The Sharpe Ratio, Part III: Comparison of current portfolio allocations

This is the final installment of our discussion of the Sharpe Ratio. Today we’ll see some representative Sharpe Ratios of portfolios allocated according to Modern Portfolio Theory (MPT) using traditional asset classes, that is, stocks and bonds (as opposed to futures or derivative instruments). We’ll also see how adding the element of market timing to classic Modern Portfolio Theory greatly increases the Sharpe Ratio by dramatically reducing overall portfolio risk.

Current Optimally Allocated Portfolios and the Sharpe Ratio
Let’s look at some current (through September 2009) classic MPT allocated portfolios and their Sharpe Ratios. The table below lists the best allocations among a set of nine diversified asset classes. Target returns shown are for the range of possible achievable returns derived from monthly data since 1928. The data reflects compounded annualized returns.

This table was computed by the SMC Analyzer software available for user subscription on the StockMarketCookBook.com web site. As you can see, the Sharpe Ratios are highest for the lower risk portfolios  predominantly made up of Treasury Bills.  This is rather misleading because as T-Bills have historically averaged those returns, they’re not producing close to the same today. This is one reason to be cautious when evaluating the Sharpe Ratio as the contributing investment return term is backward looking, i.e. what the investment has produced in the past, while the riskless investment return is forward looking, i.e. what U.S. Treasury Bills will pay over the next few months. With this caveat in mind, one can still fairly compare the Sharpe Ratios in the above table since they are all calculated in the same way and over the same time frame.

Note that higher portfolio returns come at the cost of much higher risk (as defined by the standard deviation). By definition, an increase in risk lowers the value of the Sharpe Ratio revealing the declining compensation received for assuming the risks entailed with higher targeted returns.

Market timing improves Sharpe ratio values
Let’s now look at the table for MPT allocated portfolios using the conservative Long/T-Bills strategy enhancement to MPT. In this strategy, the percentage allocated to a particular asset class (equity asset classes only) is normally held Long when that asset class is in a bullish trend. When the trend reverses, the percentage of the portfolio allocated to that particular asset class is transferred into the safety of T-bills. (Trend reversals are determined by an oscillator that is optimized according to a robust and proprietary optimization method.)

The following table was produced using the SMC Analyzer software. In the table, Long/Long is equivalent to the classic MPT approach and is always applied to the bond asset classes.

The remarkable thing to notice is that it is possible to achieve a long-term rate of return of 10% with a Sharpe Ratio greater than 1.0. Compare that with the meager 0.44 Sharpe Ratio obtained with the classic MPT strategy!

Summary
The Sharpe Ratio can be a useful tool for financial decision making when properly understood and properly applied. As mentioned in Part I, the Sharpe Ratio is not a particularly reliable measure of non-traditional hedge fund comparison because of the non-Gaussian nature of the underlying instruments. What the investor has hopefully learned here is that when properly understood and applied, the Sharpe Ratio can be a legitimate tool for fund comparison but it shouldn’t be the only tool.

This is the technical portion of this article trilogy on the Sharpe Ratio so be sure you’re armed with high-water pants and a pocket-pen protector. A slide rule is a bonus.

A little background
Modern Portfolio Theory (MPT) is a Nobel Prize winning financial concept that utilizes mathematical optimization algorithms to determine the best allocation of investments in a portfolio. For a selected desired overall rate of return, MPT will tell you how to allocate your capital among a diversified set of investment asset classes such that the variation in the periodic portfolio total returns as measured by the standard deviation (aka risk) is minimized.

Follow any discussion on MPT and you’ll run into the following graph on the Efficient Frontier:

This technical plot shows the current relationship between desired return and standard deviation. What it shows is that to get a higher return, you’d need to assume more risk (given by a higher standard deviation). The curve known as the “efficient frontier” represents those optimally efficient portfolios that provide the least amount of risk at a given level of return. The shaded area underneath the curve represents the possible space of less efficient allocations of portfolio assets. To obtain a return in the area above the efficient frontier is theoretically impossible with the given asset classes.*

Note that the efficient frontier is a curved convex line. This is a result of the lack of correlation between the various asset classes in the portfolio and how that results in an optimum portfolio characterized by an overall standard deviation less than it would be for one of completely correlated assets. For uncorrelated assets when some are up in value others are down in value and the overall portfolio therefore exhibits reduced variation. For a portfolio of totally correlated assets a combined linearly weighted standard deviation would result and the efficient frontier curve would be straight rather than curved. It is the existence of optimum allocations that pulls the line to the left for a lower standard deviation at each point that creates that convex shaped curve. The end points of the curve do not benefit from this left pulling because they are dominated by single asset class portfolios, at the high end with 100% the riskiest asset class and at the low end by mostly the least risky asset class. The convexity of the efficient frontier curve therefore graphically demonstrates the beneficial effects of diversification.

How the Sharpe Ratio fits into MPT
Let’s look at the line of Best Capital Allocation on the graph above. This line represents the best way to incorporate a riskless cash component into an already optimum portfolio thereby reducing the risk even further albeit at a reduced overall rate of return. On the graph, the annual rate of return on a 90-day T-Bill is about 1.2% (those were the days!). This is considered as safe an investment for the short term as is possible to find and is considered to be as riskless as cash. The line on the graph therefore starts at 1.2% plotted at zero standard deviation (it’s riskless!) and is drawn so that it contacts the efficient frontier curve at a point such that the slope of the line is a maximum. This is the point on the efficient frontier with the greatest Sharpe Ratio and is known as the market portfolio. The slope of the line is the Sharpe Ratio for the market portfolio and the current riskless rate of return. As seen on the plot above, the market portfolio provides about a 5.0% return which has a standard deviation of 3.4%. The Sharpe Ratio for this investment is therefore:

(5.0 – 1.2) / 3.4 = 1.12

which, as we have noted, is also the slope of the line of best capital allocation. Note that the line of best capital allocation is straight because the riskless investment has no standard deviation and there is no better ‘optimum’ allocation that will produce a resultant combined reduced standard deviation that would produce a convex curve like that for the optimum allocations of the other investment classes.

