The standard advice for personal investing that I see all around the web is to put your money into one or more low cost equity index tracking funds. Commentators also sometimes recommend an allocation to bonds (e.g. a 60/40 split between stocks and bonds), though the popularity of this advice seems to become less common with every passing month of the bull market.

However, the more I learn about investment, the more I come to think that this answer is suboptimal. To see why, let's consider a simplified world where we have exactly two assets to which we can allocate our wealth: stocks and bonds.

### Portfolio Theory

The historical evidence (see e.g. the excellent book Expected Returns) is that the returns on bonds and equities are uncorrelated, with bonds having lower volatility (aka standard deviation) than equities — US treasuries experienced an annualized volatility of 4.7% a year between 1990 and 2009, while US equities had a volatility of 15.5% over the same period.

Given their lower volatility, bonds are clearly less risky than equities. However, you would hope that if you invest in equities rather than bonds, then your willingness to accept the inherently higher risks is compensated for by a higher expected return. This idea of can be captured mathematically as the Sharpe ratio, which measures the reward you receive per unit risk taken. Specifically, the Sharpe ratio `S` is equal to the ratio between the expected "excess return" of the investment, and the standard deviation of those returns. The excess return is defined as the amount of the expected return `R` above a risk-free rate `R _{f}` (e.g. the rate of return you can get by lending overnight in the money markets). Putting it all together we get this formula for

`S`:

All other things being equal, you probably want to invest in assets with as high a Sharpe ratio as possible.

It can be tricky to figure out what the Sharpe ratio is for investments of interest, but history suggests that stocks and bonds both have similar Sharpe ratios of roughly 0.3. What's more, the correlation between their returns is close to 0. This last fact is important because portfolio theory shows that you can form an investment with high Sharpe ratio by holding a diversified portfolio of two or more uncorrelated assets with low Sharpe ratios. Assuming that historical returns, volatilities and correlations are good guides to the future, the portfolio of stocks and bonds with highest Sharpe is one that holds roughly $3 of bonds for every $1 of stocks: this ratio comes about because the volatility of stocks is approximately 3 times that of bonds. This optimal portfolio has Sharpe ratio of about 0.42 i.e. 1.41 (`= sqrt(2)`) times higher than that either asset class by itself.

You can get a feel for how this optimization process works by playing with my online portfolio theory tool.

### Leverage

What's striking about this optimal portfolio is that it's very different from the normal advice: it puts 75% of capital into bonds, much more than even the most conservative conventional advice of a 40% allocation. The obvious objection to the bond-heavy optimal portfolio is of course that it will have very low expected return compared to one with a bigger weighting on equities. This is absolutely true: we expect the volatility of the optimal portfolio to be around `0.75*(bond volatility) + 0.25*(stock volatility) = 0.75*4.7% + 0.25*15.5% = 7.4%`. Because this is roughly half the volatility of equities by themselves, we'd expect the excess return on the portfolio to therefore only be about `(0.42/0.3)/2 = 1.41/2 = 0.7` times that of a pure-equity allocation, which definitely sounds like bad news for the optimal portfolio.

However, this is a solvable problem — to recover the high expected returns we desire, we simply have to borrow money to invest greater notional amounts into the portfolio. If we borrow, then using 2x leverage (i.e. borrowing so as to invest $2 for every $1 of capital we actually control) would scale the volatility on our portfolio up from 7.4% to the equity-like levels of 14.8%. If we assume that we could borrow at the risk-free rate, then because our portfolio has Sharpe ratio higher than that of plain equities, the expected excess return in this scenario would be 41% **higher** than that of equities alone. So we earn better returns than with a pure-equity play even though we are running similar risks.

Everyone has probably been told at some point that diversification is good, but the way it is usually explained is by saying that diversification reduces your risk, which sounds worthy but sort of boring â˜º. When you realise that this risk reduction means that you free up some "risk budget" which you can use to achieve extra returns via leverage, diversification starts sounding more exciting!

Of course, in reality we are unlikely to be able to borrow at the risk-free rate, but there are ways to borrow that are only just slightly more expensive than this — depending on the currency, companies and investment funds regularly borrow at rates as low as 0.5% above the risk-free rate. Even as an individual investor, there are ways in which you can borrow quite cost-effectively — this is a topic I will cover in a future post. The cost of leverage is of course a constraint that we should bear in bind, though: the fact that we face lending costs rules out activities such as levering up short term bonds (e.g. US treasury bills) which have very low volatility (because they take on very little interest rate risk).

### Risk Parity

The approach to investing outlined above can be roughly summarized as:

- Assume that all investible assets have the same Sharpe ratio
- Therefore decide to allocate to them in an amount inversely proportional to their volatility (following the advice of portfolio theory and mean-variance optimization for Sharpe ratio maximization)
- Leverage up the resulting portfolio to achieve a particular desired level of risk

This method is also known as "risk parity". Famously, it's the strategy used by Bridgewater's All Weather hedge fund, which has returned a Sharpe of roughly 0.5 since inception in 1996. (All Weather invests in asset classes other than stocks and bonds, so we would expect it to have a higher Sharpe than our earlier prediction of 0.42, simply due to the extra diversification.)

What's particularly interesting about risk parity is that it's not actually immediately obvious that taking bigger risks with your money leads to a sufficient extra level of return to compensate you for those risks. For example, take the case of stocks and bonds. From 1990 to 2009, US equities returned a (geometric) mean of 8.5% per year while treasuries returned 6.8%: so equities did earn a higher return, but one that doesn't seem commensurate with the 3 times higher volatility experienced. Furthermore, global equities (which had similar volatility to US equities) actually only returned 5.9% i.e. considerably less than US bonds!

The fact that risk-taking is under-compensated is actually a well-known anomaly: an excellent paper on the subject is Betting Against Beta which suggests the reason may be because many investors are either unwilling or unable to use leverage. Whatever the cause, it's good news for risk parity, because this means that the low-risk assets you are levering up actually have a higher Sharpe ratio than the high-risk assets that you (relatively speaking) disprefer, so your portfolio's Sharpe ratio will be even better than you would naively expect. This subject is explored further in this readable paper on risk parity from AQR.