Finance-Theory · Unit 23 · Video 1 · Interactive Practice

Adaptive Markets: Why the Risk Premium Changes Over Time

IKey Formulas

Formula	Name	What it says
$\mathsf{E}[R_i] = R_f + \beta_i\left(\mathsf{E}[R_m] - R_f\right)$	Security market line (CAPM)	Expected return rises in a straight line with beta: $\beta_i = 0$ gives $R_f$ , $\beta_i = 1$ gives $\mathsf{E}[R_m]$
$\mathsf{E}[R_m] - R_f$	Market risk premium	The slope of the line: the extra expected return risky assets as a whole offer over the risk-free rate. Under adaptive markets it changes over time and across circumstances
$\Delta \mathsf{E}[R_i] = \beta_i \, \Delta\left(\mathsf{E}[R_m] - R_f\right)$	Shift when only the premium moves	With $R_f$ unchanged, each security's expected return moves by its beta times the change in the premium

Key Insight: The CAPM treats the security market line as one fixed relation. Experience changes preferences, as it did for the woman who lived through the Depression and still took the bus, so the line's slope moves, and in general the line need not even be straight.

IIThe Security Market Line Moves

The line always starts at $R_f$ ; its slope, the market risk premium, is what experience changes.

💡 The straight line is itself a CAPM assumption that rests on stable, rational preferences; the video adds that once preferences shift, the relation between risk and reward need not even be linear.

IIIPasture and Sheep: Why Strategies Wax and Wane

Profits draw investors in, investors use the profits up and leave, and then the profits grow back.

Time start

IVThe Sheep Left: Private-Label Mortgage-Backed Securities

How much of the 2006 peak in private-label issuance was left in 2008, and how much had come back by 2025?

VQuiz Questions

Problem 1 · Reading the Calm-Era Line

Given: In an illustrative calm era (our numbers) the risk-free rate is $R_f = 3\%$ and the market risk premium is $\mathsf{E}[R_m] - R_f = 4\%$ . Find the CAPM expected return on a stock with $\beta = 1.6$ .

✅ Correct! The line starts at

3\%

and rises

4\%

per unit of beta, so a beta of

1.6

adds

6.4\%

on top of

R_f

❌ Not quite. That is only the risk term

\beta_i(\mathsf{E}[R_m] - R_f) = 1.6 \times 4\%

. The security market line starts at

R_f

, so add the

3\%

❌ Not quite. That multiplies beta by the market's whole expected return,

1.6 \times 7\%

. Beta multiplies only the premium,

\mathsf{E}[R_m] - R_f

; the risk-free part is earned once, not scaled by beta.

❌ Not quite. That is the market's own expected return,

3\% + 4\%

, which is what a stock with

\beta = 1

earns. A beta of

1.6

sits further up the line.

Show solution

Substitute into the security market line:

\mathsf{E}[R_i] = R_f + \beta_i\left(\mathsf{E}[R_m] - R_f\right) = 3\% + 1.6 \times 4\% = 3\% + 6.4\% = 9.4\%

For comparison, the market itself ( $\beta = 1$ ) earns $7\%$ ; a stock with more market risk than the market earns more than that.

Problem 2 · Keeping the Calm-Era Line After a Crisis

Given: In the same illustrative market, a crisis leaves investors demanding a market risk premium of $9\%$ instead of $4\%$ , while $R_f$ stays at $3\%$ . An analyst treats the security market line as fixed and keeps using the calm-era line for a stock with $\beta = 0.7$ . Find by how many percentage points that understates the stock's expected return.

✅ Correct! The line pivots on

R_f

, so the stock moves by its beta times the change in the premium:

0.7 \times 5 = 3.5

points.

❌ Not quite.

5

points is how far the premium moved, which is how far a

\beta = 1

stock moves. A stock with

\beta = 0.7

moves only

0.7

times as far.

❌ Not quite.