The market portfolio has the unique property that, taken in combination with a riskless cash component, it will produce an optimal return not on the efficient frontier and outside the suboptimal allocation space. It will lie on the line of best capital allocation. The combination will offer a larger return for a given amount of risk than any of the portfolios on the efficient frontier. In this way you can easily construct a composite portfolio that is still optimum but also has a reduced standard deviation. Unfortunately, this comes at the penalty of a lower overall portfolio return. The percentage of cash to include in the composite portfolio is easy to determine since the standard deviation is reduced linearly according to the slope of the line (the Sharpe Ratio).

For example, a portfolio of assets at risk returning 5.0% with a standard deviation of 3.4% can be reduced to one with an arbitrarily selected standard deviation of 2.0%. From the line of best capital allocation on the graph, such a portfolio returns about 3.2% and is constructed by reducing the component of optimally allocated market portfolio risky assets to 64% (3.2 / 5.0) x 100% and by adding a 36% (the rest) riskless cash component. Portfolios on the line of best capital allocation above the market portfolio can only be achieved by adding leverage to the market portfolio financed by borrowing at the riskless rate of return.

Tomorrow’s focus
In the final segment, we’ll look at Sharpe Ratios for optimally allocated portfolios in the current market environment.

*A traditional MPT portfolio allocates funds among the following asset classes: Large Company Stocks, Small Company Stocks, Long-Term Corporate Bonds, Long-Term Gov’t Bonds, Medium-Term Gov’t Bonds, T-bills, REITs, International Stocks, and International Bonds.

Note that the efficient frontier curve will be different assuming a different set of asset classes.

I’ve been reading that investment professionals still use the Sharpe Ratio to evaluate fund performance which led me to ask the question of some of my peers as to what they consider to be a “good” Sharpe Ratio. The answers I received were unexpectedly all over the map; some considered anything over 1 to be good. Others thought that only double digit Sharpe Ratios were worth bothering with while still others said that the Sharpe Ratio had no impact on their investment decisions.

This left me even more confused and I decided to explore the Sharpe Ratio in detail so that I could make my own informed decision concerning its validity. To help me in my investigation, I called on Professor Pat as he’s the resident StockMarketCookBook expert on all things related to portfolio theory.

In only two days, he presented me with an exceptional treatise on the subject of the Sharpe Ratio. The only problem I had with it was the length which I felt was too long for one blog so I broke it up into three distinct parts which will be run over the next few days. What follows is essentially his writing with some of my own thoughts tossed in.

Definition of The Sharpe Ratio
The Sharpe Ratio is named after William Sharpe, a 1990 Economics Nobel Laureate who won it for his work on the Capital Asset Pricing Model (known as CAPM) which shows how the market prices individual securities in relation to their asset class. Here the discussion is focused on the Sharpe Ratio which for a particular investment is a direct quantitative measure of reward to risk.

Sharpe devised the ratio which he called the ‘reward to variability ratio’ in 1966. It later became known as the Sharpe ratio as other economists and financial professionals attributed the ratio to him. It is a measure of how much excess return an investment provides over and above a riskless investment (e.g., T-bills) considering the additional risk (σ, the standard deviation of the returns on the investment under evaluation) that is entailed. It is mathematically defined as the following:

Sharpe Ratio = [Investment Rate of Return - Risk-free rate of return] / σ

The Sharpe ratio is there used to assess how well an investment compensates an investor for assuming additional risk. Higher values of the Sharpe ratio are considered to be better than lower ones. The investment community generally considers values over 1.0 to be good, over 2.0 to be very good, and over 3.0 to be excellent, but these can vary depending on the current financial climate. [Note: As of this writing, the risk-free rate is almost zero which means that the Sharpe Ratio is essentially just the investment return rate divided by the investment risk.]

Riskless Investment Returns
The measure used in calculating the rate of return of a riskless investment is typically short-term (90-day or less) U.S. Treasury Bills. This investment is considered to be as safe an investment as you can possibly find, and will exhibit no variation in base value during the holding period. Only the rate of interest will change as the bonds mature and are rolled over.

Other measures of what is considered “risk-less” can produce significantly different Sharpe Ratios. Long-term government bonds are not an appropriate measure for a risk-less security as market interest rate changes can significantly alter their values and in some cases can drive down the total return to very small levels or even produce losses.

Hedge Funds and the Sharpe Ratio
Implicit in the Sharpe Ratio is the assumption that returns on the investment follow a normal distribution (i.e., the bell-shaped curve). While this is a very good assumption for equity based stock index funds that are large and liquid it is not so good for strategically managed hedge funds that employ dynamic trading techniques, illiquid investments, or highly leveraged instruments such as options. For example, a hedge fund strategy that sells deep out-of-the-money options will show a higher than average Sharpe ratio–that is, until the market unexpectedly moves counter to the prevailing trend and the fund is hit with large losses.

Because of the Sharpe Ratio’s mathematical nature, high values must either employ a high return and/or a low risk. As history has shown, high return/low risk situations can’t be sustained for long periods of time.

Here’s an actual example of a supposedly stable fund with a high Sharpe Ratio whose demise nearly destabilized the global financial markets.

In the 1990’s, Long Term Capital Management, the bond arbitrage hedge fund touted as mathematically safe due to the supposed low probability of incurring significant overall losses on its massive portfolio, showed a very high Sharpe Ratio of 4.35. The fund managers employed highly leveraged strategies to make money on small arbitrage spreads. A financial crisis in Russia caused by defaults on government bonds and a resultant flight to quality resulted in massive losses for the fund which controlled nearly 5% of the world’s fixed income market.

