## Asset Allocation: The Extra Credit Question of Investing

Today we talk about an important investing topic that I’ve been putting off for a while: Asset Allocation. One of the reasons I’ve been putting it off is because I wanted to finish reading what is perhaps the most authoritative source on the topic: a book called The Intelligent Asset Allocator by William Bernstein, which I had read maybe half of before but never finished. So, over the last couple weeks I got it from the library^{1}If you’re paying for books you need to go back and read some more of this blog! and read it cover to cover. Since approximately 98% of my knowledge on the topic comes from the book, this post is part book review. Also, quick warning. There’s gonna be some 9th or 10th grade level math here, mostly at the beginning, and just a little bit of it. So buckle up!

# What is Asset Allocation?

Essentially, it is dividing up the money you have invested (your portfolio) into different types of assets, or asset classes, in fixed percentages. Examples of asset classes include US Stocks, International stocks, International bonds, etc. You can further subdivide asset classes into things like US large-company stocks, European small-company stocks, US short-term bonds, and so on. As you can imagine, this level of subdividing leads to quite a large number of different asset classes, and an almost infinite number of different combinations. So one example asset allocation might be 80% US stocks, 10% US bonds, 10% International stocks. Or you can do something a little spicier like 10% each of 10 different asset classes.

# Why allocate your assets?

To answer this question, I’ll try to semi-briefly go through the example Bernstein uses in his book.

You work for Uncle Fred, who contributes $5,000 to your pension plan each year. At the end of each year, the fate of your entire portfolio is decided by a coin toss. If heads, you make +30% on everything in the portfolio. If tails, you lose 10%. Thus, on *average *the return on the coin toss investment is 10%. But is this the return you realize? Consider a representative two year period where you get one heads and one tails. Assume you start with $100 for simplicity. After the heads, you’ll have $130, or 1.3 x $100. Then, after the tails, you’ll lose 10%, which is the same as multiplying by 0.9. So 0.9 x $130 = $117.

Note that if you realized your *average *return of 10%, you’d have $121 (1.1 x 1.1 x 100; or, 110 after the first year and then 110 + 10% = 121 after the second year). In reality, you ended up with $117, which is the same as having gained 8.17% per year. This is called the *annualized *return, which is a very important concept in finance. The annualized return is the return you’d need to get every year to get from your initial investment to your ending amount. The annualized return is always less than the average return for any investment that doesn’t return exactly the same percentage all the time. How do we calculate annualized return? Here comes the high school math. It’s a geometric mean, rather than an arithmetic mean (or a regular average in street language).

The geometric mean of a series of n terms is obtained by multiplying each of the terms together and then taking the nth root (or raising to the 1/n power).

On a higher level, why is annualized return always less than average return? It’s because losses always hurt more than gains. And not just emotionally. Consider a portfolio of $100 that gains 50% and then loses 50% (or vice versa). You end up with $75, not $100, even though your *average *return is 0%. Cool, back to asset allocation.

Uncle Fred then offers you an interesting proposition. Instead of one coin toss for your entire portfolio, he gives you the option to split your money into two halves and he’ll toss one coin for each half. Should you do it?

Yes, yes you should. In this scenario, there are four possibilities, each with equal probability:

- Two heads–same outcome as one heads–portfolio goes up by 30%
- One heads and one tails–portfolio goes up by 10%
^{2}half goes up by 30%, half goes down by 10%. Think of an example with $200 if you’re unsure - One tails and one heads–same as above, +10%
- Two tails–same as one tails–down 10%

So, what’s the annualized return of this option? Let’s consider a representative 4 year period and calculate the geometric mean:

(1.3 x 1.1 x 1.1 x 0.9)^(1/4) = 1.0908

Or, an annualized return of 9.08%, which is way better than our previous return of 8.17%! How did this happen? On a very high level: because losses hurt more than gains, and we’ve reduced the chance of loss to only 25% each year, we end up with a higher return. Also note that in both cases (1 or 2 coin tosses), we have the same *average* return of 10%. But adding the second coin toss increased our *annualized* return. As it turns out, adding the second coin toss also decreased our *risk*, as measured by the standard deviation (or how far away from the average our returns are likely to be from year to year; standard deviation is a common measure of risk in finance). This example illustrates the concept that is the entire principle behind asset allocation:

Holding

uncorrelatedassets increases return while decreasing risk.^{3}Caveat: these assets have to (over time, at least) produce positive returns!

