Investment Planning and review

Overview

Suppose you find a great investment opportunity, but you lack the cash to take advantage of it. This is the classic problem of financing. The short answer is that you borrow -- either privately from a bank, or publicly by issuing securities. Securities are nothing more than promises of future payment. They are initially issued through financial intermediaries such as investment banks, which underwrite the offering and work to sell the securities to the public. Once they are sold, securities can often be re-sold. There is a secondary market for many corporate securities. If they meet certain regulatory requirements, they may be traded through brokers on the stock exchanges, such as the NYSE, the AMEX and NASDAQ, or on options exchanges and bond trading desks. Securities come in a bewildering variety of forms - there are more types of securities than there are breeds of cats and dogs, for instance. They range from relatively straightforward to incredibly complex. A straight bond promises to repay a loan over a fixed amount of interest over time and the principal at maturity. A share of stock, on the other hand, represents a fraction of ownership in a corporation, and a claim to future dividends. Today, much of the innovation in finance is in the development of sophisticated securities: structured notes, reverse floaters, IO's and PO's -- these are today's specialized breeds. Sources of information about securities are numerous on the world-wide web. For a start, begin with the Ohio State Financial Data Finder. All securities, from the simplest to the most complex, share some basic similarities that allow us to evaluate their usefulness from the investor's perspective. All of them are economic claims against future benefits. No one borrows money that they intend to repay immediately; the dimension of time is always present in financial instruments. Thus, a bond represents claims to a future stream of pre-specified coupon payments, while a stock represents claims to uncertain future dividends and division of the corporate assets. In addition, all financial securities can be characterized by two important features: risk and return. These two key measures will be the focus of this second module.

I. Finance from the Investor's Perspective

Most financial decisions you have addressed up to this point in the term have been from the perspective of the firm. Should the company undertake the construction of a new processing plant? Is it more profitable to replace an old boiler now, or wait? In this module, we will examine financial decisions from the perspective of the purchaser of corporate securities: shareholders and bondholders who are free to buy or sell financial assets. Investors, whether they are individuals or institutions such as pension funds, mutual funds, or college endowments, hold portfolios, that is, they hold a collection of different securities. Much of the innovation in investment research over the past 40 years has been the development of a theory of portfolio management, and this module is principally an introduction to these new methods. It will answer the basic question, What rate of return will investors demand to hold a risky security in their portfolio? To answer this question, we first must consider what investors want, how we define return, and what we mean by risk.

II. Why Investors Invest

What motivates a person or an organization to buy securities, rather than spending their money immediately? The most common answer is savings -- the desire to pass money from the present into the future. People and organizations anticipate future cash needs, and expect that their earnings in the future will not meet those needs. Another motivation is the desire to increase wealth, i.e. make money grow. Sometimes, the desire to become wealthy in the future can make you willing to take big risks. The purchase of a lottery ticket, for instance only increases the probability of becoming very wealthy, but sometimes a small chance at a big payoff, even if it costs a dollar or two, is better than none at all. There are other motives for investment, of course. Charity, for instance. You may be willing to invest to make something happen that might not, otherwise -- you could invest to build a museum, to finance low-income housing, or to re-claim urban neighborhoods. The dividends from these kinds of investments may not be economic, and thus they are difficult to compare and evaluate. For most investors, charitable goals aside, the key measure of benefit derived from a security is the rate of return.

III. Definition of Rates of Return

The investor return is a measure of the growth in wealth resulting from that investment. This growth measure is expressed in percentage terms to make it comparable across large and small investors. We often express the percent return over a specific time interval, say, one year. For instance, the purchase of a share of stock at time t, represented as Pt will yield P t+1 in one year's time, assuming no dividends are paid. This return is calculated as: R t = [ Pt+1 - Pt]/ Pt. Notice that this is algebraically the same as: Rt= [P t+1/ Pt]-1. When dividends are paid, we adjust the calculation to include the intermediate dividend payment: Rt=[ P t+1 - Pt+Dt]/ Pt. While this takes care of all the explicit payments, there are other benefits that may derive from holding a stock, including the right to vote on corporate governance, tax treatment, rights offerings, and many other things. These are typically reflected in the price fluctuation of the shares.

