He is currently a director of the Daiwa closed-end funds and the Aberdeen Singapore Fund. He was president of the Western Finance Association and secretary/treasurer of that organization. Beinecke Professor of Financial and Management Studies and Director of the International Center for Finance at the Yale School of Management.
He was chairman of the Western Finance Association and the European Finance Association. Recognition of the causes of the 2008 financial crisis and the financial instruments that caused the crisis. These breakthroughs include simplifying the amount and type of inputs to the portfolio problem (Chapters 7 and 8), as well as simplifying the computational procedure for finding sets of desirable portfolios (Chapter 9).
We conclude Part 2 with a discussion of the potential benefits of diversifying portfolios internationally. Each professor's preference and the course dictates will ultimately determine the final choice.
INTRODUCTION
The first part of the analysis is to determine what options the investor has. In the next two chapters, we deal with the basic notions of the investor's opportunity put at risk. While an investor can buy any of the instruments described here (and more we haven't touched on), the investor can instead choose to invest indirectly by buying shares in investment companies (mutual funds).
Very often, back-end loads decrease as a function of the time the investor holds the fund. A day order instructs the broker to execute the order by the end of the day. The amount of the accentuation depends on the percentage of the purchase that the investor paid in cash.
As with long purchases, there are margin requirements at the time of the trade (initial margin) and margin requirements that must be met at all times (maintenance margin). Short selling margin is calculated as a percentage of the market value of the short.
PORTFOLIO ANALYSIS
A measure of this is the average (overall observations) of the squared deviations below the mean. The expected return is also a weighted average of the expected returns on the individual assets. If you take the expected value of the expression just given for the return on a portfolio dividend.
Determine the expected return and variance of each of the portfolios shown below. The same is not necessarily true of the risk (standard deviation of return) of the portfolio. In Chapter 4, the standard deviation of the portfolio return was shown to be equal to.
Since the return is certain, the standard deviation of the return on the risk-free asset must be zero. Thus, the slope of the line connecting the tangent portfolio and the efficient frontier.
APPENDIX A
For this manager, the efficient frontier is likely to be the straight line segment from RF to the point of tangency and the curved shape from there to the right. Given the low return of the tangent portfolio, the choice is likely to lie on the curve to the right of the tangent portfolio. It would be important to vary the inputs over a reasonable range to see how this composition would change given reasonable changes in the inputs.
In this chapter we discussed and illustrated the use of techniques that can be used to solve for the set of all possible portfolios that are efficient. All of the solution techniques discussed are feasible and have been used to solve problems. However, the techniques require gigantic amounts of input data and large amounts of computation time.
Furthermore, the input data is in a form that is not easy for the security analyst and portfolio manager to understand. For this reason, it is difficult to obtain estimates of the input data or to get practitioners to relate them to the final output. The next logical step is to simplify the number and type of input requirements for portfolio selection and, in turn, to see whether this reduction in data complexity can also be used to simplify the calculation procedure.
The reader should note that in the Lintnerian definition of short sales, the final portfolio weights must be scaled so that the sum of the absolute values of the weights, rather than their sum, is 1.
APPENDIX B
The reader should note that in the Lintnerian definition of short sales, the final portfolio weights must be scaled so that the sum of the absolute value of the weights, rather than their sum, is 1. Two rules from calculus are needed:. The product rule states that the derivative of the product of two functions is the first function times the derivative of the second function plus the second times the derivative of the first. The derivative terms that do not involve Xkare zero (they are constants as far as Xkis are concerned).
The chain rule states that its derivative is the power, times the expression in parentheses to the power minus one, times the derivative of what is inside the parentheses. All terms not involving k are constant with respect to k, and so their derivative is zero.
APPENDIX C
When the number of equations and variables becomes large, it is usually more convenient to solve the problem by working on a tableau.
APPENDIX D
The defined Incorrect values can be scaled to sum to 1 exactly as done before so that the optimal proportions can be determined. Rather, we simply solved the system of simultaneous equations for two arbitrary values of RF. Thus, if we have the optimal portfolio for any two values of RF, we can obtain the value for C0kand C1kand then, by varying RF, obtain the full efficient frontier.
This means that solving the system of simultaneous equations for any two values of RF allows us to trace the entire efficiency frontier. Then, by varying the RF in the appropriate range, the effective limit could be traced. We showed that solving a system of simultaneous equations for any two values of RF allowed us to obtain a general
This suggests that the efficient frontier can be determined directly by simply computing two of the optimal portfolios rather than indirectly by first determining Zkas a function of RF. This is an extremely powerful result and is the preferred way to determine the effective set. In the previous chapter, we showed how to track all combinations (portfolios) of two assets.
Thus, if we find two efficient portfolios, we can use the procedures discussed in the previous chapter to trace the full efficiency frontier, that is, if we trace the efficient frontier by combining the two efficient portfolios. If we knew the covariance between the portfolios associated with RF ⫽ 5 and RF⫽2, we could trace the frontier of full efficiency by treating each portfolio as an asset and using the method discussed in Chapter 5. Knowing the variance of the expected return and the covariance, we can trace the efficient frontier exactly as we did for combinations of two assets in Chapter 5.
But for all other values of RF, it will be kept either long or short. In fact, for all RF values above this value, the fuse will only be held long or short, and vice versa for RF values. Finally, security 3 is held long for RFs greater than ⫺4 and short for RF values below ⫺4.
APPENDIX E
These results can be illustrated with a simple example. Consider the returns on a stock
- A. Compute the mean return and variance of return for each stock in Problem 1 using
- A. If the Blume adjustment equation is fit and the appropriate equation is
Now consider the values for the single index model shown in the remaining columns of the table. The reader should now understand where all the values of the single index model come from, except i. As the number of stocks in the portfolio increases, so does the importance of the average residual risk.
6 Examining the expression for the variance of portfolio P shows that the assumptions of the single index model are inconsistent with 2P 2m. The beta value for the stock is the covariance of the stock with the market divided by the variance of the market, or. 1 stands for the variance of the distribution of the historical estimates of beta over the sample of stocks.
The Effect of Firm Capital Structure on the Systematic Risk of Common Stocks, "Journal of Finance, VII, No. Factor Variability of the New York Stock Exchange," Journal of Business, 46, No. Similarly, bi2 is a measure of the sensitivity of a stock's return to this change.
There is nothing in the model's estimates that forces this to be true. It can be shown that the model Cohen and Pogue call the diagonal form of the multi-index model is identical to the form we have discussed. 6The specialized form of the model they tested was their diagonal form of the multi-index model.
In each case, using the overall mean model outperformed the single-index model, the multi-index model, and the historical correlation matrix. The second is the shape of the discount function (the rate at which the investor discounts cash flows far into the future versus the rate used to discount near-term cash flows). That part of the market index that is uncorrelated with the six indices described previously.
A Critical Reexamination of the Empirical Evidence on the Arbitrage Pricing Theory,” The Journal of Finance, 39 (June 1984), pp. The value of C* is calculated from the properties of all the securities belonging to the optimal portfolio.
