converts "yields up 25 bp" into "we lose ₹X". Duration is that number.

Definition

The Macaulay duration is the present-value-weighted average time to the cash flows:

D_{\text{mac}} = \frac{1}{P}\sum_{k=1}^{N} t_k\,CF_k\,(1+y/m)^{-mt_k}. \tag{11.4}

The modified duration is the proportional price sensitivity:

D_{\text{mod}} = -\frac{1}{P}\frac{dP}{dy}. \tag{11.5}

DV01 (dollar value of a basis point, called PV01 on Indian desks) is $P\,D_{\text{mod}}\times 10^{-4}$ , the money lost for a 1 bp rise.

Proposition 11.2

$D_{\text{mod}} = D_{\text{mac}}/(1+y/m)$ .

Proof

Differentiate (11.2):

\frac{dP}{dy} = \sum_k CF_k\,(-mt_k)\,\frac{1}{m}\,(1+y/m)^{-mt_k-1} = -\frac{1}{1+y/m}\sum_k t_k\,CF_k(1+y/m)^{-mt_k}.

Divide by $-P$ and compare with (11.4).∎

Proposition 11.3closed forms

(a) For a zero-coupon bond maturing at $T$ , $D_{\text{mac}}=T$ . (b) For a bond at par with per-period yield $j$ and $N$ periods, measured in periods,

D_{\text{mod}} = \frac{1-(1+j)^{-N}}{j},\qquad D_{\text{mac}} = \frac{1+j}{j}\left(1-(1+j)^{-N}\right). \tag{11.6}

Proof

(a) There is one cash flow, so the weighted average is $T$ . (b) Write (11.3) per unit face as $P(j) = \frac{q}{j}(1-(1+j)^{-N}) + (1+j)^{-N}$ , where $q=c/m$ is the per-period coupon. Hold $q$ fixed and differentiate in $j$ :

P'(j) = -\frac{q}{j^2}\left(1-(1+j)^{-N}\right) + \frac{q}{j}N(1+j)^{-N-1} - N(1+j)^{-N-1}.

At par, $q=j$ and $P=1$ . The last two terms cancel and $P'(j) = -\frac{1}{j}(1-(1+j)^{-N})$ . So $D_{\text{mod}}=-P'/P$ is as stated, and Proposition 11.2 (in per-period units) gives $D_{\text{mac}}=(1+j)D_{\text{mod}}$ . (c) Let $N\to\infty$ .∎

Divide by $m$ to convert periods into years.

Worked example 11.2

Take a 6.50% 10-year par G-sec, so $j=3.25\%$ and $N=20$ . Then

D_{\text{mod}} = \frac{1-1.0325^{-20}}{0.0325}\Big/2 = 7.270 \text{ years},\qquad D_{\text{mac}} = 7.506 \text{ years}.

The DV01 per ₹100 face is ₹0.0727. A position of ₹100 crore face therefore loses about ₹7.27 lakh per bp, and about ₹1.82 crore for a 25 bp rise in yield.

A 10-year zero at 6.50% has $D_{\text{mac}}=10$ and $D_{\text{mod}}=9.685$ . Coupons pull duration below maturity because part of the value comes back early.

Practitioner notes

Duration assumes a parallel shift: every yield moves by the same amount. Real curves twist. Desks also compute key-rate durations, the sensitivity to each maturity bucket separately.
Duration is additive across a portfolio by value weight. This is why a fund manager states a single "portfolio modified duration". Indian gilt funds disclose it monthly.
The duration of a floating-rate bond is roughly the time to the next reset. Floaters carry almost no rate risk, which is why banks prefer repo-linked loans.