# Bond versus Stock Portfolio Calculator

## Bond versus Stock Portfolio Calculator

When I was doing my finance courses in graduate school I had very little interest in stock and bond securities and didn’t know why I needed to know something about them (I was a marketing major at that time). Since then however I have been gainfully employed most of the time and have seen a good amount of my retirement money going into stocks and bond mutual funds and have seen my account balances gyrating to the tunes of the market. So now I am more interested!

I do recall from graduate school that often a risky investment (like a stock mutual fund) combined with a less risky investment (like a bond mutual fund) can produce a combination that is less risky than either and there is such a thing as an “optimal” risky portfolio. This is because investments such as stocks and bonds do not move in tandem –one might have a compensatory gain while the other is losing. These key results on combining investments were due to a Harry Markowitz who won a Nobel Prize for his efforts.

Two things have whetted a recent interest in portfolio theory.

I want to see, what, based on the history of the markets, would now be a good combination of stock and bond mutual funds. I use the S&P Index as the stand-in for stock mutual funds and the Barclays Aggregate Bond Index as a stand-in for bond investments.

As the second thing, I want to see if I could offer my two bits to the debate on the use of market bonds and stocks in the social security mix. I add in social security trust fund returns to the Bond and stock indexes to see if there are any possible interesting correlations with the social security trust fund returns which would make it attractive to add in the market securities.

The portfolio theory involves a trade-off between the return of an investment and it’s risk. The standard deviation is a measure of risk and tells us roughly how far lower or higher a typical return of an investment strays from the mean return. It is obtained as the square root of the variance.

The various qualities of a mix of investments are identified in the graph below. The blue triangles represent the Return-Risk co-ordinates of the S&P Index and the Bond Index. The C-shaped curve, called the Markowitz Frontier, traces the return-risk boundary of all possible combinations of stocks and bonds. Thus the upper portion of the curve provides the minimal standard deviation for any given return rate. (If you are interested in the derivation of the C curve and other points on the plot you could refer to The Calculating Investor – this blog was very helpful to me in my calculations). Points inside the C curve represent the risk, return co-ordinates of feasible combinations of the stock and bond indices.

The left tip of the C-curve is the one with the lowest risk and is appropriately referred to as “Global Minimum Variance Portfolio”.  Notice that this minimum risk combination of stocks and bonds has a lower risk than both the stock index and the bond index and it has a return higher than the bond index. So you get a free lunch, if historical correlations and returns hold, by adding in some stocks to a purely bond portfolio – the return increases and the risk decreases. You will see for the default example in the calculator that a portfolio with 6.05% in stocks, given continuing historical patterns, is likely to outperform a purely bond portfolio on both risk and return.

A little higher on the curve is the “Optimal Risky Portfolio”. One could combine it with a risk-free investment (CD’s or treasury bills) to get any desired return on the “Capital Allocation Line”. The capital allocation line goes from a co-ordinate corresponding to a return of 5% for a risk of 0% (hence risk free – I choose a somewhat higher rate to get a nicer graph, you could try more realistic rates in the calculator in this page) through the co-ordinate with risk = 7.41% and return = 9.17%. This is where the line is tangential to the Markowitz C-curve.  The line extends beyond the optimal risky portfolio co-ordinate if one could borrow at the risk free rate and invest additional borrowed amounts as well in the optimal risky portfolio. The calculator shows that this optimal risky portfolio has an empirically derived ratio of 23.5% in stocks.

Even higher on this C-curve is the “Maximum Utility Curve”. A utility curve is specified by the investor’s degree of satisfaction (or “utility”) with an investment strategy and the investors risk aversion. For a given risk aversion one has distinct non-intersecting utility curves which move upwards with increasing satisfaction or utility. The maximum utility portfolio is the highest curve which intersects the Markowitz C-curve. The risk aversion measure believed to be between 2 to 4 for the US population (see “Investments” by Bodie, Kane and Marcus, 2005, McGraw-Hill/Irwin – I drew on this reference as well for many of the concepts/computations), where aversions closer to 2 are those for aggressive investors (those who might prefer stocks) and closer to 4 for conservative investors. The calculator uses a risk aversion of 4 as default and empirically derives, based on historical patterns, a maximum utility portfolio consisting of 39% in stocks.

The default calculations of the minimum variance portfolio, the optimal risky portfolio and the maximum utility portfolio for a combination of stocks and bonds use a risk free rate of 5%, a risk aversion of 4 and use historical patterns derived from returns from 1977 to 2011. These results are in the second box from the bottom of the calculator. These numbers correspond to the graphic above.

In addition the default calculator, in the bottom box, looks at combinations of social security trust fund investments and market stocks and bonds. One thinks of social security as almost risk-free but it does have risk as conventionally defined in finance (through variability over time). If you look at the second tab to my calculator social security returns have ranged from a high of 11.6% in 1984 and a low of 4.4% in 2011. As noted earlier if the returns of social security, stocks and bonds do not move in tandem then there is a possibility of gaining from combinations.

