Section VIII – Long-term investment strategies
8.2 Constant Relative Risk Aversion (CRRA)
In this sub-section, the theory constant relative risk aversion or CRRA is introduced. An individual with CRRA preferences receives utility based on the relative relation between risk and return. Hence, an individual with the same risk aversion and the same portfolio will receive the same utility no matter the level of wealth. This utility theory is arguably applicable to our field of study as individuals, poor as rich, would to some degree prefer to sustain a constant level of consumption through life. Thus, individuals aim to save for a certain percentage of their income rather than a certain amount of wealth.
It can be discussed which utility function is suitable for such analysis; however, in this analysis, the utility function below is chosen:
𝑀𝑎𝑥 𝑈 = 𝐸[𝑅𝑝] − 0.5𝛾 ∗ 𝐸[𝜎𝑝2] (8.1)
Page 83 of 102 The utility function in formula 8.1 consists of three variables. In line with the theory by Merton (1969), gamma (𝛾) is the level of risk aversion. The level of gamma is relatively constant with a multiplier of the expected risk. On the other hand, the individual receives positive utility defined as expected return.
Return and risk are defined as:
𝐸[𝑅𝑝] = 𝑅1𝑤1+ 𝑅2𝑤2 (8.2) 𝐸[𝜎𝑝2] = 𝐸[𝜎12]𝑤12+ 𝐸[𝜎22]𝑤22+ 2𝑤1𝑤2𝐸[𝜎1]𝐸[𝜎2]𝐸[𝜌1,2] (8.3) As the investment assumptions by Forsikring & Pension predict the correlation between stocks and bonds to be 0, the formula is reduced to:
𝐸[𝜎𝑝2] = 𝐸[𝜎12] ∗ 𝑤12+ 𝐸[𝜎22] ∗ 𝑤22 (8.4) As the sum of stocks and bonds should be equal to 100 % of the portfolio and leverage is not allowed, we get the limitation:
𝑠. 𝑡. 𝑤2= (1 − 𝑤1), 𝑎𝑛𝑑 0 ≤ 𝑤2≥ 1 (8.5)
8.2.1 Mathematical proof of the optimal CRRA portfolio composition
To calculate the optimal asset allocation between stocks and bonds, we set up an optimization problem.
The objective is to maximize utility given a specific gamma. Combining the CRRA utility function with the equations for expected return and variance, utility is given as:
𝑈 = 𝑅1𝑤1+ 𝑅2𝑤2− 0.5𝛾(𝜎12𝑤12+ 𝜎22𝑤22) (8.6) By substituting 𝑤2= (1 − 𝑤1) we get:
𝑈 = 𝑅1𝑤1+ 𝑅2(1 − 𝑤1) − 0.5𝛾(𝜎12𝑤12+ 𝜎22(1 − 𝑤1)2) (8.7) The equation can then be rewritten as:
𝑈 = 𝑅1𝑤1+ 𝑅2− 𝑅2𝑤1− 0.5𝛾(𝜎12𝑤12+ 𝜎22(1 − 2𝑤1+ 𝑤12)) 𝑈 = 𝑅1𝑤1+ 𝑅2− 𝑅2𝑤1− 0.5𝛾(𝜎12𝑤12+ 𝜎22− 𝜎222𝑤1+ 𝜎22𝑤12)
𝑈 = 𝑅1𝑤1+ 𝑅2− 𝑅2𝑤1− 0.5𝛾𝜎12𝑤12− 0.5𝛾𝜎22+ 0.5𝛾𝜎222𝑤1− 0.5𝛾𝜎22𝑤12 (8.8) The utility function can then be arranged as a quadratic equation:
𝑈 = (𝑅2− 0.5𝛾𝜎22) + (𝑅1𝑤1− 𝑅2𝑤1+ 𝛾𝜎22𝑤1) − (0.5𝛾𝜎12𝑤12+ 0.5𝛾𝜎22𝑤12) 𝑈 = (𝑅2− 0.5𝛾𝜎22) + 𝑤1(𝑅1− 𝑅2+ 𝛾𝜎22) − 𝑤12(0.5𝛾𝜎12+ 0.5𝛾𝜎22) (8.9)
Now we have the full utility function. The maximum utility can be found due to the following assumptions:
A quadratic function can be described as:
𝑌 = 𝐶 + 𝐵𝑥 + 𝐴𝑥2 (8.10)
Page 84 of 102 If A is positive, the function will be convex and have a global optimum in its minimum. If A is negative, the function will have a concave shape and have a global optimum in its maximum. In the CRRA utility function, A is strictly negative as both gamma and variances are strictly positive. Hence, it is possible to find an optimum.
