Risk is the chance that our investment will not perform as intended. Why the investment underperformed could be due to an infinite number of possibilities from the mundane to the unimaginable. Some risk affects many different industries and has broad effects, such as rising interest rates. Some risks effect only one company, such as poor management.

The higher the risk - the higher the "expected" reward. Back when we looked at returns for speculative real estate, we saw that the risk we were taking on required a 20% return on the money invested. This is much higher than what is typically expected from the stock market, an average of 10%. The additional 10% premium is being required due to the higher risk for the specific real estate from high leverage, illiquidity, and management risk. Had we paid all cash for the real estate, the required return would only be around 8%. The higher the risk of an investment, the more return the market will require in order to receive any investment. This explains why investors are willing to receive less than 5% to invest in U.S. Treasuries. Being backed by the United States Government (and our tax dollars), they are as close to being a riskless asset as an investor can get.

To control our exposure to risk, we must diversify our portfolio of investments. The basic idea behind diversifying is that a loss in value in any single investment will not have such a large impact on a diverse portfolio. Intuitively we understand not to put all of our savings into one stock or one sector of the stock market, such as tech stocks. However, we have still not truly diversified our investment. Stocks tend to move with the market at least 65% of the time. Although there may be some diversification within the stock selection, we are still exposed to risks that would affect the entire stock market, known as beta risk. True diversification requires us to spread our investment capital outside the stock market into other markets such as bonds, cash, and real estate. The net effect is that higher risk investments are dampened with lower risk investments. We are trying to turn a roll-a-coaster into a merry-go-round.


Portfolio Theory

There have been numerous volumes of work done on portfolio theory. Here we are going to touch on the basics to illustrate how portfolio theory can work for us, the small investor. The essence of portfolio theory is allocating funds to investments that maximize returns while limiting the volatility (risk). It is up to the investor to determine the level of volatility they are willing to take for a given level of return. The higher the return, the higher the volatility (risk).

Volatility in an investment is measured from the standard deviation of historic returns. Technically the standard deviation is the square root of variance - a measure of dispersion of data around the mean (please reference a good statistics book). The higher the standard deviation, the more a value moves above and below the average value - higher risk. For example, Stock "A' with historic share values of $100, $75, $10, and $15 has an average of $50 with a standard deviation of $44.5. Stock "B" with historic share values of $60, $40, $45, and $55 also has an average of $50, however; the standard deviation is only $9.12. Although the average share values average the same, we can see that Stock "A" is over five times as volatile as Stock "B". Stock "A" is more risky. The same measure of volatility can be done for markets. The stock market has a volatility of about 15% with return expectations around 10%. Bonds have a volatility of 8% with a return expectation around 6%. Real estate fits in the middle with a volatility of 10% and an expected return of 9%. All of the market volatilities and returns depend on the period that they are measured. The best bet is to verify the volatilities and returns with major financial websites and see where the consensus is. The expected returns and volatilities can change depending on the period under consideration.

Plotting the risk vs. reward on a chart for the stock, bond, and real estate markets (See Figure 1 below), we can see how each market compares to each other. Risk is plotted on the horizontal axis, while the reward labeled "expected returns" is plotted on the vertical axis. Each market is represented by a point on the graph. Stocks', being the highest risk and reward, is far to the upper right of the chart. Real estate is in the middle, with bonds at the bottom left of the chart. Plotting the potential investments gives us a quick way to select the better performing investments considering risk.


Figure 1, Chart comparing risk vs. reward.
A chart showing different investments on a risk vs. reward chart.

Suppose along with our stocks, bonds, and real estate we were considering two additional investment types X and Y. On the chart, we would draw vertical and horizontal axis through each point. Next, we look to see where the investments lie with relation to the axis we drew for each point. If another investment was in the upper right quadrant of the grid of the point we were looking at, the other investment dominates. An investment dominates another when it is producing a higher return at a lower risk. Any investment occurring in the lower right quadrant is being dominated - it produces a lower return for higher risk. To illustrate, let us look at our chart. After putting an axis on each point, we notice that bonds, real estate, X, and Y are a bit clumped together. We are only interested in dominant investments, so we begin by checking on bonds. No other points in the upper-left quadrant, therefore bonds are the best risk-reward point for its place on the chart. Now let us see how real estate is doing. We see that real estate is in the upper right quadrant of point Y, so it dominates Y. However, point X is in real estate's upper right quadrant. Point X is dominating real estate. We would choose investment X over real estate and investment Y. Stocks are the dominate investment over in the upper right region of the chart. From the group of investments being considered we would choose to invest in bonds, investment X, and stocks.

