The Encyclopedia of Economic Principals

Chapter 44

Asset Pricing Models, Factors, and Empirical Puzzles

Test a return against the risk that should explain it.

Four of the chapter's worked examples, made interactive: the equity premium a consumption model can explain, prices that swing around a constant value, a losing gambler chasing break-even, and a size-value alpha after risk and costs. Change one value at a time and watch the figure, the numbers and the hand calculation respond.

Every example here is a constructed teaching example: it uses the hypothetical numbers of the chapter's worked examples, plus a few values added for comparison and labelled as such in each panel. Nothing here measures a real market, firm or household.

Demonstration 1 of 4

How big a premium does consumption risk justify?

With smooth consumption, how much equity premium does the consumption model explain?

The premium comes from the covariance between the discount factor and equity returns. With consumption growth of only 2 percent between states, m barely moves unless gamma is very large, and a very large gamma also pushes the model safe return far above 1.02.

Equation, written in LaTeX: m_{t+1}=\beta(\frac{C_{t+1}}{C_t})^{-\gamma}.

Equation, written in LaTeX: R^f=\frac{1}{\mathbb{E}[m]}\approx1.01980,

Equation, written in LaTeX: \mathbb{E}[mR^e]\approx1.04476

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m is the stochastic discount factor with beta = 1 and risk aversion gamma. Consumption grows by the chosen rate in the good state and by 0 in the bad state, with probability one half each. Equity returns 1.30 or 0.84 (mean 1.07) and the stipulated safe return is 1.02, a 5 point premium. The state return gap 1.30 minus 0.84 = 0.46 is a gross return gap; the model premium is in percentage points.

Predict first. At 2 percent good-state growth, roughly what gamma closes the 5 point gap?

Your prediction

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Figure: How big a premium does consumption risk justify? Left: bars for the model premium, 0.46 points, and the stipulated premium, 5 points, with risk aversion 2 and good-state growth 2 percent. Right: the discount factor is 0.96117 in the good state and 1 in the bad state.
Risk aversion gamma: 2, Good-state consumption growth: 2%
Constructed example: the chapter's hypothetical two-state calibration (growth 2 and 0 percent, returns 1.30 and 0.84, gamma 2); gamma 5, 10 and 25 and growth 1 and 4 percent are added.

Calculated values

m in good state
0.96117
Model safe return R^f
1.01980
Cov(m, R^e)
-0.004466
Model premium (points)
0.46
E[m R^e]
1.04476
Unexplained (points)
4.54

m in the good state = (1 + 0.02)^(-2) = 0.96117 and 1 in the bad state, so E[m] = 0.98058 and R^f = 1 / 0.98058 = 1.01980. E[m R^e] = 0.5 x 0.96117 x 1.30 + 0.5 x 1 x 0.84 = 1.04476, and Cov(m, R^e) = 1.044760 - 0.980584 x 1.07 = -0.004466. The model premium is 0.004466 / 0.980584 = 0.004554, or 0.46 percentage points, so 4.54 points of the stipulated premium remain unexplained. Keeping the state return gap of 0.46 (1.30 minus 0.84, a gross return gap, not the premium), the model-consistent returns are 1.25435 and 0.79435.

Worked steps

  1. m good = (1 + 0.02)^(-2) = 0.96117; m bad = 1
  2. E[m] = 0.5 x 0.96117 + 0.5 x 1 = 0.98058; R^f = 1 / 0.98058 = 1.01980
  3. E[m R^e] = 0.5 x 0.96117 x 1.30 + 0.5 x 0.84 = 1.04476
  4. Cov(m, R^e) = 1.044760 - 0.980584 x 1.07 = -0.004466
  5. Premium = 0.004466 / 0.980584 = 0.004554, 0.46 points
  6. Stipulated 5 - model 0.46 = 4.54 points

Use the idea

Before accepting a risk explanation for a premium, check that the risk aversion it needs is plausible and that it does not imply an implausible safe rate.

