6.7 Margin of safety as a statistical statement

Motivation

Graham's margin of safety says: buy well below a conservative estimate of value. That is a statement about uncertainty, and it can be made precise. Each input to (6.9) is an estimate with an error; propagate them and value becomes a distribution.

6.7.1Value as a distribution#

Setup

Treat $F_0, g, W, g_T$ as independent normal random variables $F_0 \sim N(1000, 100^2)$ , $g \sim N(0.15, 0.04^2)$ , $W \sim N(0.12, 0.01^2)$ , $g_T \sim N(0.05, 0.01^2)$ , truncated to $W - g_T \ge 1\%$ (rejection sampling; without truncation the terminal value has no finite mean, because $1/(W - g_T)$ has a pole). Simulate $V$ from (6.9). Code: m06_mc_valuation.simulate_intrinsic_values.

6.7.2Convexity bias#

Proposition 6.11Convexity bias

Holding the other inputs fixed, $V_0$ in (6.9) is a convex function of $W$ on $W > g_T$ . Hence if $W$ is random, $\mathbb{E}[V_0(W)] \ge V_0(\mathbb{E}W)$ , with strict inequality when $W$ is non-degenerate.

Proof

Each explicit term $a_t(1+W)^{-t}$ with $a_t > 0$ has second derivative $a_t t(t+1)(1+W)^{-t-2} > 0$ . The terminal term is $c\,f(W)h(W)$ with $c > 0$ , $f(W) = (1+W)^{-N}$ and $h(W) = (W - g_T)^{-1}$ . Both $f$ and $h$ are positive, decreasing and convex on $W > g_T$ , so $(fh)'' = f''h + 2f'h' + fh'' > 0$ because $f'h' > 0$ . A sum of convex functions is convex. Jensen's inequality completes the proof.∎

So a single-point DCF at the expected discount rate understates the mean of the value distribution. The median is close to the point estimate; the mean is pulled up by the right tail where $W - g_T$ is small. Do not mistake that right tail for upside: it is model fragility.

Worked example

200,000 draws (seed 11):

Statistic	Value (₹ cr)
Point estimate (6.3)	22,536
Mean	23,762
Median	22,611
Standard deviation	7,000
5th / 20th / 80th / 95th percentile	14,858 / 18,162 / 28,549 / 36,430

Market EV (₹ cr)	$\hat P(V > \text{EV})$
15,000	0.946
16,902 (25% below point)	0.869
20,000	0.680
22,536 (= point)	0.505
25,000	0.356
30,000	0.155

With only the WACC uncertain ( $\sigma$ = 1 pp) the mean is 23,043 against a median of 22,531: the convexity bias of Proposition 6.11.

6.7.3Margin of safety as a quantile#

The statistical margin of safety Choose a confidence level $1 - \alpha$ and buy only if $P(V > \text{price}) \ge 1 - \alpha$ , that is, if the price is below the $\alpha$ -quantile $q_\alpha$ of the value distribution. A "25% margin of safety" on this company corresponds to about 87% probability that value exceeds price under these input distributions. The same 25% rule gives very different confidence for a company whose inputs are more uncertain: the right margin is a quantile, not a fixed percentage.

Practitioner notes

The inputs are not independent. A recession lowers $F_0$ and $g$ together and raises risk premia, so growth and the discount rate tend to be negatively correlated. Because $\partial V/\partial g > 0$ and $\partial V/\partial W < 0$ , negative correlation between them widens the value distribution, and positive correlation narrows it (Problem 6.11 derives this with the delta method).
The distribution is conditional on the model. Fraud, dilution, a regulatory change or a governance failure are not in it. Forensic checks (6.8) and quality filters (6.9) address risks a DCF cannot see.
Damodaran's spreadsheets support this kind of simulation; the point is not precision but to see which inputs drive the spread.