Financial Instruments and Investment Instruments. Lecture 11: Portfolio Performance Analysis and Measurement

Transcription

1 Financial Instruments and Investment Instruments Lecture 11: Portfolio Performance Analysis and Measurement AIMS After this session you should be able to: Calculate time and money weighted returns for portfolios with cash inflows/outflows. Attribute the returns of a domestic investing fund to its components: asset allocation, sock selection components. Attribute the returns of an internationally investing fund to its components: asset allocation, stock selection and currency. Calculate and critically appraise Jensen (alpha), Treynor and Sharpe risk adjusted rates of return. Assess the market timing ability of a portfolio manager. Construct customized benchmarks for portfolios, calculate and interpret the Information Ratio for the portfolio. Reading: B. Solnik International Investments, Ch. 16 W.F.Sharpe, Asset Allocation and Management Style and Performance, Journal of Portfolio Management, Winter 1992 T. Goodwin, The Information Ratio, Financial Analysts Journal, July/August 1998 C. Luck, Style in Indexes and Benchmarks, in Performance Evaluation, Benchmarks and Attribution Analysis, pp 42-47, AIMR Brinson, G.P., Hood, L.R. and Beebower, G.L. (1986) Determinants of portfolio performance, Financial Analysts Journal, July/August,

2 PORTFOLIO PERFORMANCE ANALYSIS and MEASUREMENT The performance of a portfolio may be measured by comparing its return over some time interval with a benchmark portfolio. This will be particularly relevant for investors who have contracted their portfolio out to a portfolio manager: unit trusts (mutual funds), investment policies, pension funds. 1. RETURN CALCULATION METHODS Basic Return (R) Calculation Let initial value of portfolio = V 0 ; and final value of portfolio = V 1. Then if there have been no cash flows in or out of the portfolio the return on the portfolio is: R = V 1 /V 0 1 HOWEVER, care needs to be taken calculating returns when there are cash flows in or out of the portfolio during the period for which you are calculating return. There are two methods for allowing for cash-flows: 1) Time Weighted Return (TWR): calculates the return achieved from a time point immediately after each cash flow until the next cash flow and compound th returns to get the total return. 2) Money Weighted Return (MWR): Treat the investment as a project with positive and negative cash flows. Calculate the internal rate of return (IRR). The IRR is called the MWR when applied in this manner. Examples Case A We invest 100 for two years in a portfolio. After 1 year this grows to 115, and after one year we decide to invest an additional 6 in the portfolio. After the second year, the final value of the portfolio is 121. So R = V 1 /V 0 1 = 121/100-1 = 0.21, over two years, or annualised to 11% per year. Case B We manage the portfolio of a mutual fund for two years with an initial value of 10m. After one year the portfolio is re-valued at 9m, and investors withdraw 2m from the fund. At the end of the second year the portfolio is valued at 10m. In this case R = V 1 /V 0 1= 10/10 1 = 0.0 In both cases the calculated return does not represent the actual return earned by the fund manager. Case A overstates the return: a payment of 6m was made into the portfolio during the period Case B understates the true return: am amount of 2m was withdrawn after one year Solutions to this Paradox Case A. Time Weighted Return R 1 = 115/100 1 = 0.15 R 2 = 121/121 1 = 0.0 2

3 Compounding: TWR = (1+R 1 )(1+R 2 ) 1 = (1.15)(1.00) 1 = 0.15 or 7.23% pa Money Weighted Return IRR formula: 0 = (-6/(1+MWR)) + 121/(1+MWR) 2 MWR = 7.1% pa Case B Time Weighted Return R 1 = 9/10 1 = R 2 = 10/7 1 = 0.43 Compounding: TWR = (1+R 1 )(1+R 2 ) 1 = (0.9)(1.43) 1 = 0.29 or 13.4% pa Money Weighted Return IRR formula: 0 = (-2/(1+MWR)) + 10/(1+MWR) 2 MWR = 10.5% pa Comments: In case A the difference between TWR and MWR is small because the intermediate cash flows are small In Case B, the difference between TWR and MWR is large because the intermediate cash flow is relatively large In general TWR and MWR differ because TWR is a pure percentage return, taking no account of the actual cash value on which the percentage is based MWR is based on cash flows. In case B, the 10% loss in year 1 in made on 10m, and has more importance than the 43% gain made in year 2 on only 7m. It is not surprising in case B, that TWR is greater than MWR. So the moral is that in calculating returns ensure that cash inflows and outflows are appropriately allowed for, and then either use TWR or MWR consistently to compute returns. 3