Unable to make the loan payments on the huge debt incurred to finance its leveraged positions, they faced collapse. Had they gone into default it would have caused a worldwide financial meltdown. They were bailed out by the Federal Reserve and other creditors and taken over in 1998. So, what else is new?

Hedge fund illiquidity also works to distort the Sharpe Ratio. Investments in real estate, private equity, or mortgage backed securities for which there is no ready market can appear to be less volatile which helps their Sharpe Ratios. Fund managers tend to price these securities in a way that is, of course, favorable to the fund’s statistics and can produce an artificially high value for the Sharpe Ratio.

[Note: Illiquidity relates to how assets are valued. When there is no ready market how do you determine what something is worth? The recent mortgage crisis is an example where fund managers were valuing their own assets arbitrarily. The rate of return on an investment is related to how its value fluctuates. If it does not fluctuate then the standard deviation is low and the Sharpe Ratio is high. If that lack of fluctuation is artificial then the Sharpe Ratio is also artificial.]

Summary
Many hedge funds that currently report a high Sharpe Ratio may be employing strategies that are manageable at their current size but as they grow those same techniques may prove impractical and the Sharpe Ratio could drop dramatically. As a result, the Sharpe Ratio may not always be a good measure for evaluating a hedge fund’s risk/reward relationship.

To get a better picture of a fund’s risk/reward profile, investment professionals look at other aspects of performance including maximum portfolio draw-down and statistical measurements such as kurtosis and skewness. The Sortino Ratio, a modification of the Sharpe Ratio, focuses on downside volatility. Some think that it’s a more accurate representation of hedge fund risk but it, too, is subject to many of the same criticisms as the Sharpe Ratio. As one professor of risk management so aptly said, “Risk is one word, but it is not one number.” [Ref. 1]

Tomorrow we’ll look at the Sharpe Ratio in the context of Modern Portfolio Theory, so dust off your propeller hats!

References
1. “The Sharpe Ratio can oversimplify risk,” Investopedia.com.
2. “Risk gets riskier,” Hal Lux, Institutional Investor, October 2002.

Image by Free-StockPhotos.com

I’ve been receiving numerous inquiries into the Stock Market Cook Book’s portfolio analytical tool, the SMC Analyzer, which is a powerful software tool that applies proven market timing strategies to Modern Portfolio Theory (MPT). Although Analyzer results have been discussed in previous blogs (see 5/12-13/09, 7/07/09), I feel that some of the points made earlier need to be reiterated.

One of the most frequently asked questions is this:  How did a portfolio based on market timing perform during the recent credit crisis as compared with a traditionally long-only MPT portfolio? Although I’ve already answered this question in the May articles, I decided to update the results to this July and take an in-depth look at how both portfolios would have been allocated on a month-by-month basis beginning in July 2008.

Modern Portfolio Theory (MPT) in a nutshell
Briefly, MPT tries to maximize returns given a certain level of risk. It attempts this by allocating a portion of one’s portfolio over a spectrum of asset classes, each with varying degrees of inter-correlation. For example, an MPT-allocated portfolio would most likely contain the asset classes from both large and small-cap domestic (US) stocks, US government and corporate bonds, real-estate (typically in the form of REITs), and international stocks and bonds. Commodities such as precious metals and oil are also popular asset additions. (Hedge funds can also play a role in portfolio diversification.)

MPT attempts to achieve the desired return via the robustness (that is, the degree of return versus the associated risk) of a particular asset class combined with the degree of correlation (or rather, un-correlation) among the other candidate asset classes. It’s a complex mathematical model for which Harry Markowitz won the Nobel Prize. So far, the theory works well over a very long time-frame, but it does dismally over short, volatile periods because of the requirement that it always must be long in the asset classes determined by the model.

Not so for the SMC Analyzer!

How the SMC Analyzer is different
The SMC Analyzer uses a market timing oscillator that is specifically optimized to each asset class. When the oscillator is positive, it will tell you to be long that asset class; when it turns negative, you’ll be instructed to either go short (if that is what you selected and if that asset class is “shortable”) or else move into the safety of T-bills or another risk-free asset.

So, how did both of these approaches compare over the past year?

Portfolio comparison
Below is a chart that graphically depicts how a traditionally MPT-allocated portfolio would have fared compared with a market-timed portfolio as determined by the SMC Analyzer. In the latter portfolio, it was assumed that when the oscillator for that asset class was negative, those funds that would have been designated by MPT for that asset class would be put into the safety of Treasury bills. In both portfolios, a compounded annual return of 10% is assumed.

You can see that the tradition portfolio failed miserably—not only could it not yield a positive return but the investor would have been down 26% and at a huge risk of nearly 27%. Note, too, that the maximum draw-down was a whopping 45%, on par with the major averages.

However, had you been in a market-timing adjusted portfolio, not only would you have realized a positive return, but also at greatly reduced risk and without any significant draw-down. This is truly amazing!

To see how these differences in results were achieved, let’s look at the asset allocations recommended by MPT for each portfolio on a month-per-month basis. The top table gives the recommended allocations for the traditional MPT portfolio while the bottom portfolio gives the recommended allocations for the market timing portfolio.

Observations
Here are several observations derived from the above tables:

1. The risk, as measured by the standard deviation of possible returns, is less than half that in the market timing model.

2. The reason that MPT failed so miserably during the recent downturn is because it is looking towards the longer term and it figures that the only way the portfolio will ever be able to achieve a 10% return is by being long in the highest risk (as well as the highest returning) asset classes, i.e., stocks.

