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Year :to Reserves Journal of Actuarial Practice, Vol. Figure 1. Constant Duration Effect.. Let there be M lines of insurance or individual Langevoort, Taming Grades mailed here Street, Apt. Nov 24, — PDF. Abstract; 1. Markowitz recognized this limitation and, in fact, suggested a measure of downside risk — the risk of realizing an outcome below the expected return — called the semi-variance. The semi-variance is similar to the variance except that in the calculation no consideration is given to returns above the expected return.

Today, various measures of downside risk are currently being used by practitioners. However, regardless of the measure used, the basic principles of portfolio theory developed by Markowitz are applicable.

The second criticism is that the variance is only one measure of how the returns vary around the expected return. When a probability distribution is not symmetrical around its expected return, then a statistical measure of the skewness of a distribution should be used in addition to the variance.

It depends not only on the variance of the two assets, but also upon how closely one asset tracks the other asset. The covariance is a new term in this discussion and has a precise mathe- matical translation. However, its practical meaning is the degree to which the returns on two assets vary or change together. The covariance is not expressed in a particular unit, such as dollars or percent. A positive covariance means the returns on two assets tend to move or change in the same direction, while a negative cova- riance means the returns move in opposite directions.

The covariance is analogous to the correlation between the returns for two assets. Dividing the covariance by the product of the standard deviations simply but importantly makes the correlation a number that is comparable across different assets.

In our illustration of how to calculate the expected return, variance, and standard deviation, we used the probability distributions for the stock. In practice, the estimation of these statistical measures is typically obtained from historical observations on the rate of returns.

We show how this is done in the next chapter. This principle has important implications for managing portfolios, as we shall see below. Measuring the Risk of a Portfolio with More Than Two Assets Thus far we have given the portfolio risk for a portfolio consisting of two assets.

Hence, the variance of the portfolio return is the weighted sum of the individual variances of the assets in the portfolio plus the weighted sum of the degree to which the assets vary together. This is certainly a goal that investors should seek.

However, the question is how does one do this in practice. First, how much should be invested in each asset class? By this they mean that an investor should not place all funds in the stock of one corporation, but rather should include stocks of many corporations.

First, which corporations should be represented in the portfolio? Second, how much of the portfolio should be allocated to the stocks of each corporation? Then we will look at the effects on portfolio risk of combining assets with different correlations. However, the expected portfolio return remains This is due to the degree of corre- lation between the asset returns. The good news is that investors can maintain expected portfolio return and lower portfolio risk by combining assets with lower and preferably negative correlations.

However, the bad news is that very few assets have small to negative correlations with other assets! The problem, then, becomes one of searching among large numbers of assets in an effort to discover the portfolio with the minimum risk at a given level of expected return or, equiva- lently, the highest expected return at a given level of risk. That is, investors make deci- sions using the two-parameter model formulated by Markowitz. Second, it assumes that investors are risk averse i.

Third, it assumes that all investors seek to achieve the high- est expected return at a given level of risk. Fourth, it assumes that all investors have the same expectations regarding expected return, variance, and covariances for all risky assets. This assumption is referred to as the homogeneous expecta- tions assumption. Finally, it assumes that all investors have a common one-period investment horizon. Hence, for a portfolio of just 50 securities, there are 1, covariances that must be calculated.

For securi- ties, there are 4, Furthermore, in order to solve for the portfolio that mini- mizes risk for each level of return, a mathematical technique called quadratic programming and a computer must be used.

The basic idea behind these alternative methods is that an esti- mated relationship between the return on a stock and some common stock market index or indexes can be used in lieu of the variance and covariance of returns.

One of the approaches suggested by Sharpe uses one common stock index, and the approach is referred to as the single-index market model. See William F. Multiple indexes have been suggested in Kalman J. Cohen and Jerry A. Feasible and Efficient Portfolios A feasible portfolio is a portfolio that an investor can construct given the assets available.

The collection of all feasible portfolios is called the feasible set of port- folios. With only two assets, the feasible set of portfolios is graphed as a curve that represents those combinations of risk and expected return that are attainable by constructing portfolios from the available combinations of the two assets.

If combinations of more than two assets were being considered, the feasible set is no longer the curved line. It would be approximated by the shaded area in Exhibit 4. To see this, consider portfolio 6 in Exhibit 3. Port- folios 4 and 6 have the same level of risk, but portfolio 4 has a higher expected return. Likewise, portfolios 2 and 6 have the same expected returns, but portfolio 2 has a lower level of risk.

Thus, portfolios 4 and 2 are said to dominate portfolio 6. The question is, which is the best portfolio to hold? As explained at the begin- ning of this chapter, this preference can be expressed in terms of a utility function. In our application, the indifference curve indicates the com- binations of risk and expected return that give the same level of utility.

Moreover, the farther from the horizontal axis the indifference curve, the higher the utility. From Exhibit 5, it is possible to determine the optimal portfolio for the investor with the indifference curves shown. If this investor had a different preference for expected risk and return, there would have been a different optimal portfolio.

Unfortunately, there is little guidance about how to construct one. In general, economists have not been successful in measuring utility functions. The asset pricing models we describe in this chapter are equilibrium models. Some of these assumptions may even seem unrealistic. However, these assumptions make the theory more tracta- ble from a mathematical standpoint. The theory assumes all investors make investment decisions over some single-period investment horizon.