IV. Arithmetic vs. Geometric Rates of Return

There are two commonly quoted measures of average return: the geometric and the arithmetic mean. These rarely agree with each other. Consider a two period example: P0 = $100, R1 = -50% and R2 = +100%. In this case, the arithmetic average is calculated as (100-50)/2 = 25%, while the geometric average is calculated as: [(1+R1)(1+R2)]1/2-1=0%. Well, did you make money over the two periods, or not? No, you didn't, so the geometric average is closer to investment experience. On the other hand, suppose R1 and R2 were statistically representative of future returns. Then next year, you have a 50% shot at getting $200 or a 50% shot at $50. Your expected one year return is (1/2)[(200/100)-1] + (1/2)[(50/100)-1] = 25%. Since most investors have a multiple year horizon, the geometric return is useful for evaluating how much their investment will grow over the long-term. However, in many statistical models, the arithmetic rate of return is employed. For mathematical tractability, we assume a single period investor horizon.

Chapter II: Preferences and Investor Choice

The last chapter presented the Markowitz model of portfolio selection, but with one key element missing -- individual portfolio choice. The efficient frontier dominates all combinations of assets, however it still has infinitely many assets. How do you pick one portfolio out of all the rest as the perfect one for you? This turns out to be a big challenge, because it requires investors to express their preferences in risk-return space. Investors choose portfolios for a myriad of reasons, very few of which can be reduced to a two-dimensional space. In fact, investors are used to having the ability the CHANGE their investment decision if it is not developing as planned. The simple Markowitz model does not allow this freedom. It is a single period model, now used widely in practice for decision-making in a multi-period world. In this chapter, we will address some of the ways that one may approximate investor preferences in mean-variance space, however these methods are only approximations.

I. Choosing A Single Portfolio

How might you choose a single portfolio among all of those on the efficient frontier? One approach is to model investor preferences mathematically, using iso-utility curves. These curves express the risk-return trade-off for investors in two-dimensional space. They work exactly like lines on a topological map. They are nested lines that show the highest and lowest altitudes in the region -- except they measure altitude in units of utility (whatever that is!) instead of feet or meters. Typically, a convenient mathematical function is chosen as the basis for iso-utility curves. For instance, one could use a logarithmic function, or even part of a quadratic function to capture the essence of investor preferences. The essential feature of the function is that it must allow people to demand ever-increasing levels of return for assuming more risk.

Although the mathematics of utility functions is beyond the scope of this course, if you are interested in further investigation, I recommend visiting Campbell Harvey's Pages on Optimal Portfolios.

One way to characterize differences in investor risk aversion is by the curvature of the iso-utility lines. Below are representative curves for four different types of investors: A more risk-averse, a moderately risk-averse, a less risk-averse, and a risk-loving investor. The whole set of nested curves is omitted to keep the picture simple.

Notice that the risk-lover demands lower expected return as risk increases in order to maintain the same utility level. On the other hand, for the more risk-averse investor, as volatility increase, he or she will demand sharply higher expected returns to hold the portfolio. These different curves will result in different portfolio choices for investors. The optimization procedure simply takes the efficient frontier and finds its point of tangency with the highest iso-utility curve in the investor set. In other words, it identifies the single point that provides the investor with the highest level of utility. For risk-averse individuals, this point is unique.

The problem with applying this methodology to identifying optimal portfolios is that it is difficult to figure out the risk-aversion of individuals or institutions. Just like mapping an unknown terrain, the asset allocator must try to map the clients preference structure -- never knowing whether it is even consistent from one day to the next!

II. Another Approach: Preferences about Distributions

The Markowitz model is an elegant way to describe differences in distributions of returns among portfolios. One approach to the portfolio selection problem is to choose investment policies based upon the probability mass in the lower left-hand tail. This is called the short-fall criterion. It's simplicity has great appeal. It does not require a complete topological mapping of investor preferences. Instead it only requires the investor to specify a floor return, below which he or she wants to avoid falling. The short-fall approach chooses a portfolio on the efficient frontier that minimizes the probability of the return dropping below that floor. Suppose, for instance, your specify a floor return level equal to the riskless rate, Rf. For every portfolio on the frontier, you calculate the ratio:

Notice that the shortfall criterion is like a t-statistic, where the higher the value, the greater the probability. The portfolio that has the highest probability of exceeding Rf is the one for which this value is maximized. In fact, the similarity to a t-statistic extends even further, as we will see.

Another useful thing is that it turns out that it is quite simple to find the portfolio that maximizes the probability of exceeding the floor. You can do it graphically!