Portfolio Allocation

Now that we know what we are investing in, we need to determine how much money to place into each investment - our portfolio allocations. Generally, we are looking to maximize our return for our chosen level of risk/volatility. Furthermore, we want to choose portfolios that are not dominated by any other combinations (portfolios) of the same assets. Let us consider a portfolio of stocks, bonds, and real estate (housing futures). The expected returns and volatility for each is listed in Table 1.


Table 1, Investments in the portfolio
Portfolio
Return Volatility
Cash 4.0% 0.0%
Bond Fund 8.0% 8.0%
Housing 9.0% 10.0%
Stock Fund 12.1% 15.0%

For each investment, we have the expected return and the volatility. We then need to enter the correlation coefficient between the investments. The correlation coefficient is the measure of how well the investments move together over a period with a value ranging from 1.0 to (-1.0) or +/- 100%. The higher the correlation the more the investments move together in unison. A lower coefficient would mean the investments are less likely to move to together. To diversify we would like to choose investments with low correlation coefficients. We could compute the correlations ourselves, but it is probably best to see and use what the major financial institutions and websites are using. The correlations change over time and from period to period, so we need to verify the values every few months to be safe.


Table 2, Correlation between investments
Correlation
Stocks Bonds Real Estate
Stocks 100% 50% 25%
Bonds 50% 100% 0%
Real Estate 25% 0% 100%

We set up the correlation table (See Table 2 above) to help us in the next couple of steps to determine the optimum portfolios. From the correlation table we can see that stocks, bonds, and real estate are 100% correlated with themselves - easy enough. Stocks and bonds have a coefficient of 50%, so they move together about a half of the time. Real estate and stocks have a very low correlation of 25%. Real estate and bonds generally not very correlated, so we plug in 0% [Note: correlations between investments change depending on the period under consideration. The example has been set up with values that may not correspond to current ratios]. Now we compute the covariance between the variables as in Table 3 below.


Table 3, Covariance Expectations
Covariance
Stocks Bonds Real Estate
Stocks 0.02250 0.00600 0.00375
Bonds 0.00600 0.00640 0.00000
Real Estate 0.00375 0.00000 0.01000

The covariance measures how well the investments move together over a certain period. More importantly, the joint movement is considered the risk that cannot be diversified away, systematic risk. We will compute the covariance from the correlation coefficients using the formula:


COVijσ iσ j


COVij is the covariance, and Cij is the correlation coefficient. Sigma, σ is the standard deviation for the asset also known at the volatility that we plugged into our chart earlier. To compute the covariance in the table, it is easier to think of the data as if we are comparing two different instances of investment for each value. For example, to compute the stock covariance of 0.0225, the calculation is, COVij = (1.00)(.15)(.15). The correlation of a stock to a stock is 100%. Since we are "comparing" two stocks, each would have a volatility of 15%. Other covariance calculations are more straightforward. Between stocks and bonds, the variance is computed by COVij = (0.5)(.15)(.08) for the table value of (.006). All the combinations of investments have their coefficients computed, then complied into a table as previously shown.


Figure 2, Efficient frontier.
A chart showing the efficient frontier from the three investment classes given.

On the chart in Figure 2 above, we see our investments plotted as points on the risk-reward chart. Additionally we see curves between our investment points. The curves represent a portfolio between the two assets. For example, the curve between bonds and real estate represents a portfolio of 100% bonds at the bond point. As we trace up the curve toward the real estate point, the portfolio is reducing the amount of bonds and increasing the amount of real estate investment. Once we reach the real estate point, the portfolio consists of 100% real estate investments. What the curves show is that a combination of assets can provide a return that is higher than the least risky asset, but lower than the most risky asset. On the bond-real estate curve, we see that the Y-axis values for return are always greater than bonds, and always less than real estate. Most importantly, the curve also shows that a combination of the investments can also be less risky than the individual investments. In the case of a portfolio of bonds and stocks, it seems the risk behaves similar to reward - always in between the original investments. Remembering that lines are composed of points of data, any curve that is in the upper right quadrant of a point we are considering on another curve is dominant. Similar to our pick of investments, we desire that our portfolio composition of investments be the dominant portfolio at the level of risk we choose. The portfolio should have the highest return for the lowest volatility (risk). This is the goal of mean-variance portfolio theory (MPT). Taking into consideration how each investment behaves (volatility) and relates to each other (covariance), we solve for the dominant portfolio. Utilizing the MPT, we can compute the risk of the "optimized" portfolio at each level of "expected" return. If we plotted the results, we would end up with the efficient frontier as plotted on our chart. The efficient frontier represents the risk-reward curve of the dominant portfolios. Looking at the curve, we notice that along the efficient frontier we obtain superior returns at a lower risk than the individual investments. In addition, a portfolio composed of all three of the investments dominates the portfolios consisting of just two of the investments.