The key here is that the two coin tosses are uncorrelated–the result of one coin toss does not affect the outcome of the other. You can imagine the extreme example of the two coin tosses being perfectly correlated. If heads for the first toss meant the second toss would also be heads, we’d be back to the original situation and asset allocation would have no positive effects.

# How does this work in real life?

Unfortunately, in the real world, it’s tough to find any two assets that have zero correlation. Stocks and bonds are poorly correlated, which is good. But US stocks and international stocks are moderately correlated. Since they’re not perfectly correlated, there’s still some benefit to asset allocation.

We haven’t talked very much about bonds around here^{4}yet!, but they are generally considered to be “safer” than stocks (with the price of safety being lower returns). However, Bernstein shows that adding a tiny amount of stock to an all bond portfolio actually **decreases** risk and increases return. Adding a tiny bit more gives you even more return for the same level of risk as an all bond portfolio:

Another reason asset allocation works is that it forces you to “buy low, sell high”. If you hold 50% US stocks and the stock market crashes, your US stock allocation will fall below 50%. Thus, when you *rebalance*, you’ll buy more stock when stocks are cheaper. Similarly, if stocks experience a huge rise, you might sell some stock to buy more bonds, for example.

So what is the optimal mix of asset classes we should have? To this Bernstein replies, “you might as well ask the meaning of life”. The problem, as the book so beautifully describes, is that we can’t predict the future. We don’t know which asset classes will do well and which will do poorly. And holding an asset that never rises will not help your portfolio one bit, even if it is completely uncorrelated with your other assets! Still, even slower-growing asset classes that are uncorrelated with the rest of your portfolio can be beneficial to have. Consider that in the graph I drew above, adding a small amount of bonds to an all stock portfolio decreased risk with almost no sacrifice in returns.

# What should you do?

Honestly, if this post gave you a headache and you really don’t want to waste any more time thinking about money, **just stick with 100% Total US Stock Market **through Vanguard’s cheap, easy Total US Stock Market Index fund (VTI or VTSAX). You already know how the stock market works, why volatility doesn’t matter, and what to do when everyone is freaking out. The US has the strongest, most diverse economy in the world, and holding “only” (aka 3600 companies) US stocks is a perfectly fine approach.

What do I do? 80% Total US stock market, 10% all world minus US stock market (VEU), 10% Total US bond market (BND). The 80% represents my strong belief in the principles in the paragraph above. The inclusion of the other two is for the theoretical asset allocation benefit. But who knows, maybe 100% stocks will have been a better approach in hindsight.

# What do I think of the book?

Honestly, I loved it. If you have any interest in finance, math, or just learning about a very important topic, you should check it out (literally, at the library). What I highlighted here is just a small part of it. But the book is filled with gems about how the stock market works, why we can’t predict the future, and how paying high money management fees will crush your long term returns. Maybe I’ll do a follow up post on some of these topics depending on how this post is received.

# RIP John Bogle

Really, all of this is possible because of John “Jack” Bogle–the man who founded Vanguard and the granddaddy of index funds, who died earlier this week. If it wasn’t for him, the high fees we’d pay money managers to hold all these different asset classes would completely wipe out any additional benefit we derived from doing it. The man made it possible for any average Joe to be a top 1% investor by simply holding the entire market in a low-cost index fund. He also led a pretty incredible life, which you can read about here.

It’s amazing how difficult it is for a man to understand something if he’s paid a small fortune not to understand it.

— John Bogle, on stock brokers