3. In contrast, the SMC Analyzer saw that the market was heading down and re-allocated the monies apportioned to Small-Cap Stocks into the safety of Medium Term Government Bonds and T-bills. It wasn’t until March of 2009 that the stock oscillators began to turn positive indicating it was safe again to invest in those asset classes.

4. You can also see that in order to achieve the monthly asset allocations as recommended by the market timing model involves significantly less trades, a plus for those investors who are worried about commission costs eating into their portfolio profits.

Summary
I hope this exercise will put to rest many of the questions that have been raised about the SMC Analyzer’s market timing model. For further information regarding the SMC Analyzer, take the features tour or test drive the Analyzer for yourself. Both are located in the left-hand column of the home page, www.stockmarketcookbook.com.

Below is a chart of the progress of an investment portfolio starting in 1975 with $1 optimally allocated among ten asset classes including Gold. The magenta line reflects a portfolio always invested in each asset class according to the percentage allocations recommended by Modern Portfolio Theory and readjusted each month. The green line is a better result using a conservative combined market timing strategy that places the investor in the safety of T-Bills for the particular allocation amount that would normally be invested in that asset class when the current performance trend in that asset class is identified as significantly negative.

The results of the above chart holding a Gold component in a portfolio should be compared with the results in the next chart that repeats the same analysis but instead leaves Gold out entirely from potential consideration as a component in the portfolio.

As can be seen, the inclusion of Gold in a portfolio during this period was significantly detrimental to overall portfolio performance. Early 1970’s gains in the Gold asset class were not sustained and the effect of expecting a continuation of this performance was to hold down portfolio value. With Gold included, portfolio value is today worth about 12 to 15 times the initial investment depending on the strategy employed. But without Gold the result is higher at about 18 to 25 times the initial investment. It took until 2000 for those early Gold asset class gains to dilute sufficiently with time to the point where the recommended percentage allocations dropped to zero as seen in the allocation chart in Part 1 of this article.

Some observations:

It should be apparent that the 10% compounded average target return was not achievable over the entirety of this analysis period due greatly, but not entirely, to the recent financial meltdown. The best was a close 9.8% reached by excluding Gold as a component and using the SMC Analyzer’s conservative “Long or in T-Bills” market timing strategy enhancement. Analyses were conducted at target return percentages other than 10% but the results were similar. In fact, the more that the allure of Gold would lead the investor to invest a greater percentage of their portfolio in the metal the better off they would have been to leave it out completely. The failure of Gold to maintain the performance achieved during its first decade of public availability produced actual portfolio returns lower than predicted over the nearly 35 years of the analysis period. Other asset classes have also failed recently compared to their historical performance levels and this contributed to the inability of the recommended allocations to meet the 10% target.

Portfolio risk as measured by the standard deviation of returns was much lower with the inclusion of a Gold component but this is to be expected with the addition of a nearly completely uncorrelated asset class and with the failure of the various asset classes to produce the desired target return. The relationship between risk and reward nearly always holds true. The greater the returns the greater the risk or variation in those returns. The lower the returns the lower the variations.

The relative preservation of portfolio value during the stock market decline of 2000 to 2003 is due to an interesting result of using an asset allocation tool like the SMC Analyzer. When the history of asset class performance is limited to starting in 1970, as we did to be compatible with the start of legal Gold ownership, large stocks as an asset class are deemphasized as an advantageous portfolio component due to their poor returns in the 1970’s. Hence the investor was automatically saved from the decline at the start of this century when the market in stocks tanked. That same principle unfortunately applied in reverse for investment in Gold. It continued to be recommended based upon good 1970’s decade returns even though the overall return for the subsequent 20 years was negative.

Based on this history then it is argued that Gold has not been the sparkling portfolio component that it is frequently touted as being. Gold is sometimes utilized as a form of portfolio insurance. If all else falls then Gold is sure to go up. That is hard to dispute but the cost of that form of insurance has been very expensive.

The future:

So, given that Gold has not worked out for much of the past 30 years what does this mean for the future? Well, with the excellent recent performance of Gold during the financial crisis and the anticipation of high future inflation from all the government money being pumped today into the economy (assuming it actually gets spent) there is once again good reason to take a serious look at Gold. The SMC Analyzer’s recommended percentages for the Gold component have risen like Lazarus starting in 2006. Had you bought a small portfolio component at the $600 price level back then you would be very happy today with the price soaring to over $900 per ounce and this would have mitigated declines in some other asset classes. The current recommendation for the percent of a diversified portfolio that should be allocated to Gold (assuming a 10% overall portfolio target annual return) sits at about 12% today. If inflation once again spikes up, as it is very likely to do sometime within the foreseeable future, that allocation could come in handy as Gold prices soar far above today’s levels.

First of all I would like to thank Dr. Kris for the opportunity to act as a guest blogger this week while she is away. This article will be about what holding a component of Gold in an investment portfolio means to the overall performance that portfolio returns.

The Midas metal has long been recommended as a component of a diversified portfolio but is that really a good idea and how much Gold, if any, should the investor hold relative to other asset classes? What will be described here are the results of adding the monthly time history of the price of Gold (on the London exchange) to the other nine asset classes already included in the SMC Analyzer offered for subscription on this web site. Briefly, the SMC Analyzer is an enhanced mean-variance optimizer for determining optimum asset class allocations in an investment portfolio along the lines of Modern Portfolio Theory.

Holding Gold bullion as an asset class has several consequences. First of all it returns no dividends nor interest. In addition, holding the metal means that the investor must incur assaying, storage, and insurance costs.  It would actually be recommended that a mutual fund of Gold stocks be used as the investment vehicle for investors interested in including the precious yellow metal in their portfolios. In conducting this analysis however the straight price of Gold was preferred over stocks to provide as pure a commodity play as possible without any possible perturbations from disparate corporate performances.

