Finance-Theory · Unit 9 · Video 4 · Interactive Practice

Why the Market, Not the Company, Sets the Discount Rate on Every Dividend

IKey Formulas

Formula	Name	What it needs
$P_t = \displaystyle\sum_{k=1}^{\infty} \frac{\mathsf{E}_t[D_{t+k}]}{(1+r_{t+k})^k}$	Dividend discount model	One expected dividend and one rate per term: each denominator carries its own subscript
$r_{t+1} = r_f$ if $D_{t+1}$ is risk-free; $\ r_{t+k} > r_f$ for risky dividends	Risk-adjusted rate	A rate commensurate with the risk of the dividend above it; the riskier of two dividends gets the higher rate
$P_t = \displaystyle\sum_{k} \big(\text{market price of } D_{t+k} \text{ bought alone}\big)$	Replicating portfolio	What the market charges for a cash flow of that risk and that date: bond pricing plus one dimension, risk
$(1+r_{t+k})^k = (1+\bar r_k)^{\tau_k}$	Uneven dates (our formalisation)	Dividend $k$ due $\tau_k$ years out, priced at a per-year rate $\bar r_k$ : the horizon difference is absorbed into $r_{t+k}$

Key Insight: The company chooses the numerators, its dividends, subject to its ability to pay them. The market chooses every denominator: $r_{t+k}$ is the market's price for bearing that dividend's risk at that horizon, not the company's bond yield, a stated policy or management's say-so.

IIEach Dividend at Its Own Rate

A share is a bundle of dividends; each is priced at the market's rate for its risk, and the prices add.

Rates set by

💡 A bond yield generally misprices the stock because the stock is the residual claim, paid only after the bondholders; how leverage links the two belongs to capital structure, later in the course.

IIISame Dividend, Higher Risk, Lower Value

For the same expected dividend, a higher risk-adjusted rate means a smaller present value, and more so at long horizons.

Horizon (k years) 5 Discount rate (r) 7.3%

💡 The three rates are Lo's illustrative numbers, not estimates for any real stock; choosing among them needs a measure of risk, which comes later in the course.

IVUneven Dates, Absorbed into the Rate

When dividend dates are unevenly spaced, the formula keeps exponent $k$ and the horizon difference moves into $r_{t+k}$ .

Date of the second dividend (years) 1.75 Per-year rate for its risk

💡 The per-year rate $\bar r$ and this conversion are our formalisation; the video states only that the difference in horizon is absorbed into the size of each rate.

VQuiz Questions

Problem 1 · Two Dividends, Two Rates

Given: A claim on the next two dividends of a stock only. The first, expected $2 in one year, is genuinely risk-free, and the risk-free rate is 2.5%. The second, expected $2 in two years, is risky, and the market prices cash flows of its risk at 6.5% per year. Find the price of the claim today.

✅ Correct! Each dividend sits over its own rate: the risk-free one at

r_f

, the risky one at its risk-adjusted rate.

❌ Not quite. That discounts both dividends at the 2.5% risk-free rate. Only the first dividend is risk-free; the second must be discounted at the higher, risk-adjusted 6.5%.

❌ Close, but check the exponent. The second dividend is two years away, so its denominator is

(1.065)^2

, not

1.065

❌ Not quite. That discounts both dividends at 6.5%. The first dividend is risk-free, so it gets the risk-free rate of 2.5%, not the risky dividend's rate.

❌ Not quite. Price each dividend separately at its own rate, then add.

Show solution

Each term of the dividend discount model carries its own rate:

P_t = \frac{\mathsf{E}_t[D_{t+1}]}{1+r_{t+1}} + \frac{\mathsf{E}_t[D_{t+2}]}{(1+r_{t+2})^2} = \frac{2}{1.025} + \frac{2}{(1.065)^2}

The first term is about $1.95 and the second about $1.76 (since $1.065^2 = 1.134225$ ), so the claim is worth about $1.95 + $1.76 = $3.71.

Using a single rate for both terms, 2.5% or 6.5%, gives $3.85 or $3.64: the first overprices the risky dividend, the second underprices the risk-free one.

