Risk, return, and asset allocation (toy problem)

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trugs
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Joined: Mon Dec 20, 2021 8:40 pm

Risk, return, and asset allocation (toy problem)

Post by trugs »

This is my first post to Bogleheads.

Summary: When constructing a portfolio of a risky asset and a riskless asset, in some cases expected return can counterintuitively increase as you decrease the allocation to the risky asset.

To get through more quickly, you can skip remarks in brackets [ ].

[In physics, a toy problem is oversimplified, but makes a relevant point.]

Consider an investment that is risky, but offers a positive return, like stocks. In a toy version, once a year, the investment s is assumed to return triple or nothing with equal probability. [Double or nothing is an even bet, offering no return on average. Triple or nothing seems comparatively appealing, certainly better than anything on offer in a casino. This perhaps resembles trading options even more than buying stocks, since options often expire worthless; or venture capital.]

Should you invest some of your savings in this offering s, and if so, how much? Assume you have W_0 in initial savings invested in s. After n years it is worth

W_n = W_0 * r1 *r2 *r3 … * rn (1),

where r1 is a random variable that assumes the value 0 or 3 with equal probability, r2 is another random variable for the second year with the same distribution, etc. The simplest statistical models and theorems (like the central limit theorem) deal with additive random variables, not multiplicative random variables as in (1). To convert this to a more standard problem, consider the (natural) logarithm of net worth,

y = ln(W_n) = ln(W_0) + ln(r1) + ln(r2) + … + ln(rn) . (2)

Now the random variables are additive as in the usual case.

If you know what a logarithm is, you can skip this: [A logarithm writes a number as a power. A number u can be written as a power of 10, u = 10^p . For example, 10^1 = 10, 10^2 = 100, etc. p is then the logarithm (base 10) of u. p does not have to be an integer; if p is ½, then u is the square root of 10 = 3.1623 …. This discussion would also work with log base 10, but in fact uses the natural logarithm, written ln, which is base e, where e = 2.718…. Why would one want to use e? If you know calculus, how to solve dx/dt = x, what function has the same slope at every point as its height, you will know the answer. If not and you are interested in investing, if an investment pays an interest rate g per year, compounded continuously (daily or monthly is close enough), say g = .03 or 3% a year, after t years it will grow by a factor eg*t . This defines e. And the rule of 72, which is actually the rule of 69.3 = 100 ln(2). For the present purpose, ln(x) is defined for x > 0, and is an increasing function of x. The log of a product is given by ln(a*b) = ln(a) + ln(b). Many hand calculators have a button to calculate ln.]

Back to Eq. (2). If you put a fraction x in cash, which pays nothing, and (1-x) in the risky asset s, the random variable y1 = ln(r1) assumes the values of either ln(x) or ln(x + 3(1-x)) with equal probability. Modern portfolio theory says that x should be chosen by considering the expected return <y1> and the risk, the standard deviation of y1. Larger expected return <y1> and lower standard deviation are preferable.

What fraction x should be held in cash? Before doing the calculation, my intuition was that a smaller fraction x invested in cash would be better in the long run, if you can tolerate the risk. And that risk decreases relative to return as the time period increases. With the caveat that 100% in s is too much, because in that case you are very very likely to be left with no money at all, whatever the mean.

Mostly wrong.

Plot the average annual return <y1> = <ln(r1)> vs x. (r1 is the multiplicative factor in each time period; r1 > 1 means your money is growing, r1 < 1 means it is shrinking. ln(r1) > 0 means it is growing.)

<y1> = [ ln( x + 3(1-x) ) + ln(x) ] / 2 (3).

For later reference, the risk or standard deviation sigma for one year is

sigma = [ ln( x + 3(1-x) ) – ln(x) ] / 2 (4).

