Finance-Theory · Unit 9 · Video 5 · Interactive Practice

A Bid for One Dividend: How an Auction Reveals the Risk-Adjusted Discount Rate

IKey Formulas

Formula	Name	What it needs
$P_t = \displaystyle\sum_{k=1}^{\infty} \frac{E_t[D_{t+k}]}{(1+r_{t+k})^k}$	A share as a portfolio of one-dividend companies	Each dividend is its own company, discounted at a rate $r_{t+k}$ matched to that one cash flow's risk
$E[D] = \tfrac{1}{2}(54) + \tfrac{1}{2}(0) = 27$	Expected liquidating dividend (millions of dollars)	Every outcome and its probability; no single outcome pays the average
$15 = \dfrac{27}{(1+r)^5} \;\Rightarrow\; (1+r)^5 = 1.8$	The bid sets the price	The highest bid ($15 million), the expected payoff ($27 million), about 5 years
$r = \left(\dfrac{E[D]}{P}\right)^{1/T} - 1 = 1.8^{1/5} - 1 \approx 12.5\%$	Implied risk-adjusted rate	Price and expected payoff only; no riskless rate or credit spread needed

Key Insight: The bidders do the risk adjustment. The auction produces the paper's yield in the same way bidding produced the Treasury-bill yield on 17 September 2008, when the three-month bill fell to about three basis points (0.03%); the difference is that the paper's payoff is uncertain, so its yield is risk-adjusted.

IIThe Bid Reveals the Rate

Price and expected payoff pin down $r$ : the rate at which the bid grows into the expected $27 million.

Highest bid ($ millions) 15 Years to the payoff (T)

IIIA Fair Price Still Pays the Fair Rate

Paying $15 million for an expected $27 million earns about 12.5% a year on average, though any single paper pays 54 or nothing.

Papers bought (n) 1

IVQuiz Questions

Problem 1 · Back Out the Rate

Given: A one-payment company will pay a single liquidating dividend in 3 years, with an expected value of $32 million. At auction the highest bid is $20 million. Find the implied risk-adjusted annual discount rate.

✅ Correct!

(1+r)^3 = 32/20 = 1.6

, so

1 + r = 1.6^{1/3} \approx 1.1696

and

r \approx 17.0\%

a year.

❌ That ignores compounding. Splitting the 60% total gain evenly over 3 years gives 20%, but

1.2^3 = 1.728

, not

1.6

. Take the cube root of

1.6

instead.

❌ That is the total gain, not a rate per year.

32/20 - 1 = 60\%

is earned over 3 years; the annual rate solves

(1+r)^3 = 1.6

❌ That uses the video's 5 years.

1.6^{1/5} - 1 \approx 9.9\%

; this payment arrives in 3 years, so take the cube root.

Show solution

The bid is the price, and the price is the expected payoff discounted for 3 years (amounts in millions of dollars):

20 = \frac{32}{(1+r)^3} \quad\Rightarrow\quad (1+r)^3 = \frac{32}{20} = 1.6

1 + r = 1.6^{1/3} \approx 1.1696 \quad\Rightarrow\quad r \approx 17.0\%

Check: $20 \times 1.1696^3 \approx 32.0$ . The bidder expects to earn about 17% a year for holding this risk.

Problem 2 · The Numerator Is an Expectation

Given: A venture pays one liquidating dividend in 5 years: $60 million with probability 0.4, or nothing with probability 0.6. The highest bid today is $12 million. Find the implied risk-adjusted annual rate.

✅ Correct!

E[D] = 0.4 \times 60 = 24

, so

(1+r)^5 = 24/12 = 2

and

r = 2^{1/5} - 1 \approx 14.9\%

❌ That discounts the success payoff. The $60 million arrives only 40% of the time. The numerator is the expected dividend, which already carries the chance of a zero:

0.4 \times 60 + 0.6 \times 0 = 24

❌ The probabilities are swapped.