Some history:

The price of Gold for most of the 20th century was controlled by government regulation and international agreements. Here are some major events.

- In 1900 the Gold Standard Act placed the United States officially on the Gold standard. This meant committing the United States to a fixed currency exchange rate relative to a fixed price of Gold (then at just over $20 per ounce) with other countries also on the Gold standard.

- In 1913 the Federal Reserve Act mandated that Federal Reserve Notes be backed 40% in Gold.

- In 1933, as a response to the banking crisis at that time, the U.S. Government prohibited private ownership of all Gold coins, bullion, and certificates.

- A year later the Gold Reserve Act of 1934 gave the government ownership of all Gold money and stopped the minting of any new Gold coins. Gold certificates could only be held by the Federal Reserve Banks. At that time President Roosevelt reduced the value of the dollar by increasing the price of Gold to a regulated $35 per ounce.

- The Bretton Woods conference, held in 1944 as World War II raged, reaffirmed the regulated price of Gold at $35 per ounce and established an international framework for how the participating nations were to maintain an exchange rate for their currencies relative to that price.

- In 1968 as mounting financial pressures strained this framework a two-tiered pricing system was enacted separating official transactions between monetary authorities (at the regulated price of $35 per ounce) and other private transactions (to be conducted at a variable free market price on the London Gold Market.)

- In 1971 the framework collapsed and the United States officially went off the Gold standard and halted the redemption of foreign held dollars into Gold. This caused a great deal of financial upheaval in the world economy and established the U.S. dollar as the world’s reserve currency.

- At the end of 1974 Americans were finally permitted to privately own Gold other than as jewelry for the first time in 40 years.

The analysis:

Therefore, based on this first year Americans could legally own Gold bullion the starting year for portfolio analysis will be 1975 and history data for the price of Gold will start in 1970 when the free market for Gold came into full swing. The chart below shows how $1 invested in Gold in 1970 has performed since that time. This reflects exactly the variation in the price of Gold from about $36 per ounce in 1970 to today’s price of about $935 per ounce.

To perform portfolio analysis an overall target return must be selected. A compounded average annual return of 10% was selected as being a viable long term goal. Using the SMC Analyzer to optimize portfolio allocations among the various asset classes to achieve a minimum standard deviation of overall returns we obtain the following result for the period from January 1975 through June 2009 for the recommended percentage of Gold as an asset class to be held at each monthly point in time.

The Long/Long oscillator strategy referred to in the figure above indicates that the market timing capabilities available in the SMC Analyzer were not used to generate the above chart. You can see that the percentage peaked during the great bull market in Gold in the 1970’s when inflation raged in the U.S. reaching a whopping 13.58% during 1980. The subsequent generally falling recommended percentages of Gold holdings through 2005 reflect the flat to slightly declining price of Gold resulting from comparatively low rates of inflation in those decades.

This illustrates one problem with the strict application of classical Modern Portfolio Theory. That stellar returns in one asset class (in this case Gold) during one time period, the 1970’s, are treated as though they are still occurring (although being diluted with time) when in fact they are not. The tendency of the recommendations to keep the investor invested in that asset class even though returns are currently poor is a valid criticism of Modern Portfolio Theory which led to the development of the SMC Analyzer’s market timing strategies.

This article will continue tomorrow in Part 2.

Dr. Kris is out of town this week and is rerunning her series written by guest contributor, Professor Pat, on the topic of Modern Portfolio Theory. These posts originally appeared in the April 21-24, 2008 blogs. The content has been edited and data updated to reflect current values where appropriate.

Today is the final installment on Modern Portfolio Theory (MPT) where we’ll be looking at the correlation among asset classes. We’ll also provide you with proxies for these asset classes drawing from the mutual fund and ETF pool so that you, too, can construct your own portfolio.

Correlation Among Various Asset Classes
Additional asset classes added for consideration should preferably be uncorrelated or negatively correlated to the other asset classes for best results. The relation between how asset classes or securities move relative to each other is defined by what is known as the Pearson correlation coefficient. The table below shows the Pearson correlation coefficients for each of the asset classes to one another.

A value of +1 indicates perfect correlation while a value of -1 would reveal total negative correlation in the sense that when one asset goes up in value the other would go down in the same proportion, and vice versa. A value of zero demonstrates no correlation between the two asset classes; that is, they behave independently. (Note: The Pearson coefficient assumes a Normal distribution of data as we discussed yesterday.)

As expected, all of the stock asset classes have high correlations with each other and low correlations to government bonds, especially T-bills, where the correlation there is slightly negative. As for bonds, the Long & Medium Term Government, Corporate, and International Bond asset classes are highly correlated with each other. T-bills, on the other hand, don’t show much of a correlation to any other asset class except for Medium Term Gov’t and International Bonds.

What this table confirms is that there is a sufficient lack of correlation between the various asset classes to present a suitable set of alternatives from which to compile a portfolio that can be counted on to behave according to the precepts of MPT.

Historical volatilities & returns of the asset classes
Let’s now look at a comparison of the returns and risks of an optimally allocated portfolio with those that contain just one asset class. The table below shows the average annual return and volatility of each of the nine asset classes using data from 1928 to the end of April, 2009.

You can see that to get a higher return requires that the investor take a higher risk. To achieve higher returns, MPT would allocate the portfolio chiefly among the stock and REIT asset classes, with some component of bonds thrown in to minimize risk. This is why young investors should concentrate their portfolios into these asset classes while those nearing retirement should focus on minimizing risk to preserve needed capital. Bonds are the least risky asset classes which is where the older investor’s portfolio woukd be concentrated. But, MPT would likely throw in a small percentage of stocks in order that the desired return can be achieved.