Correlations measure this tendency to move in tandem, and is a measure going from -1 for perfect negative association to +1 for perfect positive association. The correlations from 1977 to 2011 of stocks, bonds and social security returns are in the first box of the first tab of the calculator.  Stocks have a low correlation with both bonds and social security returns which might support a mix of these. However social security has a very low risk and the minimum variance portfolio has negative allocations for bonds and stocks. The negatives can be achieved through what is called a shorting strategy. To achieve -1.4% in stocks, -7.6% in bonds and 109% in Social security trust funds, the social security trust would have to borrow \$1.40 worth of stocks and \$7.60 worth of bonds for every \$100 it receives in contributions, sell these stocks and bonds and invest the total of \$109 as it usually does in it’s non-market securities. This minimum variance portfolio has less risk (marginally) but also has a lower return. A similar shorting strategy would need to be used for the optimal risky portfolio.

The two portfolios above derive an objective mix of investments without entirely considering the investors satisfaction with his investment mix or his risk aversion. For a risk aversion of 4, the optimal mix for a maximum utility portfolio is 35.4% in stocks, 9.5% in bonds and 55.1% in conventional social security investments. With this investor satisfaction/risk aversion based mix we end up with more than thrice the risk (2.1% for current social security investments versus 6.69% for the maximum utility mix) and a gain in returns from 7.69% to 9.32%, again assuming historical patterns hold.

If reducing risk is the dominant objective then the resulting shorting strategies imply that social security as currently invested is just fine. A defined benefit program (as social security is) may be a better idea than a defined contribution plan (like 401-K) managed by those contributing to the plan, especially, since social security had originally been formulated as a social safety net. If all else fails (including our defined benefit plans which depend on the performance of market securities) then the government assured benefits would be all we could rely on.

However if personal choice in investing trumps security then there is an argument, through the not inconsiderable stock component in the maximum utility mix, for a portion of the funds to be available to investors to make their own choices. However personal choice would lead to a tripling of the risk with only a modest gain. The increased volatality could lead to an erosion of wealth through improper choices such as moves out of stock at troughs and reentry near peaks. (many of us may have done that instead of sticking it through an extended bear market till a recovery — I know I have!).  Given the increased volatality one would need, in a context allowing some choice, a portion backed by government defined benefits to meet the needs of the poor and the financially challenged/illiterate.

Edit the blue cells in the spreadsheet and enter your data and the calculations in the bottom box of the spreadsheet will refresh.  Be sure to read the caveats on using this calculator that follows the calculator.

The “optimal” mix, when looking at stocks in combination with bonds, can sometimes  favor conservative investments (favoring bonds over stocks), especially for young investors. Our investment professionals however tend to recommend a portfolio mix which is sometimes a lot less conservative. This is because investment professionals assume that you have a multi-period investment horizon while the Markowitz model is a one period model. The investment professional’s argument is that you may lose in the long run with too conservative an investment mix – one bit of advice I have heard is to invest your Age as a percentage in bonds and (100-Age)% in stocks. Thus a 25 year old investor would invest 75% in stocks. This presumes that you can ‘invest for the long term’ and that you can stomach the churning financial markets till you retire. However, if you want to be relatively safe and optimal this coming period then you might choose the Markowitz “optimal” mix.

This portfolio theory can result in investment strategies that only hedge fund managers can understand and some of these complexities are sometimes not divulged when this topic is presented. In the interest of full disclosure I would like to add that I also tried to come up with an optimal mix based on data from 1989 to 2011 on the returns of the S&P index, Barclay’s aggregate bond index, Russell 2000, MSCI emerging markets and MSCI EAFE. This resulted in a complex shorting strategy for all three portfolios and I settled for a simpler 3 investment mix, despite which, I encountered some shorting strategies. There are methods which do not allow for shorting strategies (see the Bodie, Kane and Marcus reference I mentioned earlier).

One statistical caveat to all the calculations presented in these pages are that they presume that the inputs such as the returns, standard deviations and correlations are good estimates of the what you might expect to hold in your investment horizon. However because we have a limited (100 years or so) data on financial markets, because we don’t know which 15-20 year period statistics best reflect markets in the coming periods and  because we can’t tell what the future holds, estimates, can be off.

Statistician’s like to distinguish between what is merely an estimate and a true state of nature – what we call a parameter. We do that by putting a hat (the symbol ^) for the estimate on our parameter’s symbol in order to make that healthy distinction. My calculations, while being presented to a number of decimals, are only meant to help you make a crude assessment based on crude estimates. This reminds me of a light bulb joke which goes “How many statisticians does it take to change a light bulb?” – The answer is 1.163 +/- 2.564!

If one attempts to do it right then one must think through the parameter by gauging what the market might be like in your investment horizon. Then put a hat (I give you some to choose from below!) on this parameter and see if you can get that estimate for it in the data of the not too long history of the financial markets. Good Luck!

Or as most investment professionals put it – “Past Performance may not be a good predictor of future results”.

Hats of Many Sizes and Colors.