How long that period is i. In reality, the investment decision process is more complex than that, with many investors having more than one investment horizon — such as short-term and long-term planning horizons.

Nonetheless, the assumption of a one-period investment horizon is necessary to simplify the math- ematics of the theory. It is also necessary to make assumptions about the character- istics of the capital market in which investors transact. These are covered by the last two assumptions. Capital Market Theory Earlier in this chapter we distinguished between a risky asset and a risk-free asset. The point of tangency is denoted by M. Portfolios to the left of M represent combinations of risky assets and the risk-free asset.

Portfolios to the right of M include purchases of risky assets made with funds borrowed at the risk-free rate. Such a portfolio is called a levered port- folio since it involves the use of borrowed funds. Notice that for the same risk the expected return is greater for PB than for PA. A risk-averse investor will prefer PB to PA. That is, PB will dominate PA. The investor will select the portfolio on the line that is tangent to the highest indifference curve. In the absence of a risk-free asset, it would not be possible to construct such a portfolio.

One more key question remains: How does an investor construct portfo- lio M? Eugene Fama answered this question by demonstrating that M must consist of all assets available to investors, and each asset must be held in proportion to its market value relative to the total market value of all assets.

Because portfolio M consists of all assets, it is referred to as the market portfolio. This should be done by combining an investment in the risk-free asset and the market portfolio.

The capital market line can be derived algebraically. The numerator is the expected return of the market in excess of the risk-free return. It is a measure of the risk premium, or the reward for holding the risky market portfolio rather than the risk-free asset. The denominator is the risk of the market portfolio. Thus, the slope measures the reward per unit of market risk.

Since the CML represents the return offered to compensate for a perceived level of risk, each point on the line is a balanced market condition, or equilibrium.

The slope of the line determines the additional return needed to compensate for a unit change in risk. That is why the slope of the CML is also referred to as the equilibrium market price of risk. The CML says that the expected return on a portfolio is equal to the risk- free rate plus a risk premium equal to the price of risk as measured by the differ- ence between the expected return on the market and the risk-free rate divided by the standard deviation of the market return times the quantity of market risk for the portfolio as measured by the standard deviation of the portfolio.

Based on this result, a model can be derived that shows how a risky asset should be priced. This risk measure can be divided into two general types of risk: systematic risk and unsystematic risk. This is the risk that is unique to a company, such as a strike, the outcome of unfavorable litigation, or a natural catas- trophe.

Both of these unforecastable and hence unexpected trag- edies had negative impacts on the stock prices of the two companies involved. Therefore the total risk of an asset can be measured by its variance. However, the total risk can be divided into its systematic and unsystematic risk components. Next we will show how this can be done so as to be able to quantify both components. For example, if a stock has a beta of 1.

The beta for the market portfolio is 1. The term in the market model, popularly referred to as alpha is equal to the average value over time of the unsystematic returns for the stock. For most stocks, alpha tends to be small and unstable. Recall from our earlier discussion, the total risk of an asset can be decomposed into market or systematic risk and unique or unsystematic risk.

We can use the market model to quantify these two risks. This is done by determining the variance of equation 9. Another product of the statisti- cal technique used to estimate beta is the percentage of systematic risk to total risk. The Security Market Line The capital market line represents an equilibrium condition in which the expected return on a portfolio of assets is a linear function of the expected return on the market portfolio.

This ver- sion of the risk- return relationship for individual securities is called the security market line SML. As in the case of the CML, the expected return for an asset is equal to the risk-free rate plus the product of the market price of risk and the quantity of risk associated with the security. Another more common version of the SML relationship uses the beta of a security.

To see how this relationship is developed, look back at equation The higher the beta, the higher the expected return. The beta of a risk-free asset is zero, because the variability of the return for a risk-free asset is zero and therefore it does not covary with the market portfolio. Thus, the return on a risk-free asset is simply the risk-free return. Of course, this is what we expect. The beta of the market portfolio is 1.

In this case, the expected return for the asset is the same as the expected return for the market portfolio. If an asset has a beta greater than the market portfolio i. The reverse is true if an asset has a beta less than the market portfolio.

A graph of the SML is presented in Exhibit 8. It follows that the only risk that investors will pay a premium to avoid is market risk.

Hence, two assets with the same amount of systematic risk will have the same expected return. There is one more version of the SML that is worthwhile to discuss. An asset that has a positive covariance will have a higher expected return than the risk-free asset; an asset with a negative covariance will have a lower expected return than the risk-free asset.

If the covariance is positive, this increases the risk of an asset in a portfolio and therefore investors will only purchase that asset if they expect to earn a return higher than the risk-free asset. If an asset has a negative covariance, recall from our discussion earlier in this chapter that this will reduce the portfolio risk and investors would be willing to accept a return less than the risk-free asset.

The market model is an ex post, or descriptive, model used to describe historical data. Hence, the market model makes no prediction of what expected returns should be. One bibliographic compilation lists almost 1, papers on the topic. The major implication that is tested is that beta should be the only factor that is priced by the market. Several studies have discovered other fac- tors that explain stock returns.