0.6 \times 60 = 36

attaches the failure probability to the success payoff. The venture succeeds with probability 0.4, so

E[D] = 24

❌ That ignores compounding. Doubling over 5 years is not 20% a year:

1.2^5 \approx 2.49

. Solve

(1+r)^5 = 2

Show solution

Numerator first (millions of dollars):

E[D] = 0.4 \times 60 + 0.6 \times 0 = 24

Then the bid sets the price:

12 = \frac{24}{(1+r)^5} \;\Rightarrow\; (1+r)^5 = 2 \;\Rightarrow\; 1 + r = 2^{1/5} \approx 1.1487 \;\Rightarrow\; r \approx 14.9\%

No single outcome pays $24 million; it is the average value of one payment, and the rate is measured against that average.

Problem 3 · A Share as Two One-Dividend Companies

Given: A share pays an expected dividend of $2 in one year and an expected liquidating dividend of $50 in two years, then nothing. Treat each payment as its own company: the market discounts the near, safer payment at 6% a year and the distant, riskier one at 10% a year. Find the share price.

✅ Correct!

\frac{2}{1.06} + \frac{50}{1.10^2} \approx 1.89 + 41.32 = 43.21

: the share is the sum of the prices of its one-payment companies, each at its own rate.

❌ That uses one rate for both.

\frac{2}{1.10} + \frac{50}{1.10^2} \approx 43.14

discounts the near, safer dividend at the distant one's 10%. Each company gets the rate matched to its own cash flow: 6% for the first.

❌ Check the exponent. The $50 arrives in two years, so it is divided by

1.10^2 = 1.21

, not by

1.10

❌ That adds the dividends without discounting. Each payment must be discounted for time and risk before the pieces are summed.

Show solution

Company 1 lives for one payment in one year; company 2 for one payment in two years:

P = \frac{2}{1.06} + \frac{50}{(1.10)^2} \approx 1.8868 + 41.3223 \approx 43.21

The share price is about $43.21. Discounting both at 10% would give about $43.14; the gap is the near, safe dividend's lower rate.

Problem 4 · Why Buy a Fairly Priced Paper?

Given: The video's paper: the winner pays $15 million today, and in 5 years the venture pays $54 million with probability $\tfrac{1}{2}$ or nothing. The bid is the fair price.

If the venture succeeds, what annual rate does the winner earn?

Which statement is correct?

✅ Correct! Success grows $15 million into $54 million, about 29.2% a year; failure loses everything. Priced fairly, the paper's expected payoff of $27 million is what delivers the fair 12.5%.

❌ That is the expected rate, not the success rate. 12.5% comes from the expected payoff of 27; in the success state the payoff is 54, so solve

(1+r)^5 = 54/15 = 3.6

❌ That ignores compounding. Spreading the 260% total gain evenly over 5 years gives 52%, but

1.52^5 \approx 8.1

, not

3.6

. Take the fifth root of

3.6

instead.

❌ That is the total gain over 5 years, not a rate per year.

54/15 - 1 = 260\%

; the annual rate solves

(1+r)^5 = 3.6

❌ Check the success rate. In the success state the payoff is 54, not the expected 27: solve

(1+r)^5 = 54/15 = 3.6

. Dividing the 260% total gain by 5 years (52%) ignores compounding.

❌ Check the statement. A fair price means the expected return equals the fair rate for the risk, not zero; the rate is an expectation, not a guarantee; and a belief in mispricing is an optional extra reason to buy, not a requirement.

Show solution

Success state (millions of dollars):

15(1+r)^5 = 54 \;\Rightarrow\; (1+r)^5 = 3.6 \;\Rightarrow\; r = 3.6^{1/5} - 1 \approx 29.2\%

Failure state: the payoff is 0, a loss of the whole $15 million.

On average: $E[D] = 27$ and $15(1+r)^5 = 27$ gives $r = 1.8^{1/5} - 1 \approx 12.5\%$ . That is the fair rate of return for holding this risk. A fairly priced security still earns it in expectation; you buy because you want this risk and reward, and believing the price is wrong is only an additional motive.

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