For example, we saw yesterday that the most conservative portfolio of 99.6% in T-bills would return an average of 3.7% annually with an associated risk of 0.9%. However, by increasing our risk by only 0.1% to 1.0% we can achieve a return of 4.0%, or 0.3% over the previous return, just by adding Medium Term Gov’t bonds to the mix.

Observations
You can see why it’s important to not only include a mix of asset classes of differing correlations but of differing expected returns. MPT uses the higher volatility (i.e. the higher return) asset classes to maximize portfolio return and uses the correlation coefficients to minimize portfolio risk. It’s important to note that no portfolio can achieve a higher return than that of its highest returning asset class.

There are many interesting observations to be gleaned from these tables such as the high volatility of REITs and the counterintuitive performance of Long-Term Corporate bonds versus Long-Term U.S. Government bonds where Corporate bond returns display the expected risk premium but the volatility is actually lower. Perhaps a closer examination of the actual composition of the comparative fund holdings might provide an explanation for this.

Asset Class Proxies
An investor interested in forming a portfolio along the lines of the nine asset classes discussed here will find the following Vanguard funds as suitable proxies for some of them.* Vanguard does not currently offer an International Bond fund so a T. Rowe Price no load fund with a reasonable expense ratio is listed as well as one by Oppenheimer.

The electronically traded funds listed in the following table are also suitable proxies for the named asset classes.

ETF or Mutual Funds?
This is really a subject for another blog but I thought I’d touch on it because it’s relevent. One main reason to use mutual funds, especially if you hold many funds within the same fund family, is that you can exchange among them at little or, even better, no cost (like Vanguard).

One reason to use ETFs is that you don’t have to wait until the end of the day to transfer in and out of them because they trade exactly like stocks. (But also like stocks there’s a spread between the bid and ask prices.) If you’re judicious in your trading, you could actually do better in the transaction, especially if you’re trading among negatively correlated asset classes.

The other reason to use ETFs is that, in general, they’re not subject to as many tax consequences as mutual funds. This is because that instead of selling stocks like a mutual fund is forced to do, and by so doing incur short or long-term capital gains, the equivalent ETF has the legal right to “swap” in and out of stocks avoiding capital gains altogether.

The choice is yours, and you should do your homework on both strategies.

Summary
In summary, we have shown that diversification provides real measurable benefits and that you, the investor, can use the tools of MPT to quantitatively measure reward versus risk in forming an investment portfolio that is suitable for your circumstances.

Note: All of the above calculations were determined using the SMC Analyzer. To see how the Analyzer achieved these results, click here to view the Demo.

*Disclaimer: No one at the StockMarketCookBook has any affiliation with Vanguard, but we like their low management fees and the ease with which one can move among their funds at no cost. You can check out more about them and other mutual funds and ETFs at Morningstar.

*Dr. Kris is out of town this week and is rerunning her series written by guest contributor, Professor Pat on the topic of Modern Portfolio Theory. These posts originally appeared in April 21-24, 2008. The content has been edited and data updated to current values where appropriate.

Introduction
Yesterday, we saw that the aim of MPT is to provide the investor with a desired return value while minimizing risk. It achieves this by allocating portfolio resources among appropriately selected asset classes of varying volatilities and inter correlations. Today we’ll look at some actual data and examine the results.

Note that the data presented here represents monthly data collected from 1928 to the present. You probably won’t find these results anywhere except perhaps in institutions of higher learning or in large brokerage houses. This is a rare opportunity to view it first-hand.

Portfolio Construction: Selection of Asset Classes
The core philosophy behind MPT is to construct a portfolio of uncorrelated (or relatively uncorrelated) asset classes. Two assets are said to be uncorrelated if they react differently to market events. For example, the stock market and the bond market are two asset classes that are relatively uncorrelated as bonds usually perform better when the stock market is suffering and vice versa. The point of MPT is to construct a portfolio consisting of relatively uncorrelated assets in ratios that will give the investor the return he expects while minimizing risk.

In order that the recommendations provided by MPT be robust, it’s necessary to have enough asset classes of varying risks and inter correlations. The reason to include higher risk asset classes such as small-cap/high-growth stocks is that they’re precisely the ones that can provide the investor with higher returns, if needed. It’s also important to include a mixture of relatively uncorrelated as well as negatively correlated asset classes, such as stocks and bonds. The nature of the correlations among the asset classes is the factor that minimizes risk.

Description of the Selected Asset Classes
We’ve selected nine asset classes that are not only popular investment vehicles but also represent a collection of asset classes with the required volatilities and correlations. They are the following:

1. Large U.S. Stocks as currently represented by the S&P 500 Index.
2. Small U.S. Stocks defined by the Dimensional Fund Advisors (DFA) Microcap Portfolio.
3. Long-Term Corporate Bonds measured by the Citigroup High-Grade Corporate Bond Index.
4. Long-Term Government Bonds, a 20-year Treasury Bond.
5. Intermediate-Term Government Bonds, a 5-year Treasury Note.
6. 30-day U.S. Treasury Bills
7. Real-Estate Investment Trusts (REITs) defined by the DFA US Real-Estate Securities Index
8. International Stocks per the DFA Global Equity Portfolio
9. International Bonds per the DFA 5-Year Global Fixed Income Index Fund

Optimum Asset Allocations at required levels of return
Using annual total return performance data for each of these individual asset classes from January,1928 through April, 2009 and applying the mathematics of MPT to that data, the following table can be constructed:

To use this table, you, the investor, would choose the desired rate of return (given in the first column) that lies within your tolerance for risk. The risk is defined by the standard deviation located in the second column. You should not merely chase the highest potential return but also consider your ability to accept years in which sharp draw-downs in the overall portfolio value may occur. It is of utmost importance to note that variations in returns are smoothed out and tend towards the desired return, but this usually requires a long time horizon—sometimes on the order of 40 to 50 years.

