The Time Value of Money in Finance

Almost every valuation in finance is a present value: the value today of cash flows expected later, discounted at a rate that reflects time and risk. This module applies discounting to bonds, loans and shares, runs the calculation in reverse to find the return or growth rate implied by a market price, and uses the principle of no arbitrage to derive forward interest rates, forward exchange rates and a simple option value.

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What you will learn

1. Calculate the present value of fixed-income instruments, including zero-coupon bonds, coupon bonds, perpetual bonds and amortizing loans, from their promised cash flows.

Zero-coupon bond price = FV / (1 + r)^N.
Coupon bond price = PV of the coupon annuity + PV of the face value; coupon rate = market rate gives a price of par.
Perpetual bond value = coupon / r.
Amortizing loan payment A = r x PV / [1 - (1 + r)^-N].
Price and discount rate move inversely; distant cash flows are the most sensitive.

2. Calculate the present value of equity instruments, including preferred shares and common shares with constant or two-stage dividend growth.

Preferred share (fixed dividend, no maturity): V0 = D / r.
Gordon growth: V0 = D1 / (r - g), with D1 = D0(1 + g) and g < r.
Value is very sensitive to the spread r - g.
Two-stage model: discount high-growth dividends, then a terminal value V_n = D_(n+1) / (r - g_L) discounted n periods.
The terminal value is often most of the estimated value.

3. Calculate and interpret the implied return of a fixed-income instrument and the implied required return or implied growth rate of a share, given its price and expected cash flows.

Implied return = the discount rate that sets the PV of expected cash flows equal to the price (an IRR).
Zero-coupon yield: r = (FV / PV)^(1/N) - 1; coupon bond yields are found iteratively.
Implied required return on equity: r = D1 / P0 + g.
Implied growth: g = r - D1 / P0; with D0 given, solve P0 = D0(1 + g) / (r - g).
Compare implied figures with fundamentals to judge whether a price looks optimistic or pessimistic.

4. Explain the cash flow additivity principle and the condition of no arbitrage, and use them to price instruments from spot rates and to calculate implied forward interest rates.

Cash flow additivity: PV of a stream = sum of PVs of its individual cash flows.
No arbitrage: identical future cash flows must have the same price today.
Price a bond by discounting each cash flow at the spot rate for its maturity.
Implied forward rate: (1 + S2)^2 = (1 + S1)(1 + F1,1).
Upward-sloping spot curve: forwards above spots; inverted curve: forwards below spots.

5. Apply the no-arbitrage principle to calculate forward exchange rates and the value of an option using a one-period replicating portfolio.

Covered interest parity: F = S x (1 + r_price) / (1 + r_base), with rates quoted as price currency per unit of base currency.
The higher-interest-rate currency trades at a forward discount.
A replicating portfolio of the underlying and risk-free borrowing reproduces the option payoff.
Risk-neutral probability q = [(1 + r)S0 - Sd] / (Su - Sd); option value = discounted risk-neutral expected payoff.
Real-world probabilities do not enter the no-arbitrage option price.

Module at a glance

Value = sum of expected cash flows discounted at the required return for their timing and risk.
Zero: FV / (1 + r)^N; perpetuity: PMT / r; loan payment: r x PV / [1 - (1 + r)^-N].
Gordon model V0 = D1 / (r - g); two-stage models add a terminal value discounted n periods.
Run valuation in reverse to get implied yields, implied required returns and implied growth.
No arbitrage: identical future cash flows must cost the same today.
Forward rates: (1 + S2)^2 = (1 + S1)(1 + F1,1); forward FX: F = S x (1 + r_price) / (1 + r_base).
A one-period option is worth the cost of its replicating portfolio, using risk-neutral probabilities.

This free module has 5 lessons, 13 practice questions with full explanations.

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