To solve for the portfolio allocation using mean-variance portfolio theory (MPT), we are going to need a solver that solves will run the computations until it can find the answer (if there is one). Most spreadsheet software has a solver included or that can be easily installed if they were not part of the original installation. In order to help our solver to calculate results, we need to set up a few items.

First, we need to put in some "dummy" variables for the allocation of each investment. Next, the weighted covariance for each investment pair needs to be solved. The weighted variance takes into account how much the investments move together, but in relation to how much of each investment is made. To compute the weighted covariance we multiply the covariance by the amount of the investment bought in each category. For example, the weighted covariance for stock and real estate in the table is 0.000613. It is found by multiplying the covariance (0.00375) between stocks (39.7%) and real estate (41%) by their allocations. When determining the weighted covariance between an investment and itself (i.e. stock-stock, bond-bond, etc.), we must square the value of the allocation. In order to compute the weighted covariance for the stock-stock relationship (0.003546), the computation is (39.7%) x (39.7%) x (0.0225). With our weighted covariance chart set up, we can solve for the portfolio's risk. The variance of the portfolio is simply the sum of all of the weighted covariance pairs. To compute the risk we take the square root of the variance, which is also the standard deviation of the portfolio. The last computation we need to set up is the portfolio return. To calculate the return, we take each of the investment allocations, multiply them by their expected return, and then sum them up. With all of our computations in place, we can turn to using the solver.

Our goal is to determine the investment allocations to meet our required return at the lowest risk. To do so we need to set up the goals and constraints for the solver. First, we want the minimum risk value - set the solver to find the minimum. The variables the solver will manipulate will be our allocations to stocks, bonds, and real estate. Now we need to apply appropriate constraints. The solver needs to make sure our allocations in all investments add up to 100%. The allocations in each investment must be less than 100% and greater than zero (you could have negative allocations that represent short positions). Otherwise, the allocations will be greater than 100% and perhaps negative. Finally, our calculated return for the portfolio needs to match the required return. When we run the solver, it should produce a reasonable allocation of investments (See Figure 9.1 C). Be aware that not all levels of return will have an answer - we can only make as much as the highest returning investment. If we run several iterations and plot the results on a graph of risk verses reward, we end up with the efficient frontier as seen in Figure 2.


Table 4, Portfolio allocation results utilizing variance (risk) minimizing.
Portfolio
Req'd Return 10%
Calc. Return 10%
Stocks Bonds Real Estate
Expected Return 12% 8% 9%
Volatility 15% 8% 10%
Allocation 39.7% 19.10% 41%
Correlation
Stocks Bonds Real Estate
Stocks 100% 50% 25%
Bonds 50% 100% 0%
Real Estate 25% 0% 100%
Covariance
Stocks Bonds Real Estate
Stocks 0.02250 0.00600 0.00375
Bonds 0.00600 0.00640 0.00000
Real Estate 0.00375 0.00000 0.01000
Weighted COV
Stocks Bonds Real Estate
Stocks 0.003546 0.000455 0.000613
Bonds 0.000455 0.000233 0.000000
Real Estate 0.000613 0.000000 0.001698
Risk 8.73%

Sharpe Maximizing Portfolio

The mean-variance portfolio theory (MPT) is a useful tool for considering investment allocations within our portfolio, however; there are a few drawbacks. First, we have to determine our desired level of return, which is based on our appetite for risk. Aggressive investors are willing to take higher risks for higher returns. A conservative investor is willing to take less return in order to reduce risk. Next, a riskless asset such as T-bills or cash cannot be used in our model. Any riskless asset is assumed to have zero risk (volatility). Researchers found that when including a riskless asset the efficient frontier would change. Effectively the "new" efficient frontier would go from the riskless return rate to a point of tangency on the efficient frontier. The point of tangency represented the optimal allocation of riskless and risky assets. Instead of worrying about investor preferences, the optimal portfolio allocation only depended on the expected returns. The investor could then raise or lower the portfolios return (and risk) by either lending or borrowing in the riskless asset.

For our purposes we probably are not able to borrow or lend large sums of cash, however; the inclusion of a riskless asset is important. It allows us to determine the optimal portfolio allocations, which would dominate over the efficient frontier we had computed. Requiring a dominant portfolio, we need to determine the optimal portfolio at the point of tangency. To do this we will utilize Sharpe-maximizing portfolio, which utilizes the Sharpe ratio.

The Sharpe ratio is the risk premium of an investment divided by its volatility. The risk premium is an investments return over the riskless investment - usually cash or treasuries.


