Mehra and Prescott, (1985) begin with an assumption that a standard representative household attempts to maximise its time-additive expected utility over consumption within Lucas's (1978) pure exchange economy. This is represented as
𝐸 [∑ 𝛽𝑡. 𝑈(𝐶𝑡)
∞
𝑡=0
] , 0 < 𝛽 < 1 (2.1)
Where the utility function of this representative household is assumed to be a power utility function given by
𝑈(𝐶, 𝛼) = 𝐶1−𝛼
1 − 𝛼, 0 < 𝛼 < ∞ (2.2)
and where 𝛼 is the curvature of the utility function which simultaneously controls inter- temporal substitution and risk aversion. The advantage of this utility function is that, it is strictly increasing, implying that the household prefers more consumption than low
consumption (“greedy”) and it is concave implying diminishing marginal utility of consumption and strict risk-aversion. It is also differentiable two times with 𝑈′′(. ) < 0 The household faces an inter-temporal choice where it can choose not to consume today and save and use the proceeds to buy an asset at price Pt which has a total payoff of xt+1
in time t+1 and use it to consume Ct+1 in the future. In other words, the loss in marginal
utility {𝑃𝑡. 𝑈′(𝐶𝑡)} of not consuming today and saving it and using it to buy an asset at price Pt must be at the most same as the expected gain in the marginal utility of
consumption because of the payoff xt+1 {𝐸[𝛽. 𝑈′(𝑐𝑡+1). 𝑥𝑡+1]} in the future, discounted
by investor’s impatience 𝛽 = 𝑒−𝛿. Thus we have,
𝑃𝑡. 𝑈′(𝐶
𝑡) = 𝐸[𝛽. 𝑈′(𝐶𝑡+1). 𝑥𝑡+1] (2.3)
Equation 2.3 is the first order condition of optimal consumption path which leads to the basic pricing identity for any asset.
𝐸 [𝛽.𝑈′(𝐶𝑡+1) 𝑈′(𝐶
𝑡) . 𝑅 𝑡+1] = 1
(2.4)
Equation 2.4 can also be written as
𝐸𝑡[𝑚𝑡. 𝑅𝑡] = 1 (2.5)
where, 𝑚𝑡+1≡ 𝑒−𝛿 .𝑈
′(𝐶 𝑡+1)
𝑈′(𝐶𝑡) is marginal rate of substitution or the pricing kernel or the
stochastic discount factor that captures the household’s preference to postpone Ct to Ct+1. This suggests that the household evaluate the price of an asset by discounting the
future stream of uncertain cashflows from that asset using their marginal rates of substitution as a stochastic discount factor. Equation 2.5 is the fundamental asset pricing equation which suggests that in the absence of arbitrage, there exist a strict positive SDF
(𝑚(𝜔) > 0 ∀ 𝜔 ∈ Ω) which is used to price all the tradable assets (risky and risk-free) and has finite variance (Cochrane, 2001). Equation 2.5 implies that risk averse household care about marginal utility of consumption and as such marginal utility is the appropriate indicator of risk. It suggests that assets which does not provide higher payoffs when marginal utility is higher, have low expected returns compared to assets that does provide higher payoffs when the marginal utility is higher and thus command more premium.
For a risk free asset, we have
𝑅𝑓 =
1
𝐸(𝑚) (2.6)
Now as C-CAPM implies that the pricing kernel or the SDF is the marginal rate of substitution, therefore for a household with time-additive expected utility we have
𝑚 = 𝑒−𝛿.𝑈′(𝐶𝑡+1) 𝑈′(𝐶 𝑡) (2.7) Implying that, 𝑅𝑓 = 𝑒𝛿. 𝐸 [𝑈′(𝐶𝑡+1) 𝑈′(𝐶 𝑡) ] −1 (2.8)
Equation 2.5 also implies that for any risky asset i
1 = 𝐸(𝑚). 𝐸(𝑅𝑖) + 𝐶𝑜𝑣(𝑅𝑖, 𝑚)
⟹ 𝐸(𝑅𝑖) = 𝑅𝑓−
𝐶𝑜𝑣(𝑅𝑖, 𝑚)
Now assuming that consumption growth rate 𝑅𝑐 and dividend growth rate 𝑅𝑑 of the risky asset follow the lognormal distribution and the utility function of the household is the standard power utility function given by equation 2.2.