4 2. ATTRIBUTION ANALYSIS The aim of attribution analysis is to find out the contribution to performance of different managerial abilities including: Stock selection Sector allocation Global asset allocation Currency Timing of sector or asset allocations A. Domestic stock selection and sector allocation Attribution analysis analyses the difference between the manager s portfolio (P) and his benchmark (B) R p = Σ W pi (R pi ) R b = Σ W bi (R bi ) R bi is benchmark return for sector I, and R pi is portfolio return for sector i. Rp - Rb = Σ W pi (R pi ) Σ W bi (R bi ) ie Rp - Rb = Σ W pi (R pi R bi ) + Σ (W pi - W bi ) R bi STOCK + SECTOR SELECTION ALLOCATION Example: Portfolio Return Benchmark Return Weighting Weighting Industrials Services Resources Rp =.2 x x x 20 = 14.1% Rb =.3 x x x 15 = 13.3% Rp Rb = 0.8% STOCK SELECTION =.2 x (8-6) +.7 x (15-17) +.1 x (20-15) = SECTOR ALLOCATION = (.2 -.3) x 6 + (.7 -.5) x 17 + (.1 -.2) x 15 = 1.3 Portfolio shows INFERIOR stock selection, but SUPERIOR sector allocation B. International Attribution Analysis INCLUDES ASSET AND CURRENCY COMPONENTS R bi is Benchmark return in LC for Market i. R pi is Portfolio return LC for Market i. C bi is Benchmark Currency return for Market i. C pi is Portfolio Currency return for Market i. (LC is Local Currency) 4

5 Rp - Rb = Σ W pi (R pi R bi ) + Σ (W pi - W bi ) R bi + Σ (W pi C pi - W bi C bi ) STOCK + ASSET + CURRENCY SELECTION ALLOCATION COMPONENT Example Portfolio is USD based, invested in Japan and Eurozone. Over the period the change in JPY/USD is -20% (JPY/USD = 100 at beginning, = 80 at end); and the change in EUR/USD is 10% (EUR/USD = 1 at beginning, =1.1 at end) R p = 47.83%, R b = 36.35% International Attribution Analysis PORTFOLIO BENCHMARK Weight LC USD C pi % Weight LC USD C bi % Ret % Ret % Ret % Ret % Japan Euro Calculation of Stock Selection, Asset Allocation and Currency Components a) Stock selection = 0.7 x (30-25) x (25-28) = + 2.6% b) Asset allocation = ( ) x 25 + ( ) x 28 = - 0.6% Portfolio Currency Components For Japan: R pi = 30%, but change in JPY/USD = -0.2 R pi = 1.30/0.8 1 = = 62.5% USD If we had invested $1 at the beginning, this bought JPY100. This would grow to JPY130 at the end. This would buy $130/80 = $ This is a 62.5% USD return compared with a 30% JPY return. Ie C pi = 32.5% For Eurozone: R pi = 25%, but change in JPY/USD = 0.1 R pi = 1.25/1.1 1 = = 13.6% USD If we had invested $1 at the beginning, this bought EUR 1. This would grow to EUR This would buy $1.25/1.1 = $ This is a 13.6% USD return compared with a 25% JPY return. Ie C pi = -11.4% Benchmark Currency Components For Japan: R bi = 25%, but change in JPY/USD = -0.2 R bi = 1.25/0.8 1 = = 56.3% USD If we had invested $1 at the beginning, this bought JPY100. This would grow to JPY125. This would buy $125/80 = $ This is a 56.3% USD return compared with a 25% JPY return. Ie C pi = 31.3% 5

