Equity Valuation, Risk, and Investment: A Practitioner's Roadmap

APPENDIX C

Deriving the “Equity Duration” Formula

We begin by repeating equation (3.16):

(3.16) Repeated

To find the sensitivity of valuation to changes in the real discount rate, ρE, we take the derivative utilizing the Chain Rule:

(C.1)

Substituting equation (3.16) into (C.1) allows us to cut down the order of the exponent in the denominator:

(C.2)

Dividing −V0 on both sides, and remembering that dV/dρE = ΔV/ΔV/ΔρE as ΔV ⇒ 0 and ΔρE ⇒ 0, we get the definition of duration:

(C.3)

We are able to use equation (3.23) to retrieve these results:

(C.4)

Substituting both of these expressions into equation (C.3) gives us:

(C.5)

which reduces to the simple expression found in the text:

(C.6)