Portfolio allocations for investors with shorter time frames
Investors nearing retirement will need to avoid the possibility of sharp declines happening just when they will be needing to withdraw funds. In this case, an aggressive allocation should be converted to a more conservative one. For example, the average return from the most conservative portfolio is 3.7% which is achieved with a portfolio of 0.1% Small Stocks, 0.2% Long-Term Corporate Bonds, 0.1% International Stocks, and the bulk of the portfolio, 99.6% in Treasury Bills, an essentially riskless asset class.

This represents the safest possible portfolio in that it provides the smallest variation in return from year to year. In any one year there is an approximate 68% chance (see the last installment of this article) that the actual return will be within one standard deviation. In this case, the return can be expected to vary between 2.8% and 4.6% (3.9% +/- 0.9%) In other words, the chance of a decline in overall portfolio value over any given year is small.

Portfolio allocations for those with longer time frames
A more aggressive investor with a longer time horizon, such as a person in his 20s -30s, might choose a higher yielding allocation but he must also be willing to accept the risk of short-term market declines or intermediate-term lackluster returns. This must be done so that the portfolio will be properly invested during those times where the previously underperforming assets become out-performers.

Portfolio allocations for those with intermediate time frames
Most investors will want to choose an allocation that moderates the risk while still providing a healthy average return. Returns between 7% – 9% are particularly attractive because as there’s a chance of a small decline in any one year, there will be reliable gains for most years over time. Right now, these portfolios would be allocated among the Small-cap stocks, Medium-term government bonds, and International stocks.

Observations gleaned from the above asset allocation table
Small stocks have historically produced the highest overall returns over the long haul as indicated by the table. To achieve the higher returns, the portfolio must be heavily weighted in small-cap stocks. But as MPT shows, higher returns come only with higher risk.

In 1973, for example, the loss in small stocks was about 31% followed in the next year by another 20% hit. The following nine years, however, saw a nice recovery with a whopping average annual appreciation of 36%. Regardless of your risk tolerance, the table does indicate that some exposure to small-cap, high-growth stocks is advantageous.

It is interesting to note that the above table recommends that Large-cap stocks, Long-Term Government Bonds, REITS, and International Bonds are not an attractive investment to hold in any portfolio right now.

In the next MPT installment, we’ll suggest some proxies for these asset classes. We’ll also look at the correlation table between them.

*Today’s blog and the next few to follow are reposts of those blogs written last year on the topic of Modern Portfolio Theory, or MPT. They were written by the StockMarketCookBook’s resident academic guru, Professor Pat, but Dr. Kris has edited and updated them as needed. Since MPT is an integral part of the SMC Analyzer, we felt that including several brief tutorials on the subject would be helpful to those readers interested in learning more about the technical foundations of the Analyzer.

This week, Dr. Kris is out of town playing Aunt Kris at her nephew’s high school graduation. (Aunt Kris is proud to mention that he’s graduating at the the top of his class. Looks like he’s following in his Auntie’s footsteps proving that brains as well as modesty runs deep in the family’s bloodlines.)

We’ll be back to our regularly scheduled blogging next week. Have a safe and happy Memorial Day!

Modern Portfolio Theory (MPT): The Background
Modern Portfolio Theory (MPT) was first conceived by Harry Markowitz in 1952. For his inspiration and hard work he was awarded the 1990 Nobel Prize in Economics. Markowitz proposed that investors should be concerned not just with investment returns but also with investment risk, and put forth a strategy on how to achieve optimum performance using a mathematically determined allocation of asset classes.

MPT asserts that investors could and should balance the returns they receive on their investment portfolios with the risks those investments present. As such, and for the first time, a way to actually quantify the notion of risk was available. The new idea was that risk could be equated with volatility or the variation in returns from one time period to the next. The beauty of MPT is that it was also now possible to take this one step further and mathematically determine an optimum investment portfolio that delivers the desired long-term rate of return while simultaneously minimizing risk. The benefits of portfolio diversification in terms of risk reduction could now be mathematically derived.

Correlation and volatility
An investment portfolio consists of various asset classes (e.g., stocks, bonds), some of which appreciate and some of which may depreciate within any specified time interval. MPT uses the concepts of volatility and correlation among asset classes in its computations. Briefly, volatility is the rate of change in the value of each class. Correlation is a measure of how two asset classes move in relation to each other. For example, investments that move in tandem with each other are positively correlated; those that move in opposite directions such as stocks versus bonds are said to be negatively correlated; while those whose movements are independent of each other are said to be uncorrelated. MPT takes advantage of this fact to determine not only which investments to invest in but how much of each you should hold relative to each another.

The concept of MPT as a type of portfolio insurance
MPT can also be thought of as a type of portfolio insurance. As an investor, you hope that all of your investments will increase but you also know that this is an unrealistic expectation. MPT insures your portfolio against unreasonable losses by holding asset classes that can counteract losses in one class with gains in another. The way to achieve this is to select investment vehicles that are uncorrelated with each other. A properly diversified portfolio also includes relatively low yielding assets (like treasuries) that can be counted on to produce steady and reliable gains. An important assumption of MPT is that the future will, on average, be just like the past.

[Shameless plug: This assumption is proving to be a major drawback to MPT, especially in recent years. That's why the addition of market timing to the MPT model not only increases potential portfolio returns but at significantly reduced risk. This is precisely the function of the SMC Analyzer*.]

MPT: The Theory
The techniques that are used in MPT to solve the optimum asset allocation problem is a branch of mathematics known as quadratic programming. There are two inputs to this program: 1. The actual historical returns of the various candidate assets input as a time series, and 2. The desired average rate of return that portfolio is expected to produce. The output of the calculations is the percentage to be ascribed to each asset class so that the entire portfolio will produce the desired return while minimizing risk. MPT quantifies that level of risk with a number called the standard deviation.