Where the conclusion applies

Two equally likely states, power utility with beta = 1, and stipulated returns. The calibration is hypothetical, not an estimate.

Check your understanding: At gamma 2, what is the model-consistent mean equity return holding the 0.46 state return gap?
Shift both returns by x so that E[m(R + x)] = 1: x = (1 - 1.04476) / 0.98058 = -0.04565, giving 1.25435 and 0.79435 with mean 1.02435, which is 0.4554 point above 1.01980.

Chapter 44 source: section "Equity-premium puzzle".

Demonstration 2 of 4

Prices that move more than value

Can a price that swings around a constant value be rational?

With constant dividends and a constant rate, value never moves, so any price variance is excess. Letting the discount rate vary can rationalize the same prices, which is why the variance test is conditional on the discount-rate assumption.

Equation, written in LaTeX: P^*=\frac{5}{0.05}=100\text{ dollars}.

Equation, written in LaTeX: \frac{(70-100)^2+(130-100)^2}{2}=900.

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A share pays a certain 5 dollar dividend forever. P* is its value at a 5 percent discount rate. Two equally likely sentiment states quote prices 100 minus and 100 plus the swing.

Predict first. With a swing of 15, what is the price variance?

Your prediction

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Figure: Prices that move more than value. Two price bars, $70 and $130, against a flat line at P* = $100. The discount rate is held at 5 percent.
Sentiment swing, plus or minus ($): $30, Discount rate: Constant 5%
Constructed example: the chapter's hypothetical 5 dollar perpetuity at 5 percent with prices 70 and 130; swings of 0, 15 and 45 dollars are added.

Calculated values

P* = 5 / 0.05
$100.00
Price variance
900.00
Variance of P*
0.00
Required return, state 1
7.14%
Required return, state 2
3.85%
Prices consistent with the model
no

P* = 5 / 0.05 = $100 in both states. Price variance = ((70 - 100)^2 + (130 - 100)^2) / 2 = 900.00, against zero for P*. With a constant 5 percent rate a known 5 dollar stream cannot be worth both $70 and $130, so the prices violate the benchmark.

Worked steps

  1. P* = 5 / 0.05 = 100.00
  2. Price variance = ((70 - 100)^2 + (130 - 100)^2) / 2 = 900.00
  3. Required return in state 1 = 5 / 70 = 7.14%
  4. Required return in state 2 = 5 / 130 = 3.85%

Use the idea

When prices swing more than cash flows, ask what discount-rate movement would be needed and whether that movement is believable.

Where the conclusion applies

A certain perpetual dividend, two equally likely states and no growth. The example is hypothetical.

Check your understanding: What required return makes a 5 dollar perpetuity worth 85 dollars?
5 / 85 = 5.88%.

Chapter 44 source: section "Excess-volatility puzzle".

Demonstration 3 of 4

Chasing break-even

Why does a losing gambler take a bet they would refuse at the start?

In the loss region the value function is convex, so a spread of outcomes can beat a sure loss. A win that erases the deficit is especially attractive, even when the wager's expected value is negative.

Equation, written in LaTeX: 0.5\sqrt{60}-0.5(2\sqrt{70})\approx-4.49,

Equation, written in LaTeX: v(-60)=-2\sqrt{60}\approx-15.49.

Equation, written in LaTeX: 0.5v(0)+0.5v(-130)\approx-11.40.

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L is the loss already taken, measured from a zero reference point. The wager wins the chosen amount or loses 70 dollars with probability one half each. The value function is v(z) = sqrt(z) for gains and -2 sqrt(-z) for losses.

Predict first. At a prior loss of 30 dollars, does the gamble beat stopping?

Your prediction

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Figure: Chasing break-even. Left: the S-shaped value function with points for stopping at -60, winning to 0 and losing to -130. Right: bars for the stop value -15.49 and the gamble score -11.40.
Prior loss ($): -$60, Win amount ($): $60
Constructed example: the chapter's hypothetical wager (+60 or -70) at prior losses 0 and -60; prior losses of -30 and -90 and wins of 40 and 80 are added.

