5 Valuation of Bonds and Stocks

Discounting future cash flows

Valuation estimates what a financial asset is worth today by discounting its expected future cash flows. A dollar received later is worth less than a dollar received today, and riskier or more uncertain cash flows generally require a higher return.

The general discounted cash-flow relationship is:

where $V_0$ is value today, $CF_t$ is the cash flow at time $t$, and $r$ is the discount rate per period. The timing of the cash flows and the periods used for the discount rate must match.

Bond valuation and market yields

A conventional fixed-rate bond promises periodic coupon payments and repayment of face value at maturity. Its value is the present value of both streams. For annual coupons, the bond valuation formula is:

where $C$ is the annual coupon, $F$ is face value, $n$ is the number of years to maturity, and $y$ is the market-required yield for comparable bonds. The yield reflects factors such as prevailing interest rates and the issuer’s credit risk. Coupon payments form an annuity, while face value is a final lump sum.

Example: a below-par bond

A five-year bond has a face value of $\$1{,}000$ and an annual coupon rate of $6\%$, so it pays $\$60$ each year. If comparable bonds yield $8\%$, its value is:

The bond sells below face value because its $6\%$ coupon is less attractive than the $8\%$ return available on comparable bonds.

Bond prices and market yields move in opposite directions: when required yields rise, the present value of fixed payments falls; when yields fall, it rises. A bond sells at par when its coupon rate equals its required yield, above par when its coupon rate is higher, and below par when its coupon rate is lower.

Stock valuation with dividends

Unlike a typical bond, common stock has no fixed maturity or promised payment, and its future cash flows are uncertain. One approach is the dividend discount model (DDM), which values a share as the present value of its expected future dividends.

If dividends are expected to grow at a constant rate forever, the Gordon growth model is:

where $D_1$ is next year’s expected dividend per share, $r$ is the required return on equity, and $g$ is the constant dividend growth rate. The model requires $r>g$, and its constant-growth assumption is most suitable for a mature company with a reasonably stable growth outlook.

Example: constant dividend growth

If a company is expected to pay a dividend of $\$2.10$ next year, dividends are expected to grow at $4\%$ indefinitely, and investors require a $10\%$ return, then:

When dividends grow unevenly, discount each expected dividend separately and add the present value of a later estimated selling price. A stock that does not pay dividends can be analyzed using other equity cash-flow approaches, such as discounting cash flows available to shareholders. In any approach, the cash flows and discount rate must correspond to the claim being valued.

Model values and market prices

A model’s estimated value is not the same as an observable market price. Market prices reflect buyers’ and sellers’ combined expectations about future cash flows, growth, and risk.

An analyst can compare an estimated value with the market price, but the estimate depends on assumptions. Changing expected cash flows, growth, or the required return changes the result: a higher required return lowers present value, while higher expected cash flows generally raise it. Valuation is a reasoned estimate, not a guaranteed prediction of the price at which an asset will trade.