The standard deviation, denoted in statistical terms by the Greek letter σ, is a number that provides a bracketed range for the desired return. One standard deviation represents approximately 68% of the values around the expected outcome.

A basic assumption in MPT is that investment returns follow a pattern known as a normal distribution commonly found in natural processes.** Graphically, it forms a shape that looks like a bell as depicted in the illustration below. For example, if the desired annual rate of return is 10%, MPT might reveal a certain asset allocation strategy that would result in a minimum standard deviation of 11.7%. This means that approximately 68% of the time the actual return in any one year would fall in the range from -1.7% to +21.7%. (10% +/- 11.7%)

To be very confident about the exact range of one’s returns, we can again look to the normal distribution which also tells us that about 95% of the time the return the average return will be within two standard deviations of the mean, or in the above case between -13.4% to 33.4%.

The image below depicts these concepts. The dark blue region is one standard deviation,and the light blue region extends to two standard deviations. The symbol μ on the image is the average rate of return, or 10% in our example.

Investment philosophy
MPT is a useful tool for investors who prefer to remain somewhat passive in their investment activities. That is, they are not concerned with market timing. The optimum asset allocation technique tells them that they can remain comfortable with fluctuations in the market and limit their activities to monthly or even quarterly rebalancing to preserve the correct allocations following increases or decreases in the performance of individual assets held.

When we get to presenting the reality of these results, the data will confirm what everyone already knows: there is no such thing as a free lunch. That is, the higher the rate of return you wish to achieve, the more risk you have to be willing to take on. But we will see with numbers just how much more risk we’ll have to take to achieve those higher returns or how much return we need to give up to achieve more safety.

The next installment in this series will present actual results using common investment classes and will show you how to assemble a portfolio needed to achieve your desired rate of return. Stay tuned!

*For more details on MPT, take the SMC Analyzer Features Tour.

**Post-Modern Portfolio Theory (PMPT) impudently challenges this assumption, but for right now, it’s a good-enough one.

In the previous blog we looked at how the addition of an optimized market timing scheme to the asset allocation model provided by Modern Portfolio Theory (MPT) not only increases portfolio returns, sometimes substantially, but also at lower, and sometimes greatly lower, risk.

Today we’ll be expanding on yesterday’s topic of comparing two portfolios both with assets allocated according to the tenets of MPT. The first portfolio is allocated according to the traditional buy-and-hold strategy, and the second uses a market timing oscillator, the CCI, that is specifically optimized separately for each of the nine asset classes under consideration. [See the previous blog for the list of those asset classes.] We looked at the performance of each fund starting with equal dollar amounts in January of 1990 with the requirement that the funds attempt to generate at least a 10% average annual compounded return. The analysis was run through April, 2009.*

The difference in results was staggering. The buy-and-hold portfolio, which is the one that is used in typical MPT calculations, only was able to attain an average compounded return of 4.2% compared with the market timing portfolio that achieved a comparable return of 9.8%. Moreover, the risk as given by the standard deviation was 15.8% for the first portfolio compared with only 6.9% for the market timing one—a huge difference!

Where the market timing portfolio really outperformed the other was during the dot-com bubble in 2000 and in the recent mortgage meltdown in late 2007. How did it accomplish this?

Let’s examine what the SMC Analyzer would have recommended just before and during each of these major market downturns.

Just before the dot-com bubble burst
The first period we’ll be examining is the bull market preceding the internet bubble burst. The tables below show how the SMC Analyzer would have allocated assets for a buy-and-hold portfolio which is one that is long during all market conditions; and a market timing portfolio which is long when the CCI oscillator is positive for that asset class and in T-bills for that class when the oscillator is negative.

Here, still, the risk is much less for the market timing model (the second table): 10.1% compared with 18.2%. And the table shows you one reason why: the percentage of funds allocated to stocks is much less in the market timing model as compared with the buy-and-hold one.

[Note: The first row in each table shows you the percentage of your portfolio that MPT recommends you allocate to each class; the third row tells you how that percentage should be invested. In all of the above, a long position is recommended for each class.]

Let’s see move ahead a bit in time to the middle of the dot-com decline to August, 2001 and see if there’s been any change to these allocations.

Mid-dot-com decline
The following tables show that the buy-and-hold portfolio has become riskier, mainly because in order to meet the 10% return, MPT must allocate more funds to those asset classes that historically offer higher returns. In contrast, the Long/T-bill timing portfolio is long in REITs (although only at a 3.3% portfolio exposure) and Medium-term government bonds, and in T-bills for the other asset classes (given in the third row). Moving into these conservative investments during this market downturn not only prevented loss in the portfolio but actually increased returns, although not by much.

Pre- and Mid-Mortgage Meltdown
Let’s take a brief look at how these portfolios would have been allocated before and during the recent downturn.

Pre-meltdown (June, 2007):


Mid-meltdown (September, 2008):


Again, had you applied market timing to your portfolio, you would have been spared the nasty loss in stocks as you would have been 25% in Medium-term government bonds and the rest in T-bills.

What is the Analyzer recommending now?
If you’re wondering what the Analyzer is recommending now, here’s the up-to-date table for the market timing portfolio at the desired target of a 10% average annual compounded return:

Interestingly, you’d be long 25% in small-company stocks, 42% in Medium-term government bonds with the rest in T-bills (or under your mattress). We’ll see how these allocations change next month.

Summary
I hope you’re beginning to see how adding the concept of timing can save your portfolio from major damage through periods of market adversity, and how it can provide you with your desired return over an extended period of time while minimizing risk.

*Transactions costs, margin rates, trading fees, and taxes are not considered in these analyses as these factors differ with individual circumstances. The SMC Analyzer software, however, does have the capability to take all of these factors into its calculations.