To find the maximum utility, the utility function is differentiated with respect to 𝑤1 and set equal to zero:
𝑈′= (𝑅1− 𝑅2) − 𝛾𝜎12𝑤1+ 𝛾𝜎22− 𝛾𝜎22𝑤1 (8.11) 𝑈′= 0
0 = 𝑅1− 𝑅2+ 𝛾𝜎22− 𝛾𝜎12𝑤1− 𝛾𝜎22𝑤1 (8.12)
As the objective is to find the weights of bonds and stocks, 𝑤1 is solved for:
(𝑅1− 𝑅2) +𝛾𝜎22=𝛾𝜎12𝑤1+ 𝛾𝜎22𝑤1 (𝑅1− 𝑅2) +𝛾 ∗ 𝜎22= 𝑤1(𝛾𝜎12+ 𝛾𝜎22)
𝑤1 =(𝑅1− 𝑅2)+𝛾𝜎22 (𝛾𝜎12+ 𝛾𝜎22) 𝑤1 = 𝜎22
(𝜎12+ 𝜎22)+ (𝑅1− 𝑅2) 𝛾(𝜎12+ 𝜎22)
(8.13)
As the optimization is based upon expectations rather than actual returns and variances, the final equation for the optimal portfolio composition, is given as:
𝑤1= 𝐸[𝜎22]
𝐸[𝜎12] + 𝐸[𝜎22]+ 𝐸[𝑅1]− 𝐸[𝑅2]
(𝐸[𝜎12] + 𝐸[𝜎22])𝛾−1 (8.14) and
𝑤2= 1 − 𝑤1 (8.15)
From the optimal CRRA portfolio composition, it can be concluded that;
The intercept is given as 𝐸[𝜎𝐸[𝜎22]
12]+𝐸[𝜎22] which implies the weight of asset 𝑤1 is greater when the risk of the other asset becomes relatively higher. Thus, the individual chooses assets with lower volatility when returns on both assets are equal. The slope of the optimum is a concave power function of 𝛾 and is given as
𝐸[𝑅1]−𝐸[𝑅2]
(𝐸[𝜎12]+𝐸[𝜎22]). The numerator is given as the relative relationship between the expected return on asset 1 and asset 2. Hence, the individual increases the proportion of 𝑤1 when 𝐸[𝑅1] − 𝐸[𝑅2] increases. The denominator determines the importance of the nominator. Thus, if the sum of the risk increases, the importance of the differences in expected returns decreases.
8.2.2 Results and implications of CRRA
To calculate and evaluate results from our CRRA-analysis, two graphs have been created. Both are based on F&P long-term investment assumptions where the expected return after investment costs for stocks and bonds are 6.00 % and 3.28 %. The standard deviations are 15 % and 6 % with a correlation of 0.
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Table 40: Assumptions about the expected return after costs and standard deviations (Authors’ creation)
In the first graph below, the optimal allocation of stocks and bonds are found formula 8.14 and 8.15 for optimization of CRRA.
The optimal allocations given various levels of gamma can be seen in figure 26 below. The results indicate that an increase in gamma (higher risk aversion) lead to a higher proportion of bonds. The graph also shows how a gamma of less than 1.20 will result in an optimal asset allocation of more than 100 % stocks. Due to our limitation where borrowing is not allowed, the allocation is fixed at 100 % stocks. At a gamma of 3.09, the optimal allocation if 50% in each asset class. The graph further implies how the proportion of bonds rarely do not exceed 75 %, even at gamma levels of more than 10. If increasing gamma to infinity, the proportion of bonds will converge towards 82.1%. Hence to optimize CRRA preferences, the individual would never short stocks at any level of gamma. Due to the characteristics of a CRRA, individuals should hold a constant proportion of bonds and stocks if gamma is known to the individual.
In figure 27 below, the utility given a gamma is drawn as a function of the proportion of stocks. To draw the graph, equation 8.9 is used from the previous page is used. It shows how utility increases when gamma is low. It also shows how an individual with a high gamma receives less utility from stocks.
As utility is often an intangible size, one must not conclude on the absolute value of utility. However, all individuals should maximize utility given their specific value of gamma. Hence, the black triangles on the graph show the proportion of stocks to optimize utility given a specific gamma. For the values of gamma 1.5 to 5, it is evident how the individual with a gamma of 1.5 optimizes utility with a stock proportion of 84 %.
With a gamma of 2, the optimal percentage of stocks is 68 %. As gamma increases to 5, the optimal stock proportion decreases to 38 %. Although, gamma is negatively correlated with stocks it is evident stocks should be bought at any level of gamma. However, this is, of course, subject to the investment assumptions defined by Forsikring & Pension.
0%
20%
40%
60%
80%
100%
0 1 2 3 4 5 6 7 8 9 10 11 12
Proportion of assets
Gamma (y)
Optimal allocation given gamma
Stocks Bonds
Figure 26: Optimal long-run allocation between stocks and bonds given different values of gamma (Authors' creation)
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Figure 27: Utility given proportion of stocks (Authors' creation)