One valuation date and no-arbitrage cash flows

Learning goals

• Value debt and equity cash flows and solve for the rate or growth implied by a price.
• Use replication to connect spot rates, forwards, currency and a simple option.

Draft · approximately 60 minutes, excluding further revision.

A price is a dated cash-flow sum

For a fixed annual rate r, PV = Σ CFₜ/(1+r)^t. Use spot rates by maturity when the term structure is not flat. A zero-coupon bond with face F and maturity n has P = F/(1+sₙ)^n, so its annual effective yield is (F/P)^(1/n) − 1. Coupon-bond yield is a single discount rate fitted to price, not a forecast of every reinvestment rate.

With stable perpetual dividend growth, equity value is D₁/(r−g), requiring r > g. Rearranging gives r = D₁/P + g. If next dividend is fixed, implied g = r − D₁/P. If only current dividend D₀ is fixed, D₁ = D₀(1+g), so instead g = (Pr−D₀)/(P+D₀). Identifying which dividend is given prevents an apparently small but systematic error.

Replicate before discounting

Cash-flow additivity means identical dated cash flows have equal prices in a frictionless no-arbitrage model. A one-year forward rate starting in year 1 satisfies (1+s₂)² = (1+s₁)(1+f₁,₁). For a spot exchange rate quoted as domestic currency per foreign currency, covered parity is F = S(1+r_domestic)^T/(1+r_foreign)^T, with matching maturity and compounding. Fees, funding constraints and bid–ask spreads matter outside this model.

A one-step option can be replicated with Δ shares and borrowing/lending. If an underlying moves from 100 to either 120 or 80 and a call with strike 100 pays 20 or 0, Δ = (20−0)/(120−80) = 0.5. Borrowing 40 at maturity replicates both option payoffs. At a 5% one-period rate, call value is 0.5×100 − 40/1.05 = 11.9048. This uses replication, not the analyst’s subjective probability of an up move.

Move every cash flow to the same valuation date

A fictional contract pays 50 one year from now and 50 two years from now. At a flat effective annual discount rate of 8%, PV=50/1.08+50/1.08²≈89.1632. At date 1, the first payment is already at that date and the second is one period away: value immediately before payment is 50+50/1.08≈96.2963. The two values differ by exactly one period of growth, not by a change in the promised cash flows.

An ordinary annuity pays at period-end; an annuity due pays at period-start. For otherwise identical positive payments, the annuity-due PV equals ordinary-annuity PV×(1+r). A perpetuity formula C/(r−g) requires r>g and the first cash flow one period away. Draw dates 0,1,2 before choosing a calculator mode. A negative or implausibly large value often reveals a timing error or invalid growth assumption.

Formula and assumptions

PV = Σ CFₜ/(1+sₜ)^t; r = D₁/P + g; f₁,₁ = (1+s₂)²/(1+s₁) − 1; F_X = S(1+r_d)^T/(1+r_f)^T

Forward interest from two spot rates

1. Assume s₁ = 4% and s₂ = 5%, annual effective rates. Two-year accumulation is 1.05² = 1.1025.
2. The year-2 forward rate solves 1.04(1+f) = 1.1025; f = 6.0096%, not 6% exactly.
3. A one-year FX example with S = 25 domestic per foreign, domestic rate 6% and foreign rate 4% gives F = 25×1.06/1.04 = 25.4808 domestic per foreign. Reversing the quote requires reversing both the spot and forward, not only swapping a label.

Errors to catch

• Using D₀ as D₁ or a single yield as every spot rate.
• Reversing FX quote conventions midway through parity calculations.

Original knowledge check

Answer every question before checking. Review explanations after submitting; retry starts a new attempt.

Sources and coverage status

These original lessons are drafts. Public outlines are used to check scope; the official learning-outcome text and proprietary question banks are not reproduced here. Follow the source for exam-year requirements and updates.

• Reference · www.cfainstitute.org