Diversification works in mysterious ways. But does that mean diversifying a portfolio is a hard thing to do? No! It’s actually quite simple. • The problem is that it’s not obvious exactly HOW diversification improves a portfolio’s return. It’s not rocket science. It’s simple math. But many people have never learned the ways diversification can turn a pig’s ear into a profitable silk purse.
• Part 2 of a series. Part 1 appeared on Mar. 19, 2019. •
As we saw in Part 1, the human mind is not ideally suited to guess the best asset allocation for a portfolio. That’s true even when people are asked to choose between two simple assets like a money-market fund and a predictable range-bound stock, such as a utility.
To be sure, that’s an artificial example. It illustrates — in the simplest possible terms — the way diversification can improve the performance of a portfolio beyond the returns of any of the individual assets that may be held.
What if we make the example more interesting by using stocks that rise and fall like actual market securities? Unlike our previous example, these securities aren’t predictable like utility stocks and aren’t price-stable like money-market funds.
To add an extra level of difficulty — and, hopefully, of realism — what if the stocks in today’s example actually decline during the course of the experiment? You might be surprised at the way we can construct a profitable portfolio out of two unprofitable investments.
These two stocks do not a pretty picture make
Figure 1 shows the performance of two imaginary stocks. Let’s say we’ve predetermined — by fundamental analysis or psychic readings or whatever — that both companies will decline over the next 48 months, but with a great deal of random fluctuation. The target price for Security C four years from now will be 38% lower than today, while Security D’s will be 42% lower. If our only choices in this very limited stock exchange were these two distressed stocks, what would be the most profitable allocation of our money between the two?
Figure 1. In an imaginary stock exchange with only two stocks, Security C will have an annualized rate of return of minus 11.4% and Security D will have a return of minus 12.8%.
The prices of Security C and Security D are determined each month by a coin toss, as follows:
- If Security C’s coin comes up heads, its price rises 40% in the following month. If the coin says tails, the price drops 30%. That gives Security C a negative expected return.
- If Security D’s coin comes up heads, its price rises 15% in the following month. If the coin says tails, the price drops 20%. That also gives Security D a negative expected return.
Important note: You may think that Security C has a positive expected return. After all, you might say, a 40% rise in price would lift the stock more than a 30% decline would hurt it.
Unfortunately, that’s not the way price compounding works. Our human minds didn’t evolve to handle these kinds of relationships, so let’s look at the actual numbers. If Security C started at $100 and then declined 30%, its price would be $70. If the price the following month rose 40%, the stock would wind up worth only $98. That’s a 2% loss from your original $100 investment. Because of compounding, Security C would have to rise nearly 43% to get back to even. That’s why a random series of 40% gains and 30% losses gives Secuity C a negative expected return.
Now that we understand how the two stocks operate, we’re ready to make a decision. You’re given $100 to allocate between the two securities. The question is this:
Which asset allocation will produce the highest rate of return?
a. 100% in Security C
b. 100% in Security D
c. 50% in C and 50% in D, rebalanced monthly
d. It cannot be determined
Make a note of your answer. (Don’t send it to me, it’s just a mental exercise for your benefit.)
Turning lemons into a profitable lemonade stand
Figure 2 shows the most profitable allocation for this particular scenario.
Figure 2. Given the random set of returns shown in Figure 1, a 50/50 allocation of dollars between the two securities rises in value — even though both securities declined.
The answer is “c.”
- The diversified portfolio turned $100 into $175 during the 48-month period, after transaction costs. That’s an annualized return of +15.2%. The 50/50 allocation, rebalanced monthly, produced a gain even though both of the stocks individually lost money.
- Notice how patience was required to get the benefit of diversification. For the first 24 months of Figure 2, the dollar value of Security C was actually ahead of the 50/50 portfolio’s balance.
- You had to wait two long years before your diversified portfolio pulled ahead of Security C for good and never looked back.
How did the 50/50 portfolio pull off this trick? The magic of diversification, after all, isn’t actually a conjuring show. It’s purely a mathematical fact.
The diversification of the 50/50 portfolio worked because Security C and Security D have no correlation with each other. Both securities’ prices are determined by the flip of a coin. One stock goes up or down with no relationship to whether the other stock went up or down.
Because of the randomness of these stocks’ price movements, each security will tend to revert to average. After a “hot streak,” each security will appear to have a “cold streak.” Over long periods, a fair coin will come up heads about the same number of times as it comes up tails.
When one security represents more than 50% of the dollar value of the portfolio, you sell a few shares of that security. You use the cash to buy a few shares of the other security, thereby rebalancing the portfolio to 50/50. Selling some of each month’s winner and buying some of each month’s loser turns out to be a very good bet.
Actual markets aren’t exactly like our artificial example
In a true stock exchange, most stocks are highly correlated with each other. You don’t find many stocks that have a 0% correlation to the S&P 500 or the Wilshire 5000. That’s why you can’t rebalance your money 50/50 between two arbitrary stocks and get a better return than either of them. Diversification works better when you have choices that are negatively correlated or uncorrelated with each other, such as stocks and bonds and precious metals, respectively.
So will a diversified portfolio always outperform both of any two stocks? Absolutely not! Despite the case we’ve seen in Figures 1 and 2, a 50/50 portfolio isn’t guaranteed to win.
Flipping Security C’s coin 48 times and Security D’s coin 48 times would generate over 79 trillion quadrillion scenarios. (Think of that as 79 with 27 zeros after it.) In one of those scenarios, Security C might come up heads 48 times in a row, and its price would rise like a shot. If Security D came up tails 48 times in a row, its price would sink like a stone. Rebalancing to a 50/50 allocation each month wouldn’t help you at all in that case.
There are quadrillions of possible scenarios in which a diversified portfolio would underperform either Security C or Security D or both. It all depends on how the coin flips worked out.
But in the example laid out in Figure 1, the magic of diversification pulled a rabbit out of a hat. It constructed a portfolio that made you a profit using nothing but two securities that had negative returns.
This is a class of problems known as “The Joy of Volatility.” An academic paper by that name was published by three British university professors in 2008 in the prestigious journal Quantitative Finance. Read their explanation to see the underlying math, specifically the fact that volatility is required for diversification to work. (You can’t use diversification to beat two money-market funds, for instance.)
In actual stock exchanges, of course, we don’t know how any two securities will move for the next 48 months or almost any period of time. For this reason, the next part of this series will examine known assets with historical data going back 43 years in the US market.
With great knowledge comes great responsibility.
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