Section VIII – Long-term investment strategies
8.3 Human capital investment strategy
Page 86 of 102
Figure 27: Utility given proportion of stocks (Authors' creation)
Page 87 of 102 8.3.2 Mathematical proof of optimal portfolio composition
Jagannathan and Kocherlakota (1996) showed the optimal portfolio depended not only on expected return, standard deviation, and risk aversion but also on human capital. The relative size and characteristics of human capital define the optimal portfolio of an individual and can be explained by an example based on the calculations in the previous section of CRRA:
Assume an individual with CRRA preferences and a gamma of 2. Hence, the optimal proportion of stocks at any time is given by equation 8.14.
𝑤1= 7%2
15%2+ 7%2+6.00% − 3.28%
15%2+ 7%2 2−1= 68%
Hence, the optimal fraction of stocks is 68 %.
As human capital is the expected future wealth, it can be included in the decision making today. Therefore, the individual will seek to maximize both the current wealth and future wealth:
𝑀𝑎𝑥 {𝐸 [𝑊1 𝑊0] −𝛾
2𝑉𝑎𝑟 [𝑊1
𝑊0]} (8.17)
Where
𝑊 = 𝐿 + 𝐹 (8.18)
L denotes the human capital, F the financial wealth today, and 𝑊1 the total wealth after period 0. In the relation to the CRRA-approach the individual now cares about the total wealth rather than only the financial wealth.
The optimal choice of portfolio composition at time t now depends on the risk of human capital. Let us assume three possible scenarios:
1. The human capital has risk properties of a bond 2. The human capital has risk properties of a stock
3. The human capital has risk properties of 50 % bond and 50 % stock.
Assume now the financial wealth today (𝐹), is DKK 100,000. Also, assume the human capital (L) is DKK 100,000. The total wealth is then DKK 200,000 according to equation 8.18. Based on the three scenarios of riskiness, the optimal portfolio using equation 8.14 is reported in table 41:
Table 41: Optimal asset allocation for gamma=2 and different risk properties of human capital (Authors’ creation)
For example: As the optimal fraction of stocks is 68 % of the total wealth, the optimal fraction will be DKK 136,000. If human capital is characterized as a bond, more than the financial wealth should be put in stocks – Hence the individual should borrow DKK 36,000 and place it in stocks. On the other hand, if human capital is characterized as stocks, the individual should only invest DKK 36,000 in stocks and the rest in bonds.
Optimal portfolio y=2
Stocks Bonds Stocks Bonds
L =100 % Bond 136,000 -36,000 100,000 -L = 100 % Stock 36,000 64,000 36,000 64,000 L = 50 % Bond, 50 % Stock 86,000 14,000 86,000 14,000 No-restrictions No-shorting allowed
Page 88 of 102 To put the example into mathematical terms, the individual maximizes total wealth given a risk aversion in a mean-variance model according to equation 8.17.
Suppose the individual can invest in a risk-free bond, 𝑟𝑓, and in a number of risky assets given by the vector 𝑟. 𝜇 denotes the expected return. 𝜋 denotes the vector of financial wealth allocated to the risky assets. In the end of the first period the wealth will be:
𝑊1= 𝐹0(1 + 𝑟𝑓+ 𝝅(𝑟 − 𝑟𝑓𝟏)) + 𝐿0(1 + 𝑟𝐿) (8.19)
where 𝑟𝐿 is the discounting rate on human capital.
Hence, the change is given as:
𝑊1
𝑊0= 𝐹0
𝐹0+ 𝐿0(1 + 𝑟𝑓+ 𝝅(𝑟 − 𝑟𝑓𝟏)) + 𝐿0
𝐹0+ 𝐿0(1 + 𝑟𝐿) (8.20)
The expectations of both expected return and variance of the wealth can thus be expressed as:
𝐸 [𝑊1
𝑊0] = 𝐹0 𝐹0+ 𝐿0
(1 + 𝑟𝑓+ 𝝅(𝜇 − 𝑟𝑓𝟏)) + 𝐿0 𝐹0+ 𝐿0
(1 + 𝑟𝐿) (8.21)
(8.22)
By combining equation 8.17, 8.21 and 8.22 the equation for maximization can be established (Munk, 2017):
(8.23)
Note, 𝑙 =𝐿0
𝐹0 denotes the human capital relative to the financial wealth. Hence, if human capital is less risky than the portfolio, the amount of stocks will decrease as the fraction of human capital decreases.
8.3.3 Optimal portfolios across lifecycles
Based on the mathematical approach presented above, a lifecycle investment strategy can be made for individuals with a known gamma (𝛾). As the calculations rely on a series of assumptions, these are presented before the results.
8.3.3.1 Assumptions
It is assumed the investor has a CRRA preference with a known risk aversion (𝛾). The individual can invest in bonds or stocks at time=t. The expected return, standard deviation, and correlation of stocks and bonds are equal to the assumptions made by Forsikring & Pension. It is assumed these assumptions do not change in the future.