6 For Eurozone: R bi = 28%, but change in JPY/USD = 0.1 R bi = 1.28/1.1 1 = = 16.4% USD If we had invested $1 at the beginning, this bought EUR 1. This would grow to EUR This would buy $1.28/1.1 = $ This is a 16.4% USD return compared with a 28% JPY return. Ie C pi = -11.6% c) Currency Component = (0.7 x x 31.3) + (0.3 x x -11.6) = 9.48% R p R b = = 11.48% Check: R p = Portfolio USD return = 0.7 x x 13.6 = 47.83% R b = Benchmark USD return = 0.5 x x 16.4 = 36.35% R p R b = = 11.48% as before 3. RISK ADJUSTED PERFORMANCE ANALYSIS In making the comparison in terms of realised returns it is important to make an adjustment for the risk of the portfolio. What is the appropriate measure of risk? If the portfolio is the only portfolio held by the investor then the total risk of the portfolio would be the measure of risk. If the portfolio is held along with a number of other portfolios, then the nondiversifiable risk of the portfolio will be the appropriate measure. (i) Excess Return to Variability: Sharpe Measure S = (r p - r f )/ σ p where r p is the return on the portfolio, r f is the risk-free rate and σ p is the standard deviation of the returns on the portfolio. To assess the porfolio returns compute the Sharpe measure for the portfolio under consideration and also compute it for the benchmark portfolio. The Sharpe measure is using the Capital Market Line as a benchmark and is appropriate for investors who have invested all their wealth in the one portfolio under consideration. ER A r f B σ 6

7 In the diagram portfolio A is preferred to portfolio B, since portfolio A has a higher expected return per unit of risk (measured by standard deviation). That is combinations of A and the riskless asset give higher returns for the same level of risk than combinations of B and the riskless asset. (ii) Excess Return to beta: Treynor Measure T = (r p - r f )/ β p where β p is the beta of the portfolio. The Treynor measure uses the Security Market Line as a benchmark and is appropriate for investors who have invested their wealth in a number of portfolios. ER A r f B β In the diagram portfolio A is preferred to portfolio B, since portfolio A has a higher expected return per unit of risk (measured by beta). That is combinations of A and the riskless asset give higher returns for the same level of risk than combinations of B and the riskless asset. (iii) Jensen Differential Performance Index J = α p = r p - {r f + β p (Er m - r f )} where Er m is the expectected return (sample average) on the market portfolio. Alternatively we may regress the excess return (r p - r f ) of a portfolio against the excess return on the market and interpret the intercept as the Jensen measure. The Jensen measure specifically evaluates the active fund management by the portfolio manager, as opposed to the passive fund management of investing in the risk-free asset and the market portfolio. 7

8 ER B A A M Security Market Line r f B β The diagram shows that the Treynor measure would rank B better than A. But Jensen s differential performance index would rank A above B. The portfolios A and B show the expected returns from a passive strategy of investing in the risk-free asset and the market portfolio with the same level of risk as at A and B. The differential return at the portfolio manager s chosen risk level (AA and BB ) measures how much better the manager did that the passive strategy. The Jensen measure ranks A above B because the distance AA is greater than BB. This is because the manager of A was able to outperform a passive strategy at A (with the same level of risk as at A) by more than the manager of B was able to outperform a passive strategy at B. Understanding the Performance measures as Investment Returns S shows the return per unit of volatility to the zero investment portfolio formed by borrowing $1 and investing it in the portfolio T shows the return per unit of risk to the zero investment portfolio formed by borrowing $1 and investing it in the portfolio J shows the return earned by the zero investment portfolio formed by: - selling SHORT β P dollars in the market portfolio and (1-β P ) dollars in the risk free asset - investing $1 in the portfolio Measuring Fund Performance with multiple risk factors Jensen s technique is easily extended to multiple risk factors: regress the excess returns on the individual fund (above the risk free rate), against the excess return on the market (R mt - r ft ) plus any additional factors F t R it - r ft = α i + β i (R mt - r ft ) + γ i F t + ε it for each fund i over the t data periods, and save the coefficients α, β and γ. Under the null hypothesis of no-abnormal performance the α coefficient should be equal to zero. For each fund we may test the significance of α as a measure of that funds abnormal performance. In addition we may test for overall fund performance, by testing the significance of the mean α. 8