0.7 \times 9\% = 6.3\%

is the stock's whole risk term after the crisis. The analyst already counts

0.7 \times 4\%

of it on the calm-era line; the understatement is only the difference.

❌ Not quite.

9.3\%

is the stock's expected return on the post-crisis line, a level rather than a gap. Subtract the calm-era value the analyst is using.

Show solution

Calm-era line: $\mathsf{E}[R_i] = 3\% + 0.7 \times 4\% = 5.8\%$ .

Post-crisis line: $\mathsf{E}[R_i] = 3\% + 0.7 \times 9\% = 9.3\%$ .

9.3\% - 5.8\% = 3.5 \text{ percentage points} = 0.7 \times (9\% - 4\%)

Because $R_f$ does not move, the shift is $\beta_i$ times the change in the premium. A security market line treated as a constant, like gravity, misprices every risky asset once the premium has moved.

Problem 3 · Reading the Premium Off One Stock

Given: After a crisis, in an illustrative market (our numbers), $R_f = 3\%$ and a stock with $\beta = 0.7$ has a CAPM expected return of $9.3\%$ . Find the CAPM expected return on a stock with $\beta = 1.6$ in the same market.

✅ Correct! The

\beta = 0.7

stock pins down the slope: a

9\%

premium. The

\beta = 1.6

stock then sits at

3\% + 14.4\%

❌ Not quite. That scales the whole

9.3\%

1.6 / 0.7

, as if the line went through the origin. It goes through

R_f = 3\%

: scale only the part above

R_f

❌ Close, but…

14.4\%

is the risk term

1.6 \times 9\%

. Add the risk-free rate the line starts from.

❌ Not quite.

12\%

is the market's expected return,

3\% + 9\%

, which belongs to

\beta = 1

. A stock with

\beta = 1.6

sits above the market on the line.

Show solution

Step 1: the premium from the $\beta = 0.7$ stock.

9.3\% = 3\% + 0.7\left(\mathsf{E}[R_m] - R_f\right) \;\Rightarrow\; \mathsf{E}[R_m] - R_f = \frac{6.3\%}{0.7} = 9\%

Step 2: the $\beta = 1.6$ stock on the same line.

\mathsf{E}[R_i] = 3\% + 1.6 \times 9\% = 3\% + 14.4\% = 17.4\%

Two points fix a line: $R_f$ at $\beta = 0$ and any one priced security give the slope, the market risk premium in force at that moment.

Problem 4 · A Strategy That Stopped Paying

Given: An illustrative trading strategy (our story) earned unusually large profits for several years. As more funds adopted it, its profits shrank to almost nothing and many of those funds closed. Years later, with few funds left using it, the strategy is profitable again. Which reading matches the adaptive markets hypothesis?

✅ Correct! This is the sheep and the pasture: investors arrive, use the profits up and leave, and the pasture grows back. Free lunches exist from time to time; what is far harder to find is one that lasts.

❌ Not quite. The profits disappeared for years, so they were not repeatable year after year. A free-lunch program is exactly what the adaptive markets hypothesis says is far harder to come by.

❌ Not quite. If efficiency were permanent, the strategy could not pay again later. Its return shows efficiency changing with who is trading.

❌ Not quite. The profits shrank as more funds adopted the strategy and returned once they left. That is a pattern driven by competition, not a run of luck.

Show solution

Two of the adaptive markets implications describe this story. Limited arbitrage exists from time to time: some group of participants finds a profitable opportunity, and over time other arbitrageurs compete those profits away. Strategies wax and wane: once the profits are gone the investors leave, and with fewer of them the opportunity grows back, as the pasture does when the sheep decline.

What the hypothesis rules out is the steady version, consistent arbitrage year after year (option b), and a permanent switch to efficiency (option c). Option d ignores the link between the number of funds using the strategy and its profits, which is the mechanism the video describes with private-label mortgage-backed securities.

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