Identify the floor return level on the Y axis. Then find the point of tangency to the efficient frontier. In the figure, for instance, the tangency point minimizes the probability of having a return that drops below R floor. One particular floor value is of interest -- that is the floor given by the riskless rate, Rf. The slope of the short-fall line when Rf is the floor is called the Sharpe Ratio. The portfolio with the maximum Sharpe Ratio is the one portfolio in the economy that minimizes the probability of dropping below treasury bills. By the same token, it is the one portfolio in the economy that has the maximum probability of providing an equity premium! That is, if you must bet on one portfolio to beat t-bills in the future, the tangency portfolio found via the Sharpe Ratio would be it.

The "safety-first" approach is a versatile one. In the above example, we maximized probability of exceeding a floor by maximizing the slope, identifying a point of tangency. You can also find portfolios by other methods. For instance, you can check the feasibility of a desired floor and probability of exceeding that floor by fixing the Y intercept and fixing the slope. Either the ray will pass through the feasible set, or it will not. If it does not, then there is no portfolio that meets the criteria you specified. If it does, then there are a number of such portfolios, and typically the one with the highest expected return is the one to choose.

Another approach is to find a floor that meets your probability needs. In other words, you ask "Which floor return may I specify that will give me a 90% confidence level that I will exceed it?" This is equivalent to setting the slope equal to the t-statistic value matching that probability level. Since this is equivalent to a one-tailed test, you would set the slope to 1.28 (i.e. the quantile of the normal distribution that gives you 90% to the left, or 10% in the right side of the distribution. For a 95% chance, you would choose a slope of 1.644. For a 99% chance you would choose a slope of 2.32.

Once you choose the slope, then move the line vertically until it becomes a tangent. This will give you both a floor and a portfolio choice.

III. A Note on Value at Risk

The safety first approach can be used to calculate the value-at-risk of the portfolio. Value-at-risk is an increasingly popular measure of the potential for loss over a given time horizon. It is applied in the banking industry to calculate capital requirements, and it is applied in the investment industry as a risk control for portfolios of securities.

Consider the problem of estimating how big a loss your portfolio could experience over the next month. If the distribution of portfolio returns is normal, then a three standard deviation drop is possible, but not very likely. Typically, the estimate of the maximum expected loss is defined for a given time horizon and a given confidence interval. Consider the type of loss that occurs once in twenty months. If you know the mean and standard deviation of the portfolio, and you specify the confidence interval as a 5% event (1 in twenty months) or a 1% event (1 in a hundred months) it is straightforward to calculate the "Value at Risk."

Let Rp be the portfolio return and STDp be the portfolio standard deviation. Let T be the t-statistic associated with the confidence interval. T of 1.64 corresponds to a one in 20 month event. Let Rvar be the unknown negative return portfolio return that we expect to occur one in twenty times.

The equation for the line is: Rp = Rvar + T*STDp and thus, Rvar = Rp - T*STDp. Rvar multiplied times the value of the assets in the portfolio is the Value at Risk.

Suppose you are considering the VAR of a $100 million pension portfolio over the monthly horizon. It is composed of 60% stocks and 40% bonds, and you are interested in the 95% confidence interval.

Let us assume that the monthly expected stock return is 1% and the expected bond return is .7%, and their standard deviations are 5% and 3% respectively. Assume that the correlation between the two asset classes is .5. First we calculate the mean and standard deviation of the portfolio:

Rp = (.6)*(.01) + (.4)(.007) = .0088STDp = sqrt[ .6^2*.05^2 + .4^2*.03^2 + 2*.5*.6*.4*.05*.03] = .038Then, Rvar = .0088 - 1.64*.038 = -.054Thus, the monthly value-at-risk of the portfolio is ($100 million)(.054) = $5.4 million.

Note that, despite the terminology, this does not really mean that $94.6 is not at risk. The analysis only means that you expect a loss at least as large as $5.4 million one month out of 20.

This approach to calculating value-at-risk depends on key assumptions. First, returns must be close to normally distributed. This condition is often violated when derivatives are in the portfolio. Second, historically estimated return distributions and correlations must be representative of future return distributions and correlations. Estimation error can be a big problem when you have statistics on a large number of separate asset classes to consider. Third, returns are not assumed to be auto-correlated. When there are positive trends in the data, losses should be expected to mount up from month to month.

In summary, value at risk is becoming pervasive in the financial industry as a summary measure of risk. While it has certain drawbacks, its major advantage is that it is a probability-based approach that can be viewed as a simple extension of safety-first portfolio selection models.