9 1 N i = 1 α = α i N The appropriate t-statistic is t = 1 N N i = 1 α i SE( α ) i Studies of Fund Performance Jensen (1968) examined the performance of 115 mutual funds over the period , He used the Security Market Line as the basis for a comparison (which allows for the riskiness of a fund) 1) Estimate the position of the SML using the S&P500 as a proxy for the market portfolio 2) Estimate the beta for each fund. 3) Examine the abnormal return for each fund net of expenses. He found that the the average abnormal return across funds was approx -1% per annum. If expenses were added back into the gross return, average abnormal return was approx. zero. Hence market professionals do not appear to be able to beat the market. However criticism of Jensen's methodology is that benchmark may be inappropriate, especially if fund managers engage in market timing. Market timing is when fund managers take an aggressive position in a bull market, but a defensive position in a bear market. In stage 2) above this would involve estimating a separate betas for the bull and bear markets Evidence on performance of portfolio managers Research into the performance of mutual funds [Jensen (1968), Ippolito (1989), Blake, Lehman and Timmerman (1999)] found that they do not generate abnormal returns, which is consistent with strong form efficiency. Jensen, M. (1968) The performance of mutual funds in the period , Journal of Finance, 23, Blake, D. Lehmann, B. & Timmermann, A. Performance Measurement Using Multi- Asset Portfolio Data; A Study of UK Pension Funds , Journal of Business, Carhart, M. (1997) On Persistence in Mutual Fund Performance, JOF March

10 Malkiel, B.G. (1995) Returns From Investing in Equity Mutual Funds 1971 to 1991, JOF June

11 Market Timing Market timing is when fund managers take an aggressive position in a bull market, but a defensive position in a bear market. When portfolio managers expect the market portfolio to rise in value, they may switch from bonds into equities and/or they may invest in more high beta stocks. When they expect the market to fall they will undertake the reverse strategy: sell high beta stocks and move into defensive stocks. If managers successfully engage in market timing then, returns to the fund will be high when the market is high, and also relatively high when the market is low r i -r f x x x x x x x x x x r m -r f If there was no market timing then a regression of r i -r f against r m - r f would produce points scattered around the dotted linear line. However if managers successfully market time, then a quadratic plot would prove a better fit (Treynor-Mazuy test). So test of market timing is a significant value of γ p in the following regression r pt - r f = α p + β p (r m -r f ) + γ p (r m - r f ) 2 + ε pt An alternative test of market timing suggested by Merton-Henriksson is r pt - r f = α p + β p (R mt - r f ) + δ p (R mt - r f ) + + η pt where (R mt - r f ) + = Max(0, R mt -r f ) Treynor, J.L. and Mazuy, F. (1966) Can mutual funds outguess the market?, Harvard Business Review, (44), Henriksson, R. and Merton, R. (1981) On market timing and investment performance. II. Statistical procedures for evaluating forecasting skills, Journal of Business, 54(4),

