Some Formulas neglected in Anderson, Sweeny, and Williams, with a Digression on Statistics and Finance

Transcription

1 Some Formulas neglected in Anderson, Sween, and Williams, with a Digression on Statistics and Finance Transformations of a Single Random variable: If ou have a case where a new random variable is defined as a linear transformation of a another random variable, there is an eas wa to get the mean and variance of the new random variable. Assume ou are given Y = a+ bx, where a and b are constants, and Y is a new random variable being defined as this linear transformation of the random variable X. If we know the mean and variance of the random variable X, we can easil find the mean and variance of the random variable Y using the following two formulas. µ = a+ bµ. = b A word of warning: this shortcut onl works if Y is a linear transformation of X. For instance, if Y = X it is not true that µ = µ. If Y = X (or for an non-linear transformation) we must use the equation that Eg = gf. An Eample of a Non-Linear Transformation c bgh bgbg Suppose the random variable X has the following distribution: f() It is easil verified (we did the eample in class) that the mean of the random variable is E( X) = f ( ) = 0.9. Suppose g( ) ( ) E( ) E = 0 =. One might epect that = = 0.9 = However, this is WRONG. The correct calculation is f() f() ( ( )) ( ) E g = f = Wh is this important? It is central in the economics of uncertaint. When an agent s utilit depends on an uncertain outcome (a ver common case in the real world) the utilit achieved becomes random, so the agent can t be thought of as maimizing utilit. The simpliest generalization of utilit theor that can cover this case is to assume that under uncertaint, agents maimize epected utilit. If utilit, U, depends on a 1

2 random variable X which for concreteness ou can think of as wealth then the agent maimizes EU ( ( ) ) = U( f ) ( ). The fact that E( g( ) ) g( E( ) ) has important implications in economic theor. For instance, it implies that when wealth is uncertain, an epected utilit maimizer ma not wish to select a strateg that maimizes epected wealth. A concrete eample of this phenomenon is given as additional problem number one in the chapter 5 homework. The covariance of two random variables defined: In an earlier chapter, ou were introduced to the covariance and correlation coefficient, both of which measure the degree of linear association between two variables X and Y. If ou have a joint probabilit distribution of two random variables, fb, g, ou can define a population covariance and population correlation coefficient between these two random variables, in a wa that is ver similar to the definition ou saw before. The formulas (for discrete random variables) are: = µ µ f, and ρ =. b gd i b g One of the important uses of the covariance is to give us a generalization of the formula above to the case where a random variable Z is defined as a linear combination of two random variables, X and Y. A simple eample. Suppose the joint distribution of X and Y is given b Where X takes on the values and 4, and Y takes on the values 1 and. It is eas to calculate that µ =, µ =. Therefore, using the formula above, we discover that = ( )( 1 )(.) + ( 4 )( 1 )(.) + ( )( )(.) + ( 4 )( )(.) = =. The marginal distributions of X and Y are both ver simple. The are f() f() In both cases it is eas to compute the standard deviation of both X and Y, and each is equal to one. So in this special case the correlation coefficient is also equal to.0. Linear Combinations of two random variables: an application: This eample introduces the formulas for the mean and variance of a linear combination of two random variables, and shows how the plas an important role in finance theor. The formulas I am referring to sa that if a random variable Z is defined as a linear combination of two other random variables, X and Y, so that Z = ax + by

3 then the mean and variance of Z are given b: EZ b g = aex b g + bey b g. z = a + b + ab For eample, suppose ou are investing, and have two stocks to choose between. The first stock is a "safe" stock. Its return is a random variable, X, with a mean return of 10% and a variance of 4. The second stock is a "risk". Its return is a random variable, Y, with a mean return of 15% and a variance of 5. The returns on the two stocks are correlated, with a correlation coefficient of.0. That implies the covariance of the returns is: ρ =. 0 = = 0. 5 You don't have to hold all our wealth in a single stock, however. You can divide our wealth between X and Y, putting a fraction ω in stock X and a fraction 1 ω in stock Y. If ou divide our wealth in this wa, the return on our portfolio will be a random variable, Z, equal to Z = ωx + b1 ωg Y. The mean and variance of the return on our portfolio, then, depend on ω, the fraction of our wealth ou put in the safe stock, and can be calculated from the two formulas given above. For instance, if ou put 0% of our wealth in the safe stock, our average return is 1.5%. EZ bg= aex bg+ bey bg 15. = b. 10g+ b. 7 15g The variance of our return is = a + b + ab z c h c h b g = We can find a sort of "production possibilities frontier" b considering all the different choices of ω and repeatedl using these formulas. I have done that in a spreadsheet program, with these results.

4 ω E( Z ) Z What's especiall interesting about this is that the safest portfolio, that is, the one with the smallest variance, is not the one in which ou hold onl the safest stock! In fact, the safest possible portfolio is the one with 9% of our wealth in the safe asset, and 8% in the risk asset. 1 All portfolio choices with more than 9% of our wealth in the safe asset are dominated and can never be optimal, because ou can alwas achieve a higher return without accepting an more risk. Plotting the points that can be achieved through different portfolio choices traces out a sort of production possibilities frontier of risk and return. 1 The variance of return, for an arbitrar fraction ω in the safe asset, is given b: ( ) 4 5( 1 ) 4 ( 1 ) ω ω ω ω ω = + +. We can use calculus to find the share, ω, that minimizes the variance of the portfolio return. d ( ω) = 8ω 50( 1 ω) + 4 8ω = 0 dω 50 = 46 ω ω =.9 4

5 The Risk-Return Frontier (Data Labels show portfolio fraction in safe asset) Epected Return Standard Deviation of Return 4 5 Strictl speaking, the points showing more than 9% of assets in the safe securit are not part of the risk-return frontier, but the are shown here nonetheless for purposes of illustration. One useful result of the analsis is to show that some portfolios are inefficient, and should never be selected. In looking at the diagram, ou might feel that this is of little practical importance, since onl a small fraction of all portfolios are ruled out in this wa. However, this is because we onl have two assets in the model. In a more complicated version of this model, with several assets, the fraction of all portfolios that are ruled out as inefficient can be quite large. Having derived the risk return frontier that can be achieved b different portfolio choices, we onl need to introduce agent s preferences towards risk and return to identif their preferred portfolio. 5

6 We can summarize an agent s preferences with an indifference map. The map will look different from those ou usuall see in Econ 01, since risk as measured b the standard deviation of return is considered a bad. The individual will want to move to the northwest in the diagram; that is, to an area of high return and low risk. The optimal portfolio can be found b superimposing the indifference map and the production possibilities frontier pictured above. As in Econ 01, the optimum is found at the tangenc of an indifference curve with a production possibilities frontier. Determining the Optimal Portfolio 15 R E T U R N Risk-Return Frontier Optimal Portfolio Indifference Curves 10 RISK 4 5 The optimal portfolio for this individual ields a return of about 1.75%, which (judging from the table) corresponds to a portfolio where 45% of the agent s wealth is in the safe asset, and 55% is in the risk asset. 6