The Time Value of Money in Finance

Every fixed-income and equity valuation collapses to one idea: a security is worth the present value of its expected cash flows. The exam tests the inverse just as hard. Given a price and cash flows, solve for the implied return or implied growth.

A bond promises fixed coupons plus face value at maturity. You calculate its price as the present value (PV) of those cash flows, each discounted at the market yield. Higher yield means lower price, and longer maturity amplifies the effect.

$$P = \sum_{t=1}^{N} \frac{C}{(1+r)^t} + \frac{F}{(1+r)^N}$$

KEY: At par, coupon equals yield. At a discount, yield exceeds coupon. At a premium, yield is below coupon.

The same present value idea covers three other fixed-income shapes. A zero-coupon bond keeps only the face value term. A perpetual bond pays its coupon forever and never repays face value. An amortizing bond or loan pays a level amount each period that covers interest and part of the principal, so you calculate its value with the annuity formula.

Common mistakes

• Using D0 instead of D1. Gordon growth needs the dividend one period ahead. In Example 4, D0 = $2.00 and g = 4%, so D1 = $2.08. Trap: 2.00 / 0.05 = $40.00 instead of $41.60.
• Discounting the terminal value one period too far. $P_2$ = $30.24 is a year-2 value, even though it uses D3. Trap: discounting it three years gives $25.00 for Company H instead of $27.27.
• Applying Gordon growth when g ≥ r. The formula requires r > g. Plug g = 10% and r = 9% and the denominator goes negative, so the constant-growth value is invalid. A multi-stage model helps only if the high growth is temporary and the long-run growth rate falls below r, as in Company H.

Bottom line

• Bond PV = sum of discounted coupons + discounted face. YTM is the rate that matches price; find it by trial and interpolation or with the calculator. Par: YTM = coupon; discount: YTM > coupon; premium: YTM < coupon
• Zero-coupon yield $(F/P)^{1/N} - 1$; perpetual bond $P = C/r$; an amortizing loan's payment = principal / annuity factor
• Gordon growth: $P_0 = D_1/(r-g)$, valid only when $r > g$. Preferred stock: $P_0 = D/r$
• Multi-stage DDM: discount explicit dividends, plus a Gordon terminal value at year n discounted n periods

Exam shortcut

For coupon vs. yield: "Par equals, Discount yield higher, Premium yield lower." For Gordon rearrangement: "r equals yield plus growth, g equals r minus yield." If the stem says "just paid," grow the dividend one period before you divide. In a multi-stage stock, the terminal value sits at the last explicit year, so discount it that many periods.

The full lesson (about 4,261 words, 28 min read) adds 11 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• time value of money in finance

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