It is assumed the human capital has the risk characteristics of bonds. Hence it can be argued to be a bond.
The human capital is calculated using the lifecycle income model presented in section 6. The average lifecycle income is assumed to be risk-free. Hence, the discounting rate is set equal to inflation. Thus,
Page 89 of 102 human capital is the sum of contributions in real terms. The mean income in each year is used when
simulating the lifecycle income model 100,000 times.
It is assumed that the individual cannot lever the investment. Thus, gearing and borrowing are prohibited.
8.3.3.2 Results of lifecycle investment strategies
Based on the assumptions above, the optimal portfolio choice can be made for each level of risk-aversion through life.
Figure 28 shows the allocation to stocks across a lifetime. For all levels of risk aversion (gamma), the optimal allocation to stocks the first 12 years are 100 %. This is due to the small amount of financial wealth relative to the total wealth (inclusive of human capital). For all four shown values of risk aversion, the optimal asset allocation converges towards the mean-variance optimal portfolio.
Figure 28: Asset allocation for different levels of risk aversion across the lifecycle (Authors' creation)
Hence, the risk of each portfolio is high in the early stage, as the nominal risk is low. As human capital decrease and financial wealth increase, the risk of each portfolio decreases. For higher gammas, it is evident how the risk-reduction begins earlier. For gamma = 10 the risk reduction begins at age 37 whereas the risk reduction for gamma = 2 starts at age 49. Additionally, the speed of risk reduction is more
significant for higher gammas.
Examining gamma = 2, the fraction allocated to stocks at retirement is 68 %. At this point, human capital equals 0, and the asset allocation is precisely equal to the calculation of CRRA preferences in section 8.2 The reason behind the high proportion of stocks in the early stages is the ratio between human capital and financial wealth. In figure 29 below, the ratio is shown using the same assumptions as above with a gamma of 2.
The financial wealth has a limited effect on the total wealth in the first years as the ratio is as high as 100.
At age 44 the financial wealth is expected to be equal to human capital. At the end of age 71, the human capital is 0, why the ratio is 0. Hence, it is assumed that state pensions are not part of the human capital – Although, this could be implemented to the model.
0%
20%
40%
60%
80%
100%
25 30 35 40 45 50 55 60 65 70
Allocation to stocks
Age
Lifecycle allocation to stocks given risk aversion
Gamma 2 Gamma 3 Gamma 5 Gamma 10
Page 90 of 102
Figure 29: Ratio between human capital and financial wealth (Authors’ creation)
8.3.4 Comparison to pension forecasts
As the previous results are theoretically optimal lifecycle investment strategies, it seems relevant to compare the results to the real-life. The theoretical strategies will be compared the asset allocations presented in figure 25 in section 7.2.2. Figure 25 is based upon an average of asset allocations from 5 pension funds. As part of the comparison, it has been assumed gamma 10 equals low-risk, gamma 5 equal medium-risk and gamma 3 equals high-risk.
To summarize the development, the fraction of stocks at age 25, 50 and retirement is presented in table 42 below:
Proportion of stocks Age 25 Age 50 Age 72 (Retirement)
Risk profile Gamma Real-life Theory Real-life Theory Real-life Theory
Low-risk 10 44 % 100 % 39 % 41 % 21 % 28 %
Medium-risk 5 63 % 100 % 57 % 55 % 36 % 38 %
High-risk 3 86 % 100 % 77 % 73 % 50 % 51 %
Table 42: Comparison of the asset allocation of real pension funds and theoretical weights (Authors’ creation)
From the table, it is evident how the theory proposes a higher stock allocation in the early phase. Whereas none of the real-life portfolios have 100 % allocation, all portfolios at age 25 have 100 % stock-allocation. At age 50, the asset allocation of stocks is higher in real-life than theory expects (for medium- and high-risk). Thus, it can be concluded theory expected the allocation of stocks to be significantly higher in the early phase but decrease more rapidly than in real life.
Overall it seems that the development in asset composition between an actual case and theory match quite well given the gamma assumptions above.
Conclusively, this analysis can be used as a tool to optimize the utility of an individual rather than the coverage ratio. In section 7, the differences in coverage ratios often determined the effect of a variable.
However, by including utility, a high average coverage ratio is not necessarily the most optimal choice.
In section 7, all analysis led to the conclusion that by acquiring more risk, the coverage ratio would increase. However, if the objective is to maximize utility, this analysis showed all three risk-profiles might be appropriate if the risk-aversion, gamma, is known.
0,00 0,01 0,10 1,00 10,00 100,00
25 30 35 40 45 50 55 60 65 70
Log scaled ratio
Age
Human capital to financial wealth (ratio)
Page 91 of 102