4.4.5 The Lucas Asset Pricing Model and the Consumption Capital asset Pricing Model
4.4.6.1 The Equity Premium Puzzle
The Sharpe ratio shows the risk premium return per volatility unit. Hansen and Jagannathan (1991) derived an upper bound for the Sharpe ratio called the HJ-bound. It is given by . This relation is derived as follows.
Let the intertemporal budget constraint be:
∑
4-40
where is the income, is the wealth, is the consumption and is the individual’s savings at time t.
The individual’s maximization problem can then be stated as:
[ ]
∑
Subject to
∑
4-41
43 The Lagrange function is:
[ ( ∑
)] (∑
)
[ ] (∑
) 4-42
The first-order conditions with respect to is:
[ ∑
]
[ ∑
]
∑
[ ] 4-43
The first-order conditions with respect to is:
[ ]
}
[ ]
4-44 Plugging 4-44 into 4-43 we get:
∑
4-45
Substituting for into 4-44 gives:
44 [ ] [ ] 4-46 Let:
(
) (
) (
) (
)
4-47
Then:
[ (
)]
[ ( )]
4-48
So:
[ ( )
] 4-49
Normalizing the price of the asset to be equal to 1 and setting
we have:
[ (
)]
[ ] [ ] [ ] [ ]
[ ] ( [ ] [ ]
[ ] ) 4-50
The relation between the risk free rate and the stochastic discount factor is as follows:
[
] [ ] 4-51
45 From 4-50 and 4-51 we have:
[ ] [ ]
[ ] 4-52
( ) [ ]
[ ] [ ]
[ ] 4-53
Equation 4-53 (Pennacchi 2008 p. 86) shows that the expected return of asset is equal to the risk free rate minus the ratio of covariance between the marginal utility of consumption at time divided by the expected marginal utility of consumption at time . Assets that pay off relatively higher when consumption is low are more attractive than assets that pay off relatively lower when consumption is low. This is based on the assumption that consumers prefer to smooth out consumption. Mehra (2006, pp. 10-11) gives insurance policies as an example of such assets.
Exchanging covariance with correlation we have:
( ) [ ] ( )
[ ]
→
( )
[ ] ( )
[ ]
| ( )
|
[ ] 4-54
46 Consider the basic asset pricing equation:
( )
4-55 where is the stochastic discount factor and is the asset’s pay off.
Dividing both sides by we get:
(
) ( )
( ) ( ) ( ) 4-56 where is the return on asset .
Since
( )
4-57 where is the correlation coefficient.
From 4-56 and 4-57 we have:
( ) ( )
4-58 hence
( ) ( ) ( ) ( )
( )
( )
( )
( )
( ) ( )
( ) 4-59
47 Because we conclude that
( )
( )
( ) ( )
( )
| ( )
( )| ( )
( ) ( )
( )
| ( )
|
( ) 4-60
The above inequality is called the Hansen-Jagannathan (HJ) bound.
Here is a numerical example. Assume that:
( )
( ) ( )
( )
( )
( )
48 Table 4-4: Hansen- Jagannathan bound, numerical example
( ) | ( )
| ( )
-1 0,16667 0,16667
-0,7 0,11667 0,11667
-0,4 0,06667 0,06667
-0,1 0,01667 0,01667
0 0 0
0,2 -0,03333 0,03333
0,5 -0,08333 0,08333
0,8 -0,13333 0,13333
1 -0,16667 0,16667
Figure 4-4: The Hansen – Jagannathan bound
An important property of this bound is that it is independent of the form of the utility function and can be used to test whether a pair of a discount factor and a utility function may serve as reasonable model for analyzing a market. The HJ-bound can be used to evaluate if a specific asset pricing model is a reasonable approximation to market by testing it's predictions against market data (Cochrane 2005, pp. 455-484).
-0,2 -0,15 -0,1 -0,05 0 0,05 0,1 0,15 0,2
-1,5 -1 -0,5 0 0,5 1 1,5
| ( ) |
( )
( )
The HJ-bound
49 Observe that:
( ) ( ) ( ) ( )
4-61
Then:
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) 4-62
and
( ) ( )
4-63
The HJ bound can be written as:
( )
4-64
The following notation is used:
( ) is the expectation of the stochastic discount factor is the standard deviation of the stochastic discount factor
is the expectation of excess return
Mehra and Prescott (1985) researched the return of the US stock market over the period 1889-1994. They found that the equity return is so high compared to the alternative risk free
investment (bonds) that in order to explain it by standard asset pricing models one has to assume extremely high risk aversion or too high stochastic discount rate. The excess returns
50 required by investors cannot be explained by models like CCAPM (Consumption CAPM).