I would have thought that if there is a riskless asset and a profitable risky asset, expected return would decrease as you put more money into the riskless asset. Wrong. From Eq. (3), for x=0 <y1> = minus infinity, since ln(0) = - infinity. [The logarithm of x is singular at x=0.] Expected annual return <y1> increases (!?) with increasing cash x, but still remains negative for [0 < x < .5]. Why? Since y1 is a log, extremely poor returns are severely penalized; ln(x) approaches minus infinity as x approaches 0. Increasing the cash allocation prevents a severe penalty, increasing the average return.

One will lose money on average for any x < 0.5. Continuing to map out <y1> vs x, as x increases above .5, expected return <y1> finally turns positive, and continues to increase to a maximum value at x = .75 . The maximum expected return at x = .75 is only .0589, less than 6%, not impressive. As x increases from .75 to 1, expected return <y1> decreases, becoming zero at x=1, all cash. The expected return is nonmonotonic, first increasing with x up to x=.75, and then decreasing. The risk decreases monotonically with x as expected, reaching 0 at x=1. From modern portfolio theory, everyone should choose x > 0.75 . How much above 0.75 depends on personal preference, tolerance for risk, time horizon, etc. Since 0.75 is a local maximum, increasing x above 0.75 cash decreases return only slightly at first (quadratically), while risk decreases more (linearly). An “average” investor, say one who can accept the risk of target retirement 2030, might put about 90% into cash, x = 0.9, and only 10% into the risky asset s. This yields only a modest expected annual return 3.85%, with annual risk sigma (standard deviation) of 14.4%. [At the maximum return, x = .75, the risk sigma = 34.7%. For comparison, an S&P 500 index is more attractive, historically returning on average about 10%, with sigma ~ 20% for one year. Aside: Is there any understanding why the return and risk of a modern economy are those numbers, and not, for example, twice or half as big?]

It was argued that looking at the expectation of the log of wealth <ln(W)> was better than <W>, because it turned the problem of multiplicative random variables into the more standard additive random variables. This is a big assumption, but perhaps a justifiable one, for other reasons. In standard economic theory, it is often supposed that the “utility function” of wealth is ln(W), or some other function with similar convexity properties. An extra dollar is worth much more to a pauper than to a millionaire. If one were to maximize the expected wealth <W> rather than the expection of the log of wealth, the conventional wisdom becomes true again. Expected wealth is maximized by putting all your money in the risky asset. (But expected log wealth <ln(W)> is not.) However, maximizing <W> is not the right thing for an investor. A person whose retirement plan has <W> of a million dollars might be on track, but not if all of the money is invested in a lottery ticket that has one chance in a million of paying out a trillion dollars.

Comments: In this example, it is assumed that there is an initial investment, but no new investments are made at later times. In the long run, with x > 0 in cash, the final distribution of wealth is log-normal. Some comments have been posted on Boggleheads stating that the investor will not presently put money into cash or bonds, since the real return on bonds is zero or negative. In this toy example, one should put money into cash, even if its return is zero or slightly negative. There are also assertions that one should put 100% (or more, using levarage) into risky assets, and that in the long run this will pay off, if one can tolerate the risk and not panic and sell everything after a correction. Not in this example. In this case, less than 25% should be put into the risky asset, even in the long run, even with perfect self control.

Note added: I am grateful to the Boggleheads boardmember who observed that I have “rediscovered the Kelly criterion,” which I was previously unaware of. https://en.wikipedia.org/wiki/Kelly_criterion Note also that Daniel Bernoulli argued in 1728 that one should maximize the expectation of the log of wealth rather than wealth itself in situations like this. And that this example is relevant to real world investing: Warren Buffet, Bill Gross, and Jim Simons all use Kelly methods.

Request: Some Bogglehead posts (and even titles) contain a blizzard of obscure abbreviations, so the posts are meaningless except to the cognoscenti. Not everyone wants to memorize the ticker symbol of every stock, mutual fund, and ETF, and the abbreviation for every possible investment and withdrawal strategy. I/we would be grateful if abbreviations are defined the first time they are used in a post. There is a chance that someone not in ones own sub-field may have something interesting to add, or something to learn. I did not find the function on google translate for Boggleheadish to English.
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