We have, ln(1 + 𝑟𝑐) = 𝑅𝑐 ≡ ln [𝐶𝑡+1 𝐶𝑡 ] ≈ 𝑁(𝑟̅, 𝜎𝑐 𝑟𝑐2 ) ln(1 + 𝑟𝑑) = 𝑅𝑑 ≡ ln [𝐷𝑡+1 𝐷𝑡 ] ≈ 𝑁(𝑟̅ , 𝜎𝑑 𝑟𝑑2 ) (2.10)
And equation 2.2 implies that, 𝑈′(𝐶 𝑡+1) 𝑈′(𝐶 𝑡) = ( 𝐶𝑡+1 𝐶𝑡 ) −𝛼 = 𝑒𝑥𝑝 [−𝛼. 𝑙𝑛 (𝐶𝑡+1 𝐶𝑡 )]
Now, if a variable 𝑥 ∼ 𝑁(𝑎, 𝑠2) then we know that, 𝐸[𝑒−𝑘𝑥] = exp [−𝑘𝑎 +1 2. 𝑘2. 𝑠2] Thus we have, 𝐸 [exp (−𝛼. 𝑙𝑛 (𝐶𝑡+1 𝐶𝑡 ))] = 𝑒𝑥𝑝 (−𝛼. 𝑟̅ +𝑐 1 2. 𝛼2. 𝜎𝑟𝑐2)
And from equation 2.8 we have
𝑅𝑓 = 𝑒𝑥𝑝 (𝛿 + 𝛼. 𝑟̅ −𝑐 1
2𝛼2. 𝜎𝑟𝑐2 )
⇒ ln(𝑅𝑓) ≡ 𝑟𝑓 = 𝛿 + 𝛼. 𝑟̅ −𝑐
1
2𝛼2. 𝜎𝑟𝑐2 (2.11)
⇒ 𝑙𝑛[𝐸𝑡(𝑅𝑖,)] = 𝛿 + 𝛼. 𝑟̅ −𝑐 1
2𝛼2. 𝜎𝑟𝑐2 + 𝛼. 𝑐𝑜𝑣(𝑟𝑐, 𝑟𝑑) (2.12)
Equation 2.12 minus Equation 2.11 gives log of ERP
ln[𝐸(𝑅𝑖,𝑡+1)] − ln(𝑅𝑓) = ln(𝐸𝑅𝑃) = 𝛼. 𝑐𝑜𝑣(𝑟𝑐, 𝑟𝑑) (2.13)
In equilibrium 𝑟𝐷 → 𝑟𝑖 and continuously compounded growth rate in consumption approaches to that of growth rate in dividends or return on equity. Thus, we have, 𝑟𝑖 → 𝑟𝑐
∴ 𝑙𝑛𝐸𝑅𝑃 = ln(𝑅𝑒) − ln(𝑅𝑓) = 𝛼. 𝜎𝑟𝑐2 (2.14)
Equation 2.14 implies that the log of ERP is product of coefficient of relative risk aversion and variance of consumption growth rate.
Mehra and Prescott (1985) report following empirical data for the US economy for the period 1889-1978. The actual value of US ERP is 6.18% which is far more than 0.35%, the value that is implied by standard economic theories (Equation 2.14) of asset pricing. The risk free rate in Table 2.1 is the nominal yields on 3-month T-bills rate (for the period 1931-1978), Treasury Certificate (for period of 1920-1930) and sixty and ninety- day Prime Commercial Paper (prior to 1920).
Table 2.1 Data of US Economy
Risk Free Rate Return on S&P 500 index ERP Consumption Growth Rate.
Mean 0.8% 6.98% 6.18% 1.8%
Standard Deviation 5.67% 16.54% 16.67% 3.6%
In order to get ERP of 6.18 % the coefficient of risk aversion (α) should be around 46 in equation 2.14, which is implausible based on Arrow (1971), Friend and Blume (1975) and Kydland and Prescott (1982) since they imply that α ≤ 5. This shows that consumption growth rate in the US is not volatile enough to generate ERP of 6.18%. This is the ERP puzzle. As Mehra (2003) emphasise that ERP puzzle is a quantitative puzzle not a qualitative, meaning that the puzzle does not disregard the risk-return trade- off, however the puzzle questions the mismatch between quantity of reward (premium) that one actually gets and the premium which is implied by theoretical models. It supports that fact that assets that pays off well in good times i.e. high consumption are less desirable than the assets which pays similar cashflows in the bad times i.e. low consumption. The puzzle motivates to improve the existing conventional economic theories and the preference structures of the agents to build more accurate models so that the mismatch between the actual observed ERP and the one implied by theory could be overcome. However, the puzzle does not focus why equities offer so high premium within the standard representative-agent-based utility maximisation framework.
Figure 2.1 gives a visual snapshot and the classification of the literature that I will evaluate in the following sections.
***Please insert Figure 2.1 about here***