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. Its price is the sum of those cash flows 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.

YTM is the single discount rate that equates the bond's market price to the PV of its cash flows. Coupon bonds require a financial calculator or trial. For a zero-coupon bond, the math is clean.

$$r={\left(\frac{F}{P}\right)}^{1/N}-1$$

Equity has no maturity and no fixed cash flows. The dividend discount model treats a stock as a perpetual dividend stream.

Common mistakes

• Using current dividend D0 instead of next dividend D1. Gordon growth needs the dividend one period ahead. If D0 = $2.00 and g = 4%, the correct D1 is $2.08, not $2.00.
• Applying Gordon growth when g ≥ r. The formula requires r > g. Plug g = 10% and r = 9% and the denominator goes negative. Use a multi-stage model instead.
• Reversing the forward FX ratio. $F_{P/B}=S_{P/B}\times (1+r_P)/(1+r_B)$. The price currency's rate is in the numerator. Flipping the ratio makes the high-yielder trade at a forward premium instead of a discount.

Bottom line

• Bond PV = sum of discounted coupons + discounted face. Solving for the discount rate that matches price gives the implied yield (YTM)
• Coupon-yield link: a par bond has YTM = coupon, a discount bond YTM > coupon, and a premium bond YTM < coupon
• Zero-coupon yield $(F/P{)}^{1/N}-1$: the single rate that grows price $P$ to face $F$ over $N$ periods
• Gordon growth: $P_0=D_1/(r-g)$, valid only when $r>g$. Rearrange: $r=D_1/P_0+g$; implied $g=r-D_1/P_0$

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." For forward rates: treat the two-year compound as a stack of two one-year factors, and the second factor must be the implied forward rate that makes the stack equal $(1+s_2{)}^2$.

The full lesson (about 1,729 words, 12 min read) adds 2 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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