1. EQULIBRIUM, MUTUAL FUNDS AND SHARPE RATIO Luis A Pons Perez Christian Robles Econometría y Modelos de Finanzas Dr. BalbinoGarcía 1

2. Optimal Portfolio The optimal portfolio concept falls under the modern portfolio theory. The theory assumes (among other things) that investors fanatically try to minimize risk while striving for the highest return possible. The theory states that investors will act rationally, always making decisions aimed at maximizing their return for their acceptable level of risk. 2

3. Optimal Portfolio The optimal portfolio was used in 1952 by Harry Markowitz, and it shows us that it is possible for different portfolios to have varying levels of risk and return. Each investor must decide how much risk they can handle and than allocate (or diversify) their portfolio according to this decision. 3

4. Optimal Portfolio 4

5. Two-fund Separation Is a theorem stating that, under certain conditions, any investor's optimal portfolio can be constructed by holding each of certain mutual funds in appropriate ratios, where the number of mutual funds is smaller than the number of individual assets in the portfolio. Here a mutual fund refers to any specified benchmark portfolio of the available assets. There are two advantages of having a mutual fund theorem: a. First, if the relevant conditions are met, it may be easier (or lower in transactions costs) for an investor to purchase a smaller number of mutual funds than to purchase a larger number of assets individually. b. Second, from a theoretical and empirical standpoint, if it can be assumed that the relevant conditions are indeed satisfied, then implications for the functioning of asset markets can be derived and tested. 5

6. MEASURES OF RETURN MEASURES OF RETURN complicated by addition or withdrawal of money by the investor percentage change is not reliable when the base amount may be changing timing of additions or withdrawals is important to measurement 6

7. MEASURES OF RETURN TWO MEASURES OF RETURN Dollar-Weighted Returns uses discounted cash flow approach weighted because the period with the greater number of shares has a greater influence on the overall average 7

8. MEASURES OF RETURN TWO MEASURES OF RETURN Time-Weighted Returns used when cash flows occur between beginning and ending of investment horizon ignores number of shares held in each period 8

9. MEASURES OF RETURN TWO MEASURES OF RETURN Comparison of Time-Weighted to Dollar-Weighted Returns Time-weighted useful in pension fund management where manager cannot control the deposits or withdrawals to the fund 9

10. MAKING RELEVANT COMPARISONS PERFORMANCE should be evaluated on the basis of a relative and not an absolute basis this is done by use of a benchmark portfolio BENCHMARK PORTFOLIO should be relevant and feasible reflects objectives of the fund reflects return as well as risk 10

11. THE USE OF MARKET INDICES INDICES are used to indicate performance but depend upon the securities used to calculate them the calculation weighting measures 11

12. THE USE OF MARKET INDICES INDICES Three Calculation Weighting Methods: price weighting sum prices and divided by a constant to determine average price EXAMPLE: THE DOW JONES INDICES 12

13. THE USE OF MARKET INDICES INDICES Three Calculation Weighting Methods: value weighting (capitalization method) price times number of shares outstanding is summed divide by beginning value of index EXAMPLE: S&P500 WILSHIRE 5000 RUSSELL 1000 13

14. THE USE OF MARKET INDICES INDICES Three Calculation Weighting Methods: equal weighting multiply the level of the index on the previous day by the arithmetic mean of the daily price relatives EXAMPLE: VALUE LINE COMPOSITE 14

15. ARITHMETIC V. GEOMETRIC AVERAGES GEOMETRIC MEAN FRAMEWORK GM = (HPR)1/N - 1 where = the summation of the product of HPR= the holding period returns n= the number of periods 15

16. ARITHMETIC V. GEOMETRIC AVERAGES GEOMETRIC MEAN FRAMEWORK measures past performance well represents exactly the constant rate of return needed to earn in each year to match some historical performance 16

17. ARITHMETIC V. GEOMETRIC AVERAGES ARITHMETIC MEAN FRAMEWORK provides a good indication of the expected rate of return for an investment during a future individual year it is biased upward if you attempt to measure an asset’s long-run performance 17

18. RISK-ADJUSTED MEASURES OF PERFORMANCE THE REWARD TO VOLATILITY RATIO (TREYNOR MEASURE) There are two components of risk risk associated with market fluctuations risk associated with the stock Characteristic Line (ex post security line) defines the relationship between historical portfolio returns and the market portfolio 18

19. TREYNOR MEASURE TREYNOR MEASURE Formula wherearp = the average portfolio return arf = the average risk free rate bp= the slope of the characteristic line during the time period 19

20. TREYNOR MEASURE THE CHARACTERISTIC LINE 20 SML arp bp

21. TREYNOR MEASURE CHARACTERISTIC LINE slope of CL measures the relative volatility of portfolio returns in relation to returns for the aggregate market, i.e. the portfolio’s beta the higher the slope, the more sensitive is the portfolio to the market 21

22. TREYNOR MEASURE THE CHARACTERISTIC LINE 22 SML arp bp

23. THE SHARPE RATIO THE REWARD TO VARIABILITY (SHARPE RATIO) measure of risk-adjusted performance that uses a benchmark based on the ex-post security market line total risk is measured by sp 23

24. THE SHARPE RATIO SHARPE RATIO formula: where SR = the Sharpe ratio sp = the total risk 24

25. THE SHARPE RATIO SHARPE RATIO indicates the risk premium per unit of total risk uses the Capital Market Line in its analysis 25

26. THE SHARPE RATIO 26 CML arp sp

27. THE JENSEN MEASURE OF PORTFOLIO PERFORMANCE BASED ON THE CAPM EQUATION measures the average return on the portfolio over and above that predicted by the CAPM given the portfolio’s beta and the average market return 27

28. THE JENSEN MEASURE OF PORTFOLIO PERFORMANCE THE JENSEN MEASURE known as the portfolio’s alpha value recall the linear regression equation y = a + bx + e alpha is the intercept 28

29. THE JENSEN MEASURE OF PORTFOLIO PERFORMANCE DERIVATION OF ALPHA Let the expectations formula in terms of realized rates of return be written subtracting RFR from both sides 29

30. THE JENSEN MEASURE OF PORTFOLIO PERFORMANCE DERIVATION OF ALPHA in this form an intercept value for the regression is not expected if all assets are in equilibrium in words, the risk premium earned on the jth portfolio is equal to bj times a market risk premium plus a random error term 30

31. THE JENSEN MEASURE OF PORTFOLIO PERFORMANCE DERIVATION OF ALPHA to measure superior portfolio performance, you must allow for an intercept a a superior manager has a significant and positive alpha because of constant positive random errors 31

32. COMPARING MEASURES OF PERFORMANCE TREYNOR V. SHARPE SR measures uses s as a measure of risk while Treynor uses b SR evaluates the manager on the basis of both rate of return performance as well as diversification for a completely diversified portfolio SR and Treynor give identical rankings because total risk is really systematic variance any difference in ranking comes directly from a difference in diversification 32

33. CRITICISM OF RISK-ADJUSTED PERFORMANCE MEASURES Use of a market surrogate Roll: criticized any measure that attempted to model the market portfolio with a surrogate such as the S&P500 it is almost impossible to form a portfolio whose returns replicate those over time making slight changes in the surrogate may completely change performance rankings 33

34. CRITICISM OF RISK-ADJUSTED PERFORMANCE MEASURES measuring the risk free rate using T-bills gives too low of a return making it easier for a portfolio to show superior performance borrowing a T-bill rate is unrealistically low and produces too high a rate of return making it more difficult to show superior performance 34

35. 35 ¡Gracias y Éxito!