In the article he inserts one of those pungent analytical comments I have grown to love in the ret-fin literature: "For a fixed lifespan, a constant relative risk aversion utility of consumption function, asset classes exhibiting geometric Brownian motion, and a risk free asset class, both Samuelson (1969) and Merton (1969) famously showed the optimal asset allocation is a static allocation independent of age and wealth. Their work destroyed the long held folk wisdom that one's asset allocation should decline as you age."
He goes on to provide the math: "In the absence of human capital the solution to Merton's portfolio problem may be used to determine the optimal asset allocation. The optimal solution is independent of age and portfolio size. The proportion of equities that should be held is given by: π = (μ - r) / (σ2 . γ) where μ and σ are the equity geometric Brownian motion drift [that's a random compounding/trending process i.e. a stock market random walk] and volatility, r is the risk free rate of return, and gamma the coefficient of relative risk aversion. If ra and σa are the mean annual equity rate of return and volatility: μ = ln(1 + ra ) [and] σ = sqrt(ln(1 + σ2 / ((1 + ra ) 2 )) For my scenario, after taking into account management expenses, μ = 6.2% and σ = 16.2%. This then gives 34% and 67% equity for coefficients of relative risk aversion 6 and 3 respectively. By way of approach verification, in the absence of any human capital related income, my numerical approach also gave values of 34% and 67%."
This is cool because it more or less lines up with a relatively uninformed bias I have about equity allocation having a sweet spot between 40 and 70% depending on age/longevity and level of wealth among other things. Mr. Irlam looks at the Merton math for his scenario and comes up with 34 and 67% for different levels of risk aversion and then goes on to show that when human capital comes into play from either work and/or social income, the allocations can go much higher really fast. But before he goes there, I first wanted to see what the allocation curve looked like for a vector of risk aversion coefficients (his scenario only) other than just 3 and 6 before he moved on to the human capital thing. It was easy enough to do that by plugging in different coefficients so I did. Here is what it looks like (his scenario only, though it is perhaps more plausible than not on average):
Irlam's optimization effort using the concept of human capital is way more interesting and useful than this but the question is: what little can I take from this enhanced pre-step before moving on? Well, for one thing, for this one Irlam scenario, it is clear that very low risk aversion can tolerate leverage; I hadn't thought about that. For another, for a "normal" risk aversion range (Irlam uses 3 and 6; other retirement writers use a coeff of 4 as a behaviorally tested "normal" factor; in general it is a slippery concept since no one really knows their own and only interviews and questionnaires can really suss it out) the allocation ranges from ~25% to ~100%. That is not too surprising I guess since that is pretty close to the range from a min-variance portfolio to max on an Markowitz efficient frontier. That means that for a risk aversion of 4 (i.e., some researchers' "normal") the allocation would be a little over 50% which is not too far from where I was last year. I guess that makes me normal, though I personally know people that would disagree. I suppose that a last take away is that there is a diminishing change in risk avoidance behavior via allocation maneuvers as risk aversion goes up. Not sure how useful all that is but it was fun to check it out.
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