## Stocks, Bonds, and Risk

I learned some important lessons about risk recently while reading William Bernstein’s excellent The Intelligent Asset Allocator.

For our discussion we’re going to need historical annual returns for both the US bond market and the US stock market. The easiest way to get these is to use the freely available historical returns for index funds that closely mirror both markets.

VBMFX is an index fund that tracks the entire U.S. bond market.
VTSMX is an index fund that tracks the entire U.S. stock market.

From here forward, whenever I write about bond market performance, I’m referencing the performance of VBMFX. The same goes for the stock market being represented by VTSMX.

Let’s begin by examining the annual returns for the stock and bond markets over the last 10 years.

Market 2006 2005 2004 2003 2002 2001 2000 1999 1998 1997
US Stocks 15.5% 6.0% 12.5% 31.4% -21.0% -11.0% -10.6% 23.8% 23.3% 31.0%
US Bonds 4.3% 2.4% 4.2% 4.0% 8.3% 8.4% 11.4% -0.8% 8.6% 9.4%

Simple arithmetic on the above numbers yields…

Market Geometric
Mean
Arithmetic
Mean
Arithmetic
Std. Dev.
US Stocks
8.6%
10.1%
17.7%
US Bonds
6.0%
6.0%
3.6%

The geometric mean gives us what yearly average return we would have achieved if we made an initial investment 10 years ago, and then left the money in to compound. In this case, US stocks would have grown our money at 8.6% per year versus US bonds delivering 6.0%.

But what we’re more interested in for this article is the arithmetic mean and its standard deviation. This mean tells us what return stocks and bonds average in any particular year, and the standard deviation gives us an idea as to how far the individual yearly returns will usually be from that mean.

Assuming annual returns follow roughly a normal distribution (assumption based upon the central limit theorem), and assuming future distributions match the past, 68% of the time any future yearly return should be within +/- 1 standard deviation from this mean.

So for US stocks, usually (that is, 68% of the time) the yearly return should be between -7.6% and 27.8%, and average 10.1%. Similarly the yearly return of US bonds should usually be between 2.5% and 9.6%, and average 6.0%.

And the numbers above show us why bonds are generally less risky than stocks. We can predict with greater accuracy what their rate of return should be in any given year due to their low standard deviation.

Here’s where we get to the first not-so-intuitive point about risk. From the above data you might conclude that those who are risk-adverse (but want to invest in stocks and/or bonds) should skip stocks and put 100% of their money into bonds in order to have the most predictable returns each year.

But that conclusion would be wrong. At least based upon what has happened historically.

Take a look at what we get when we compute the mean and standard deviation for portfolios that are all bonds, all stocks, and a mixture of the two.

Stock
Allocation
Bond
Allocation
ROR
Mean
ROR
Std.Dev
100%
0%
10.1%
17.7%
90%
10%
9.7%
15.8%
80%
20%
9.3%
13.9%
70%
30%
8.9%
12.0%
60%
40%
8.5%
10.1%
50%
50%
8.1%
8.3%
40%
60%
7.7%
6.5%
30%
70%
7.2%
4.8%
20%
80%
6.8%
3.5%
10%
90%
6.4%
3.0%
0%
100%
6.0%
3.6%

And here’s a graph of the data above:

As expected, we see both the average and standard deviation of the annual return drop as we progress from an all-stocks portfolio to all-bonds. But look closely at the last 3 rows. The portfolio with the lowest standard deviation is not all-bonds, but rather 90% bonds and 10% stocks!

What’s more, with this portfolio you achieve a slightly higher average return too! That’s an example of why diversity is sometimes referred to as the only free lunch in finance. ðŸ™‚

But what about the aggressive, risk-tolerant investor – does 100% stocks make sense? Well according to our data, 100% stocks does indeed achieve the highest return, but it’s also important to see that even a small about of bonds can significantly reduce risk while only marginally reducing rate of return.

Above we see that roughly for every 1% of average return that an investor is willing to give up, risk is reduced not also by 1% but by 5%.

Not a free lunch in this case, but a real bargain nonetheless. ðŸ˜‰