11.4 Why long-duration assets are rate-sensitive: equities and gold

Motivation

In 2022 the Nasdaq fell 33.1% while US 10-year real yields rose by about 2.8 percentage points (data in 11.16). The bond-maths idea that explains it is that equities have duration too, and growth stocks have a lot of it.

11.4.1Equity duration#

Setup

The Gordon model (M06 · 6.2) values a stock whose dividends start at $D_1$ next year and grow at $g$ forever, discounted at $r>g$ :

P = \sum_{t=1}^\infty \frac{D_1(1+g)^{t-1}}{(1+r)^t} = \frac{D_1}{r-g}. \tag{11.10}

Proposition 11.5equity duration

For the Gordon stock,

D_{\text{mod}}^{eq} \equiv -\frac{1}{P}\frac{\partial P}{\partial r} = \frac{1}{r-g} = \frac{P}{D_1}, \qquad D_{\text{mac}}^{eq} = \frac{1+r}{r-g}. \tag{11.11}

Proof

From the closed form, $\partial P/\partial r = -D_1/(r-g)^2$ , so $-P^{-1}\partial P/\partial r = 1/(r-g)$ . For the Macaulay version, differentiate the series term by term (it converges uniformly near any $r>g$ ):

\frac{\partial P}{\partial r} = -\sum_t t\,\frac{D_1(1+g)^{t-1}}{(1+r)^{t+1}} = -\frac{1}{1+r}\sum_t t\cdot PV_t.

So $D_{\text{mac}} = \sum_t t\,PV_t/P = (1+r)\cdot(-P^{-1}\partial P/\partial r) = (1+r)/(r-g)$ .∎

So equity duration is the forward price-to-dividend ratio. A stock on a 1% forward yield has a modified duration of 100. Growth stocks are long-duration because most of their value lies in distant cash flows: $r-g$ is small.

Worked example 11.4

Take two stocks with $r=12\%$ . The "value" stock has $g=4\%$ , so $D=12.5$ . The "growth" stock has $g=10\%$ , so $D=50$ . Raise $r$ by 100 bp, holding $g$ :

value stock: $P$ falls from $D_1/0.08$ to $D_1/0.09$ , which is −11.1%;
growth stock: $P$ falls from $D_1/0.02$ to $D_1/0.03$ , which is −33.3%.

The first-order estimates are −12.5% and −50%. The large gap for the growth stock is convexity: $P$ is hyperbolic in $r-g$ .

11.4.2Limits of the Gordon duration#

Where the Gordon duration breaks

Taken literally, an index on a 1.3% forward dividend yield would have a duration of about 77 ( $1/0.013$ ). Measured equity sensitivities to bond yields are far smaller. The reason is that $g$ and $r$ are not independent. A rise in nominal rates caused by higher expected inflation also raises nominal $g$ if firms can pass inflation through (11.14). A rise in real rates caused by tighter policy or a higher term premium does not raise $g$ . Real-rate shocks are what hurt long-duration equities. That was the 2022 pattern.
The discount rate $r$ is the risk-free rate plus an equity risk premium. If the premium falls when bond yields rise (as it did in 1999, and arguably in 2021), price and yields can rise together.
Duration is a local measure. With $r-g$ near zero the convexity term dominates, and small $r$ moves cause very large price moves.

11.4.3Gold and real yields#

Gold as a long-duration real asset Gold pays no coupon. Holding it means giving up the real yield on a safe bond. A minimal model treats gold's value as the present value of a perpetual, non-pecuniary "service flow" $s$ (insurance, collateral, reserve diversification), discounted at the real rate $r^{\text{real}}$ plus a premium $\pi_g$ : $P_{\text{gold}} = s/(r^{\text{real}}+\pi_g)$ . By Proposition 11.5 its duration with respect to real yields is $1/(r^{\text{real}}+\pi_g)$ , which is large when real yields are low. This toy model explains:

why gold tracked the inverse of US 10-year TIPS yields before 2022;
why that link "re-asserted itself" in 2026: the 10-year real yield rose from 1.76% on 2 Mar to 2.13% on 27 Mar and to 2.95% on 5 Oct, above every daily value since 2008¹, and gold fell 11.8% in March despite a war (M10 · 10.6.2; [S]: Reuters report, republished by MarketScreener).

What it does not explain is 2023–25, when gold doubled with real yields near 2%. That residual is attributed to central-bank buying, which is a change in $s$ and $\pi_g$ , not in $r$ (WGC commentary [S]). The model is a lens, not a forecast.