**The Role of Bonds**

In my view, simpler is better when it comes to investing. I think a 100% allocation to an S&P 500 index fund (or total U.S. stock market fund) is a perfectly reasonable retirement strategy for most individual investors. More complicated strategies are too often time-consuming, speculative, and prone to substantial losses due to trading costs and other expenses.

But if you're nearing retirement or for some other reason cannot tolerate the full volatility of the S&P 500, adding exposure to bonds is a great way to reduce risk while preserving as much growth potential as possible.

**Choosing a Bond Fund**

Keeping with the idea of simplicity, I usually prefer combining an S&P 500 index fund with a single bond fund. With two funds, it's still relatively easy to look at historical data and understand how various allocations affect characteristics of the fund (e.g. expected returns, volatility, Sharpe ratio).

Of course, there are thousands of bond funds out there, from Fidelity, Schwab, Vanguard, and so on. Regardless of which company you go with, the most important aspect of the bond fund is its average duration. In general, longer duration bond funds have greater yields, higher expected returns, higher volatility, and lower Sharpe ratios than shorter duration bond funds. Longer duration bond funds are also more sensitive to changing interest rates.

**Two-Fund Portfolio Optimization**

Let's consider a scenario where we want to combine Vanguard's S&P 500 mutual fund, the Vanguard 500 Index Fund Investor Class (MUTF:VFINX), with either the Vanguard Total Bond Market Index Fund Investor Shares (MUTF:VBMFX) or the Vanguard Long-Term Bond Index Fund (MUTF:VBLTX).

VBMFX has an average duration of 5.8 years and an SEC yield of 2.33%, while VBLTX has an average duration of 14.7 years and an SEC yield of 4.07%.

In general, I think the best way to build a portfolio is to define your maximum level of tolerated volatility, and then choose the funds and allocations to maximize expected returns (or equivalently Sharpe ratio) at that level of volatility.

The figure below shows historical Sharpe ratio vs. volatility for VFINX/VBMFX and VFINX/VBLTX portfolios with various VFINX allocations, using data from Feb. 28, 1994, to Dec. 31, 2015.

The data points on the graph represent 10% VFINX allocation increments. For both curves, the rightmost point is 100% VFINX, the second rightmost is 90% VFINX/10% bond fund, and so on to the other endpoint which is 0% VFINX/100% bond fund.

So if you are willing to tolerate half of the volatility of the S&P 500, or a SD of about 0.6%, you can see that the maximum Sharpe ratio is achieved on the blue curve with approximately 50% VFINX, 50% VBLTX.

VFINX/VBLTX beats out VFINX/VBMFX over the volatility range (0.49, 1.20). So as long as your maximum tolerated volatility is 40.8% of the S&P 500 or greater, you're better off using the long-term bond fund VBLTX rather than the total bond fund VBMFX. For lower risk levels, VBMFX is the better choice.

**Avoiding the Dip**

Naturally, as you reduce your level of tolerated volatility, you'd like for your Sharpe ratio to increase. That's the whole point of using bonds rather than cash: to improve risk-adjusted returns while reducing risk, rather than keep risk-adjusted returns constant.

But notice in the previous figure that at a maximum tolerated volatility of 0.49%, your best possible Sharpe ratio is 0.067 with 25.9% VFINX/74.1% VBLTX (the highest and leftmost point on the blue curve). Just to the left of that, say at a volatility of 0.47%, you have to move to the red curve, where the best you can do is a Sharpe ratio of 0.062. Notice the "dip" in Sharpe ratio that occurs on the volatility interval (0.42%, 0.49%).

Luckily, you don't have to move to the red curve here. The better solution is to "lock in" the Sharpe ratio of 0.067 with 25.9% VFINX/74.1% VBLTX, and simply add a cash allocation to reduce the volatility to 0.47%.

Graphically, adding a cash allocation to VFINX/VBLTX shifts the entire curve to the left. Take a look:

Above, the rightmost gray curve is 5% cash, and the remaining 95% in various allocations to VFINX/VBLTX; the next is 10% cash, and the leftmost is 15% cash.

With cash, the dip is gone. Now, you can use cash/VFINX/VBLTX rather than VFINX/VBMFX to achieve a higher Sharpe ratio on the target volatility interval (0.42%, 0.49%). But once you reach 0.42% and lower volatilities, VFINX/VBMFX still wins out.

**A Better Solution: Blend the Bond Funds**

The gray curves on the next figure show Sharpe ratio vs. volatility for VFINX/VBLTX portfolios with various fixed allocations to VBMFX rather than cash. Graphically, adding a VBMFX allocation shifts the VFINX/VBLTX curve upwards and to the left.

To clarify, the rightmost gray curve is 10% VBMFX and the remaining 90% in various allocations to VFINX/VBLTX; the next is 20% VBMFX, and so on until the leftmost gray curve which is 90% VBMFX.

Now, not only is there no "dip" in the volatility range (0.42%, 0.49%), but as you move to the left from 0.49% you continue to increase the Sharpe ratio rather than decrease it (by switching to VBLTX) or keep it the same (by adding cash). This is what you usually expect when you take on less risk: better risk-adjusted returns.

Also, now over the entire volatility range (0.25%, 0.49%) you can achieve a higher Sharpe ratio using both bond funds rather than only VBMFX.

In summary, as much as I like the simplicity of a two-fund stocks and bonds portfolio, you can often do much better by using two bond funds rather than one.

**Conclusions**

Combining stocks and bonds is a great way to build a portfolio that maximizes expected returns for a given level of risk. For many investors, pairing an S&P 500 index fund with a single long-term bond fund is the best way to do that.

However, more risk-adverse investors, say those who can only tolerate 30% of the S&P 500's volatility, may have to use a short-term, intermediate-term, or total bond market fund to achieve their target volatility. The main finding of this article is that such investors can do much better by combining a long-term bond fund with a shorter-term one to achieve their target volatility, rather than using the shorter-term one alone.

In fact, the allocation to the long-term bond fund should be as high as possible. You only want as much of the shorter-term bond fund as is necessary to reduce your risk to your target level. Remember, adding a shorter-term bond allocation shifts the Sharpe ratio vs. volatility curve upwards and to the left, similar to how adding cash shifts the curve to the left. You only want to shift the curve as much as is necessary to reach your target volatility.

In the bigger picture, I would now recommend 100% S&P 500 index fund for most investors not near retirement; a two-fund S&P 500/long-term bond fund for investors willing to sacrifice some growth potential for better risk-adjusted returns; and a three-fund S&P 500/long-term bond fund/shorter-term bond fund for conservative investors, such as those nearing retirement or already retired.

**Disclosure:** I/we have no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours.

I wrote this article myself, and it expresses my own opinions. I am not receiving compensation for it (other than from Seeking Alpha). I have no business relationship with any company whose stock is mentioned in this article.

**Additional disclosure: **The author used Yahoo! Finance to obtain historical stock prices and used R to analyze the data and generate figures. Any opinion, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.