IV. Conclusion

Creating an efficient frontier from historical or forecast statistics about asset returns is inherently uncertain due to errors in statistical inputs. This uncertainty is minor when compared to the problem of projecting investor preferences into mean-standard deviation space. Economists know relatively little about human preferences, especially when they are confined to a single-period model. We know people prefer more to less, and we know most people avoid risk when they are not compensated for holding it. Beyond that is guess-work. We don't even know if they are consistent, through time, in their choices. The theoretical approach to the portfolio selection problem relies upon specifying a utility function for the investor, using that to identify indifference curves, and then finding the highest attainable utility level in the feasible set. This turns out to be a tangency point. In practice, it is difficult to estimate a utility function, and even more difficult to explain it back to the investor.

An alternative to utility curve estimation is the "safety-first" technology, which is motivated by a simple question about preferences. What is your "floor" return? If you can pick a floor, you can pick a portfolio. In addition, you can identify a probability of exceeding that floor, by observing the slope of the tangency line. Safety-first also lets you find optimal portfolios by picking a floor and a probability, as well as simply picking a probability.

Value at risk is becoming increasingly popular method of risk measurement and control. It is a simple extension of the safety-first technology, when the assets comprising the portfolio have normally distributed returns.

IV. Epilogue

Notice that the introduction of a genuine risk-free security simplifies the portfolio problem for all investors in the world. Their optimal choice is reduced to the problem of choosing proportions of the riskless asset and the risky portfolio T (tangency). MRA (More Risk Averse) investors will hold a mix of tangency portfolio and T-bills, LRA (Less Risk Averse) investors will borrow at the riskless rate and invest the proceeds in the tangency portfolio.

If we could only figure out what the tangency portfolio is composed of, we could solve everyone's investment decision with the same product! What do you think T is composed of? The answer is in the next chapter.

For more information about the utility approach to risk, see the excellent write-up by Campbell Harvey on Optimal Portfolios.. For a comprehensive hyper-text book on investment decision-making, see William Sharpe's Macro-Investment Analysis

Chapter IV: The Portfolio Approach to Risk

I. The Quest For the Tangency Portfolio

In the 1960's financial researchers working with Harry Markowitz's mean-variance model of portfolio construction made a remarkable discovery that would change investment theory and practice in the United States and the world. The discovery was based upon an idealized model of the markets, in which all the world's risky assets were included in the investor opportunity set and one riskless asset existed, allowing both more and less risk averse investors to find their optimal portfolio along the tangency ray.

Assuming that investors could borrow and lend at the riskless rate, this simple diagram suggested that everyone in the world would want to hold precisely the same portfolio of risky assets! That portfolio, identified at the point of tangency, represents some portfolio mix of the world's assets. Identify it, and the world will beat a path to your door. The tangency portfolio soon became the centerpiece of a classical model in finance. The associated argument about investor choice is called the "Two Fund Separation Theorem" because it argues that all investors will make their choice between two funds: the risky tangency portfolio and the riskless "fund".

Identifying this tangency portfolio is harder than it looks. Recall that a major difficulty in estimating an efficient frontier accurately is that errors grow as the number of assets increase. You cannot just dump all the means, std's and correlations for the world's assets into an optimizer and turn the crank. If you did, you would get a nonsensical answer. Sadly enough, empirical research was not the answer, due to statistical estimation problems.

The answer to the question came from theory. Financial economist William Sharpe is one of the creators of the "Capital Asset Pricing Model," a theory which began as a quest to identify the tangency portfolio. Since that time, it has developed into much, much more. In fact, the CAPM, as it is called, is the predominant model used for estimating equity risk and return.

II. The Capital Asset Pricing Model

Because the CAPM is a theory, we must assume for argument that ...1. All assets in the world are traded 2. All assets are infinitely divisible 3. All investors in the world collectively hold all assets 4. For every borrower, there is a lender 5. There is a riskless security in the world 6. All investors borrow and lend at the riskless rate 7. Everyone agrees on the inputs to the Mean-STD picture 8. Preferences are well-described by simple utility functions 9. Security distributions are normal, or at least well described by two parameters 10. There are only two periods of time in our world

This is a long list of requirements, and together they describe the capitalist's ideal world. Everything may be bought and sold in perfectly liquid fractional amounts -- even human capital! There is a perfect, safe haven for risk-averse investors i.e. the riskless asset. This means that everyone is an equally good credit risk! No one has any informational advantage in the CAPM world. Everyone has already generously shared all of their knowledge about the future risk and return of the securities, so no one disagrees about expected returns. All customer preferences are an open book -- risk attitudes are well described by a simple utility function. There is no mystery about the shape of the future return distributions. Last but not least, decisions are not complicated by the ability to change your mind through time. You invest irrevocably at one point, and reap the rewards of your investment in the next period -- at which time you and the investment problem cease to exist. Terminal wealth is measured at that time. I.e. he who dies with the most toys wins! The technical name for this setting is "A frictionless one-period, multi-asset economy with no asymmetric information."