This is called the equity premium puzzle.
The equity premium puzzle can be illustrated by the following example (Pennacchi 2008, pp.
89-90): Let be the return on a well-diversified portfolio of say S & P 500 U.S. stocks.
Using historical data over the past 75 years one finds that a reasonable estimate of this portfolio’s annual excess return over the risk free interest rate is 8,3 %:
[ ]
4-65 The portfolio’s annual standard deviation for the same period is estimated to 17 %.
Then the Sharpe ratio is:
[ ]
4-66
Assuming a power utility function of the form can we calculate the HJ-bound from 4-64 as follows:
( ) [ ]
√ [ ( )]
[ ( )]
√ [ ( )] [ ( )]
[ ( )]
√
[ ( )] [ ( )]
√ √
⏞
√ √ where is an approximation of e for small x.
4-67
51 For a power utility function of the form is
| [ ]
| 4-68
Assuming a broadly diversified portfolio, for example the stock index, we can write:
[ ]
[ ] 4-69
So for a broadly diversified portfolio the equity premium is proportional to , and
. Equation
4-69 can be rewritten as:
[ ]
[ ]
4-70
The annual standard deviation of consumption growth in USA for 1933-2008 (Pennacchi 2008, p. 89) has been estimated to be between 0,01 and 0,0386 and the equity premium
[ ] , where is the S&P 500. Then the risk aversion is between:
[ ]
4-71
This result doesn’t harmonize with the expected range for risk aversion calculated from other sources being between -5 to -1. The volatility of consumption growth is too low compared to the premium demanded by investors for holding stocks.
52 Deriving an equation for the volatility of the stochastic discount factor
4.4.6.1.1
Let the regression equation for estimating be (Cochrane 2005, p.94):
4-72
where and are vectors
( )
→
[ ]
→ ∑
[ ] or [ ]
4-73
Taking variance from both sides of 4-72:
[ ] [ ] [( ) ]
[ ] [ ] → [ ] [ ]
[ ( )]
√
} √[ ( )]
| ( )
|
4-74
Equation 4-74 is useful for constructing the HJ-bound.
53 4.4.6.2 The Risk-Free Rate Puzzle
From Lucas model we derived in 4-67 that:
( )
[ ] 4-75
Here we follow the presentation by Pennacchi (2008, pp.80-90). Starting with the Lucas tree asset pricing model (
4-36) we get the following relation:
[
]
}
[ ] 4-76
Equation 4-76 shows the relationship between the stochastic discount factor m and the risk free discount factor .
For a risk free asset equation 4-76 becomes:
[ ] [ ] 4-77
where is the stochastic discount factor, is the subjective discount factor and C is the consumption.
Let the utility function be (Pennacchi 2008, p. 83):
, 4-78
where is the risk aversion equal to (Mehra 2006, p. 13).
54 Table 4-5: The relation between and
Relation between and
So ,
Then:
( ) 4-79
This can be rewritten as:
( ) 4-80
where ( ) is the logarithmic growth in consumption.
Let be lognormally distributed, then:
( ) and 4-81
The probability density function of is:
( )
√ ( )
[ ( ) ]
4-82
So:
55 [ ( )]
4-83
By the same token:
( )
4-84
and:
[ ( )
]
4-85
Plugging the above equation to [ ] we get:
[ ] [ ( )
] [ [ ( )
]
]
( )
( )
( ) 4-86
Assuming 0,01 (1%) of time preference is . The historical annual growth rate of consumption is . Plugging in and we get:
( ) ( )
So is calculated in this setting to be 14,2%. This is too high a number and doesn’t conform to the empirical data since the risk free interest rate for the same period averaged 1% in the United States.
56 Figure 4-9: The risk free rate as a function of the relative risk aversion .
The equation used in the diagram is derived as follows:
The net risk free rate is given by:
( )
57 Then:
( )
4-90
For high values of , for instance , the risk free rate is “reasonable” at 2,39%. High values of risk aversion imply high changes of the term for small changes in the expected consumption growth . This would imply high changes in the risk free rate, which doesn’t agree with empirical data (Mehra 2006, p. 21).