Problem 2 · The Bond-Yield Shortcut

Given: A company's bonds yield 5% per year. The market prices cash flows as risky as the company's dividends at 8.9% per year. A claim pays the single dividend expected in two years, $4. Find the price of the claim.

✅ Correct! The rate is the market's price for the dividend's risk, 8.9%, not the company's bond yield.

❌ That is the tempting shortcut. Discounting at the 5% bond yield works only if the bond and the stock carry comparable risk, and they generally do not: the stock is the residual claim, paid after the bondholders.

❌ Close, but check the exponent. The dividend is two years away, so the denominator is

(1.089)^2

❌ Not quite. That divides by

1 + 2 \times 0.089 = 1.178

, which is simple interest. Discounting compounds: the denominator is

(1.089)^2 = 1.185921

❌ Not quite. Discount the $4 for two years at the market's rate for its risk.

Show solution

The rate in the denominator is set by the market for cash flows of this risk:

\frac{\mathsf{E}_t[D_{t+2}]}{(1+r_{t+2})^2} = \frac{4}{(1.089)^2} = \frac{4}{1.185921} \approx 3.37

The claim is worth about $3.37. The bond-yield shortcut gives $4/(1.05)^2 = 4/1.1025$ , about $3.63, overstating the price by about $0.26, because it uses a rate that belongs to a safer claim.

Problem 3 · An Unevenly Spaced Dividend

Given: The second term of the model, $\mathsf{E}_t[D_{t+2}]/(1+r_{t+2})^2$ , belongs to a dividend due only 1.5 years from today. The market requires 7.3% per year for cash flows of its risk (our per-year framing). Find the rate $r_{t+2}$ that belongs in that denominator, keeping the exponent 2.

✅ Correct! The shorter horizon is absorbed into a smaller

r_{t+2}

, so the term still discounts the dividend for exactly 1.5 years.

❌ Close, but that scales the rate linearly.

7.3\% \times 0.75 = 5.475\%

ignores compounding. Match the discount factors:

(1+r_{t+2})^2 = 1.073^{1.5}

❌ Not quite. Using 7.3% with exponent 2 discounts the dividend for two full years, but it arrives after 1.5. The difference in horizon has to be absorbed into the rate.

❌ Not quite.

1.073^{1.5} - 1 \approx 11.15\%

is the total return over 1.5 years, not a rate for one period of the formula. The formula still squares

1 + r_{t+2}

, so take the square root first.

❌ Not quite. Set

(1+r_{t+2})^2

equal to the true discount factor for 1.5 years.

Show solution

The dividend's true discount factor is $1.073^{1.5} \approx 1.1115$ . The formula writes it as $(1+r_{t+2})^2$ , so

(1+r_{t+2})^2 = 1.073^{1.5} \quad\Longrightarrow\quad r_{t+2} = 1.073^{0.75} - 1 \approx 0.0543

So $r_{t+2} \approx 5.43\%$ . Check: $1.0543^2 \approx 1.1115$ . A gap shorter than a year before this dividend shows up as a rate below the per-year 7.3%.

Problem 4 · Who Sets the Rate?

Given: A CFO announces: "Our bonds yield 5%, so from now on the company values its dividends at 5%." The following month the market comes to regard the company's dividends as riskier than before. In the frictionless theory, what happens to the rate $r_{t+k}$ in each dividend's denominator?

✅ Correct! Like the economist's salary, the rate is set by the marketplace, not by the institution: the company only chooses the dividends.

❌ Not quite. The discount rate is not set by fiat or by announcing a corporate policy. A riskier dividend has to promise a higher return to attract capital.

❌ Not quite. Horizon is one reason each term has its own rate; risk is a second. More risk at the same horizon means a higher rate.

❌ Not quite. The numerator is the expected dividend; its risk is priced in the denominator, and more risk raises the rate.

❌ Not quite. Ask what fair rate of return the market demands for a cash flow of this risk.

Show solution

In the frictionless theory each $r_{t+k}$ is the market's price for bearing that dividend's risk at that horizon. Neither the company's bond yield nor its announced policy enters: the bond is a different, safer claim, and a policy is fiat.

When the market judges the dividends riskier, investors demand a higher return to hold them, so every affected $r_{t+k}$ rises. For the same expected dividends, each present value, and so $P_t$ , falls.

Solved: 0 / 4