Calculated values

Stop value v(L)
-15.49
Gamble score
-11.40
Choice
gamble
Expected value of the wager
-$5.00
Expected final position, gamble
-$65.00
Final position, stop
-$60.00

Stopping is worth v(-60) = -15.49. The gamble ends at 0 or -130, so its score is 0.5 x 0.000 + 0.5 x (-22.804) = -11.40. The gamble scores higher, so it is taken. The wager's expected value is 0.5 x 60 + 0.5 x (-70) = -5.00, so the expected final position is -65.00 against -60 from stopping.

Worked steps

  1. v(-60) = -2 sqrt(60) = -15.49
  2. v(0) = sqrt(0) = 0.000
  3. v(-130) = -2 sqrt(130) = -22.804
  4. Gamble score = 0.5 x 0.000 + 0.5 x (-22.804) = -11.40
  5. Expected final position = -60 + -5.00 = -65.00

Use the idea

After a loss, compare a new bet on its own terms, not on whether it could get you back to even.

Where the conclusion applies

The illustrative value function with loss weight 2, outcomes integrated with the prior loss and a fixed zero reference point. All numbers are hypothetical.

Check your understanding: At L = -60 and a win of 80, what is the gamble score?
0.5 sqrt(20) + 0.5 x (-2 sqrt(130)) = 2.236 - 11.402 = -9.17, against -15.49 from stopping, so gamble.

Chapter 44 source: section "Break-even effect".

Demonstration 4 of 4

Size-value alpha after costs

Does a small-value return spread survive risk adjustment and trading costs?

Alpha is the return left after the CAPM charge for market risk. A higher beta raises the charge, and implementation costs come off what remains, so a gross spread need not reach the investor.

Equation, written in LaTeX: R_f+\beta(R_M-R_f)=2\%+1.0(6\%)=8\%

Equation, written in LaTeX: 11\%-8\%=3\%.

Equation, written in LaTeX: 3\%-4\%=-1\%.

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R_f = 2% is the safe return and R_M - R_f = 6% the market premium. Portfolio S (small value) earns 11% and portfolio B (large growth) 8%. Beta is S's market beta and costs are S's extra implementation cost over the period.

Predict first. If S's beta were 1.5, is its gross alpha still positive?

Your prediction

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Figure: Size-value alpha after costs. Bars for the CAPM prediction 8.0%, gross alpha 3.0%, costs of 4.0% shown below zero, and net alpha -1.0%, with beta 1.0.
Extra implementation cost of S: 4%, Beta of S: 1
Constructed example: the chapter's hypothetical portfolios (R_f 2%, premium 6%, beta 1.0, S 11%, B 8%, costs 4%); costs of 0 and 2% and betas of 0.8, 1.2 and 1.5 are added.

Calculated values

CAPM required return
8.0%
S return
11.0%
S minus B spread
3.0%
Gross alpha
3.0%
Costs
4.0%
Net alpha
-1.0%

CAPM requires 2% + 1.0 x 6% = 8.0%. Portfolio S earns 11%, so gross alpha is 11% - 8.0% = 3.0%, which is positive. Net of 4.0% in costs, alpha is 3.0% - 4.0% = -1.0%. After costs the implementer loses relative to CAPM. The raw S minus B spread stays 11% - 8% = 3%.

Worked steps

  1. CAPM = 2 + 1.0 x 6 = 8.0%
  2. Gross alpha = 11 - 8.0 = 3.0%
  3. Net alpha = 3.0 - 4.0 = -1.0%

Use the idea

Before chasing a factor premium, subtract the risk charge and your own trading costs.

Where the conclusion applies

One hypothetical period, CAPM as the benchmark and costs that do not change with beta.

Check your understanding: With beta 1.2 and costs of 2%, what is net alpha?
CAPM = 2 + 1.2 x 6 = 9.2%; alpha = 11 - 9.2 = 1.8%; net = 1.8 - 2 = -0.2%.

Chapter 44 source: section "Size and value effects".