Sharpe Ratio = (expected investment return - riskless rate) / expected investment volatility


The ratio is a good measure of risk-adjusted return. The higher the ratio computed, better the investments return relative to its risk and the riskless investment. A lower ratio would mean we are being inefficient, receiving less return relative to the risk. For allocating our funds into different investments, we begin with our prior setup for calculating the mean-variance. A riskless asset is added to our possible allocations. Next, we calculate the Sharpe ratio for each return. Instead of trying to allocate investments to meet a desired return goal, we use the solver to maximize the Sharpe ratio. We want to get the highest return per the risk. Let us look at an example.

Jane has $30,000 to invest. Understanding the importance of diversifying her investments she looks to put money into different asset classes; stocks, bonds, real estate, and cash. Speaking to her friendly stockbroker for consultation, she has decided to utilize mutual funds that specialize in stocks and bonds in lieu of a direct investment. For her real estate exposure, she will compare REITs with the housing futures. Actual real estate assets are not considered since she does not want to incur another mortgage. Any cash would be put into her bank account or another higher interest account.

After plotting the stock and bond funds on a chart, she eliminated the dominated investments down to one stock fund and one bond fund. Now she had to consider whether to utilize property derivatives or a REIT - real estate investment trust. REIT's are corporations or trusts that buy, sell, and manage income-producing properties. Many of the REIT's trade on the stock market and offer the investor the benefit of liquidity. Using a REIT stock, our exposure to real estate is through the decisions and actions of the REIT management. We are effectively investing in a real estate company. If the REIT makes poor management decisions or market sentiment is negative, the stock value can suffer. In fact, REITs correlate with stocks about 64% of the time. Compared to REITs, Property derivates are illiquid due to a low volume of trading. Sue is not looking to trade thousands of contracts, so it is not too big a concern. Since derivative pricing is based on the S&P Case-Shiller indices, the derivatives reflect the true movements in actual real estate value. Also, she needs to make sure she buys her future contract at a reasonable price (See derivatives and synthetics). Real estate prices correlate less than 25% of the time with stocks. If Jane wants market diversification in her portfolio, she would be better off with the property derivatives - which she does.

Table 5 below shows Jane's calculations and results. Based on research, she had to decide on what the expected returns would be for each category. Although not perfect, Jane figured the long running average returns would be safe. The volatilities also were based on longer-term information. Due to the housing bubble, credit crunch, and market crash (if that were not enough) the recent volatilities were very high - upwards of thirty-five percent. Feeling that the bubble was not likely to occur again in the next few months, she used the long-term volatilities. The only bit of information she had revised from our earlier example were the correlations, specifically between stocks and bonds.


Table 5, Sharpe maximizing portfolio calculations and results.
Portfolio
Investment $30,000
Return Volatility Sharpe Allocation
Cash 4.00% 0% 25.1% $7,540
Bond Fund 8.00% 8% 0.5 31.3% $9,377
Real Estate 9.00% 10% 0.50 24.2% $9,377
Stock Fund 12.12% 15% 0.54 19.4% $5,812
Portfolio 8.04% 0.73 100.0% $30,000
Correlation
Stocks Bonds Real Estate
Stocks 100% 50% 25%
Bonds 50% 100% 0%
Real Estate 25% 0% 100%
Covariance
Stocks Bonds Real Estate
Stocks 0.02250 0.00600 0.00375
Bonds 0.00600 0.00640 0.00000
Real Estate 0.00375 0.00000 0.01000
Weighted COV
Stocks Bonds Real Estate
Stocks 0.000845 0.000167 0.000176
Bonds 0.000167 0.000625 0.000152
Volatility 5.52%

Jane set up her spread sheet solver to maximize the Sharpe ratio for the portfolio. To do so the solver would change the allocations of each investment, including the cash. The constraints needed to be added to get a reasonable answer. First, since we were not going use large short positions, the allocations had to be greater than zero. Next, the allocations also had to be less than one (100%). Finally, the total of all the allocations had to add up to 100%. The solver would determine the maximum Sharpe ratio for the portfolio. Based on the allocation results, Jane would have a portfolio with a return of 8.00% expected with a volatility of 5.52%. The portfolio's Sharpe ratio calculates out to 0.73, which is greater than the bonds, housing, or stocks alone. For a higher return Jane would reduce the amount of cash, for a lower volatility she would increase it. This can be done through the solver. The other allocations move around a bit, but the cash position seems to drive the return changes. Now Jane can go to her broker with her allocations and work out how to put the actual portfolio together.

As we have seen, using portfolio theory, we are able to build a portfolio that meets our return and risk requirements. Although our calculations are far from any guarantee of being correct, we have a way to measure how our investments affect our perceived outcomes. Just remember to dominate.



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