( )
Table 4-6: Higher values of and changes in
50 0,018 0,9
50 0,019 0,95 0,05
Table 4-7: Lower values of and changes in
10 0,018 0,18
10 0,019 0,19 0,01
We conclude that high risk aversion leads to unreasonably high risk free rate. This is called the risk-free rate puzzle.
58 4.4.6.3 The Correlation Puzzle
The HJ-bound is set under the assumption that the absolute value of the correlation between the consumption growth for non-durables and services and stock returns is equal to one. Tests with US data show that the empirical risk free rate is 0,01, the empirical Sharpe ratio is
, the empirical volatility of consumption growth 0,01 and the empirical correlation is 0,2.
That implies:
( )
( )
( ) Using ( ) we find that:
( )
Approximating with ( ) we have:
which is a huge risk aversion coefficient.
When the correlation between the consumption growth for non-durables and services and stock returns is not perfect, the HJ-bound becomes much more difficult to fulfill. For empirically observed small volatilities in the consumption growth for non-durables and services, the risk aversion coefficient has to be excessively big to satisfy the HJ-bound.
(Cochrane 2005, p. 457)
59 4.4.6.4 The Volatility Puzzle
The stock’s intrinsic value at time t is:
∑
4-91
where is the dividend cash flow of stock at time t, and r is the discount rate.
In finite time:
∑
4-92
where and is the fundamental value of the stock.
Shiller (1981) investigated whether the stock prices' volatility is greater than the dividend cash flows' volatility by testing equation 4-92 in ex-post fashion; with T as the present time and t some time points in the past. Because the test was ex post the cash flows were known.
Shiller used the present asset price as an approximation of . Utilizing the market data, Shiller calculated an estimate for and compared it with the actual asset price at time t, . The prediction of EMH is that is the optimal forecast, i.e. the best predictor, at time t of . A best predictor ̂ of a variable has the following property (Ruppert 2004, pp.443-444):
( ̂) ̂ 4-93
which implies that ̂ and ̂ are uncorrelated.
From 4-93 follows the following inequality:
( ̂) 4-94
Had been an optimal forecast of , should be less volatile than . Contrary to this expectation, the market data shows that the volatility of the asset prices is much greater than the volatility of the asset’s dividend cash flows. This is called the excess volatility puzzle.
60 Shiller (1981) concludes that asset prices are not optimal forecasts of the present value of discounted future dividends.
Shiller discusses in his book “Irrational Exuberance” (Shiller 2005, pp. 74-76, 147-156 and 189-194) possible explanations of excess market volatility such as crowd psychology and naturally occurring Ponzi schemes. Excess Volatility can be interpreted as overreaction to new information (Daniel, Hirshleifer and Subrahmanyam 1998, p. 1841).
The dynamic Gordon growth model allows for a high volatility of stock prices for small changes in the expected stock return. This is under the assumption of the logarithm of the required rate of return following a persistent process (Campbell et al. 1997, p. 265).
Models of stochastic volatility 4.4.6.4.1
The assumption of share prices having constant volatility might be true in the short run.
However, is not unreasonable to assume that the volatility of share prices is not constant in the long run. So a more realistic model of share prices is formed by relaxing the assumption of constant volatility. There are periods of high volatility and periods of low volatility. The variability of volatility is called volatility drift.
In order to describe the volatility of share prices as a stochastic variable, two stochastic differential equations are needed (Hens and Rieger 2010, pp. 329-332):
4-95
4-96
where and are Brownian motions.
The square root stochastic volatility version is also used:
√
4-97
4-98
61 The rational for using the square root of the standard deviation is that it is yields analytical solutions in option pricing (Ishida and Engle 2002).
Another stylized fact is that the share price volatility is mean reverting. This is modelled in the following way:
( ) 4-99
( ) 4-100
where is a long term volatility mean and is a constant.
The constant shows the speed of adjustment to mean . Exponent is a constant.
Some popular models in this framework are the following:
● The Heston model, where .
● The Generalized Autoregressive Conditional Heteroscedasticity model (GARCH), where .
One can run empirical tests to find the value of γ.
Some other stylized facts about the stock price volatility is that it is higher in bear markets and lower in bull markets. This is called “the leverage effect” and doesn't fare well with the
known assumption of risk return being proportional to volatility. Volatility and stock prices are correlated negatively in losses but not necessarily so in gains. This is called “volatility asymmetry”. Stylized facts explained by models of interacting agents are summarized by Lux (2009).
62 4.4.7 Attempts to explain the puzzles