The CAPM argues that these assumptions imply that the tangency portfolio will be a value-weighted mix of all the assets in the world

The proof is actually an elegant equilibrium argument. It begins with the assertion that all risky assets in the world may be regarded as "slices" of a global wealth portfolio. We may graphically represent this as a large, square "cake," sliced horizontally in varying widths. The widths are proportional to the size of each company. Size in this case is determined by the number of shares times the price per share.

Here is the equilibrium part of the argument: Assume that all investors in the world collectively hold all the assets in the world, and that, for every borrower at the riskless rate there is a lender. This last condition is needed so that we can claim that the positions in the riskless asset "net-out" across all investors.

From the two-fund separation picture above, we already know that all investors will hold the same portfolio of risky assets, i.e. that the weights for each risky asset j will be the same across all investor portfolios. This knowledge allows us to cut the cake in another direction: vertically. As with companies, we vary the width of the slice according to the wealth of the individual.

Notice that each vertical "slice" is a portfolio, and the weights are given by the relative asset values of the companies. We can calculate what the weights are exactly: weight on asset i = [price i x shares i] / world wealthEach investor's portfolio weight is exactly proportional to the percentage that the firm represents of the world's assets. There you have it: the tangency portfolio is a capital-weighted portfolio of all the world's assets.

III. Investment Implications

The CAPM tells us that all investors will want to hold "capital-weighted" portfolios of global wealth. In the 1960's when the CAPM was developed, this solution looked a lot like a portfolio that was already familiar to many people: the S&P 500. The S&P 500 is a capital-weighted portfolio of most of the U.S.'s largest stocks. At that time, the U.S. was the world's largest market, and thus, it seemed to be a fair approximation to the "cake." Amazingly, the answer was right under our noses -- the tangency portfolio must be something like the S&P 500! Not co-incidentally, widespread use of index funds began about this time. Index funds are mutual funds and/or money managers who simply match the performance of the S&P. Many institutions and individuals discovered the virtues of indexing. Trading costs were minimal in this strategy: capital-weighted portfolios automatically adjust to changes in value when stocks grow, so that investors need not change their weights all the time -- it is a "buy-and-hold" portfolio. There was also little evidence at the time that active portfolio management beat the S&P index -- so why not?

IV. Is the CAPM true?

Any theory is only strictly valid if its assumptions are true. There are a few nettlesome issues that call into question the validity of the CAPM: Is the world in equilibrium? Do you hold the value-weighted world wealth portfolio? Can you even come close? What about "human capital?" While these problems may violate the letter of the law, perhaps the spirit of the CAPM is correct. That is, the theory may me a good prescription for investment policy. It tells investors to choose a very reasonable, diversified and low cost portfolio. It also moves them into global assets, i.e. towards investments that are not too correlated with their personal human capital. In fact, even if the CAPM is approximately correct, it will have a major impact upon how investors regard individual securities. Why?

V. Portfolio Risk

Suppose you were a CAPM-style investor holding the world wealth portfolio, and someone offered you another stock to invest in. What rate of return would you demand to hold this stock? The answer before the CAPM might have depended upon the standard deviation of a stock's returns. After the CAPM, it is clear that you care about the effect of this stock on the TANGENCY portfolio. The diagram shows that the introduction of asset A into the portfolio will move the tangency portfolio from T(1) to T(2).

The extent of this movement determines the price you are willing to pay (alternately, the return you demand) for holding asset A. The lower the average correlation A has with the rest of the assets in the portfolio, the more the frontier, and hence T, will move to the left. This is good news for the investor -- if A moves your portfolio left, you will demand lower expected return because it improves your portfolio risk-return profile. This is why the CAPM is called the "Capital Asset Pricing Model." It explains relative security prices in terms of a security's contribution to the risk of the whole portfolio, not its individual standard deviation.

VI. Conclusion

The CAPM is a theoretical solution to the identity of the tangency portfolio. It uses some ideal assumptions about the economy to argue that the capital weighted world wealth portfolio is the tangency portfolio, and that every investor will hold this same portfolio of risky assets. Even though it is clear they do not, the CAPM is still a very useful tool. It has been taken as a prescription for the investment portfolio, as well as a tool for estimating an expected rate of return. In the next chapter, we will take a look at the second of these two uses.