Investment Analysis and Portfolio Management Lecture 4 Gareth Myles

Efficiency An efficient investor will choose the portfolio with the maximum return for a given level of risk Or the minimum risk for a given return Any other choice is inefficient To explore this idea it is necessary to determine the frontier of efficient portfolios The set of portfolios with maximum return for given risk

Two-Asset Portfolios Consider a portfolio composed of two assets A and B The expected return on the portfolio is The standard deviation of return is The relationship between return and standard deviation is now derived

No Short Sales Assume initially that there are no short sales The holdings of both assets must be non- negative This implies the asset proportions satisfy Now consider the standard deviation for various values of the correlation coefficient

Perfect Positive Correlation Case 1: Perfect positive correlation The standard deviation becomes The term in the bracket is a perfect square

No Short Sales Taking the square root The return is This pair of equations determine a linear relation between expected return and standard deviation The trade-off between risk and return is described by a straight line

Perfect Positive Correlation An example This gives the linear relations Expected return %Standard deviation % Allied Motors146 Brown Engineers83

Perfect Positive Correlation A B ( X A = 1, X B = 0 ) ( X A = 0, X B = 1 )

Perfect Negative Correlation Case 2: Perfect negative correlation This equation has two solutions For the example

Perfect Negative Correlation Notice that at The existence of a portfolio with zero risk is a general property when Notice also that So

Perfect Negative Correlation A B Efficient Inefficient ( X A = 1, X B = 0 ) ( X A = 0, X B = 1 )

Intermediate Values Case 3: In the intermediate case the frontier must lie between that for the two extremes It must be curved with no straight segments

Intermediate Values A B Efficient Inefficient Minimum variance portfolio ( X A = 0, X B = 1 ) ( X A = 1, X B = 0 )

Minimum Variance Portfolio Note that for every value of there is a portfolio with minimum variance This is the one furthest to the left. It is found be choosing to minimize Taking the first-order condition and solving gives

An Example

pp rprp _

The Efficient Frontier The efficient frontier is the set of portfolios that have a higher return than the minimum variance portfolio With additional assets the frontier is traced out be considering all possible portfolios Any portfolio below the frontier is dominated by one on the frontier There are no portfolios that allow risk/return combinations above the efficient frontier The frontier is always concave

The Efficient Frontier A B Inefficient C Efficient

Allowing Short Sales With short sales, no restrictions are placed on the proportions and except that Allowing short selling introduces negative proportions This extends the frontier at both ends It extends indefinitely

Allowing Short Sales X A = 1, X B = 0 X A > 1, X B < 0 X A = 0, X B = 1 X A 1

Riskfree Borrowing and Lending Consider combining a portfolio, A, and the riskfree asset in proportions X and 1 - X This gives expected return And standard deviation But and so

Riskfree Borrowing and Lending Hence Giving The attainable risk and return combinations lie on a straight line

Risk-free Borrowing and Lending Hence the efficient frontier is a straight line It is tangential to the frontier without the risk- free asset a b c d

Risk-free Borrowing and Lending X f = 1, X T = 0 X f 1 X f = 0, X T = 1 rfrf Efficient frontier Portfolio T : a mix of A and B

Borrowing and Lending Rates The above has assumed the interest rate for borrowing = interest rate for saving In practice the borrowing rate is higher Explained by asymmetric information Borrowers other than governments cannot be treated as risk-free Risky borrowers must pay a higher rate of interest Borrowing rate r b > lending rate r f

Borrowing and Lending Rates X A = 1, X B = 0 X A > 1, X B < 0 X A = 0, X B = 1 X A 1 rfrf Efficient frontier rbrb Efficient frontier now in three sections