12 A New Measure M2 (Modigliani and Modigliani) Money Magazine March 1997 p. 49 TOP FUNDS: WHERE TO EARN WITHOUT GETTING BURNED So your stock fund streaked to a market-beating 25% return over the past 12 months. Congratulations. But before you uncork the Korbel, ask yourself this: Just how much risk did you incur to achieve that lofty return? "Total returns in the absolute are not too meaningful, since they don't tell you how far you stuck your neck out to get them," says noted financial planner Harold Evensky of Evensky Brown & Katz in Coral Gables, Fla. (see Last Word in the Wall Street Newsletter on page 72). To help investors identify which funds are rewarding them for the risks they take, Leah Modigliani, a U.S. equity strategist at Morgan Stanley, and her grandfather, Franco Modigliani, a Nobel-prizewinning economist, have devised a new measure of risk-adjusted return that Morgan Stanley has labelled the Modigliani 2, or M². Below, we highlight funds in five different categories with top M2 ratings. Here's some background on how the new measure works. First, each fund's portfolio is adjusted mathematically so that it exactly matches the volatility of a benchmark such as the S&P 500 or Russell (Volatility is the degree by which an investment's returns vary from one period to the next.) "The goal is to put each fund on the same risk scale," says Leah Modigliani. Then the formula computes the total return the fund would have earned with its hypothetical adjusted portfolio. By comparing this M² return to the performance of the fund's benchmark, you can determine whether you have been rewarded for the risks it took. The results can be enlightening. For example, AIM Constellation, an aggressive growth fund, posted a 16.6% total return during the past five years, topping the S&P 500's 15.2%. But because its raw returns varied so widely from quarter to quarter, Constellation's M² return equals a more modest 10.6%. By contrast, Fidelity Puritan, a staid balanced fund that gained only 14.8%, beat both Constellation and the S&P with a risk-adjusted return of 19.7%. "With this measure, investors can more easily see in real return figures how funds performed given the risk they took," says Leah Modigliani. The M2 performance measure compares the portfolio to an adjusted benchmark return. The benchmark will usually have a different volatility to the benchmark. To solve this problem M2 suggest that you do not use the return of your portfolio as it stands: form a new zero investment portfolio adjusted t have the same volatility as the benchmark. Compare this adjusted return with the benchmark If the benchmark has a standard deviation of S b and the portfolio S p, let K = S b /S p, and if we borrow K dollars and invest in the portfolio, the new zero investment portfolio has a return of R NEW = K(R p R f ) The Standard deviation of R NEW = KS p = S b, so that R b and R NEW have the same standard deviation 12

13 The M2 performance measure then compares two zero investment portfolios with the same standard deviation: M2 = RNEW (R b R f ) INFORMATION RATIO The information ratio is similar to the Sharpe Ratio but compares a Portfolio to its BENCHMARK. The BENCHMARK can be a single index (eg FTSE 100, S & P 500, MSCI World, MSCI US Growth etc) or because STYLE is important a customised benchmark developed specifically for the portfolio in question. For example, if the portfolio holds on average 20% government bonds, 80% US equities, we may form a customised benchmark as 0.8 x S & P x Merrill Lynch Govt Bond Index Usually the customised benchmark is based on an effective-mix style analysis. Suppose we perform an effective-mix style analysis on a mutual fund and find the following style exposures: 0.4 large value stock; 0.2 large growth stock; 0.3 small stock; 0.1 government bond; then we can form a customised benchmark for the fund R b = 0.4LV + 0.2LG + 0.3S + 0.1GB The TRACKING ERROR for a PORTFOLIO IS Err = R p R b The INFORMATION ratio is IR = Mean ERR/Standard Deviation of ERR i.e. the Information Ratio compares the return over the benchmark with the risk taken - here risk = deviation from benchmark Examples Data: MONTH FUND RET FTA RET VALUE RET GROWTH RET Example 1 - Using FTSE All-Share as Benchmark MEAN ERR = 0.35, Standard deviation of ERR = 1.61, IR = 0.24 Example 2 - Using Customised benchmark, 0.5 Value, 0.5 Growth MEAN ERR = 0.13, Standard deviation of ERR = 0.54, IR = Note ERR, MSE and IRR are all reduced by using a customised benchmark. 13

14 Conditional Performance Evaluation Ferson and Schadt (1996) advocate allowing for the benchmark parameters to be conditioned on economic conditions. Jensen regression becomes R it - r ft = α i + β i (Z t-1 ) (R mt - r ft ) + γ i F t + ε it where Z t-1 is a vector of instruments for the information available at time t (and is therefore specified as t-1) and β i (Z t ) are time conditional betas, and their functional form is specified as linear β i (Z t ) = b 0 + B z t-1 where z t-1 = Z t-1 - E(Z) is a vector of deviations of the Zs from their unconditional means. Implementing this approach involves creating interaction terms between the market returns and the instruments Instruments used are: lagged treasury bill rate, dividend yield, default premium (difference between low and high quality corporate bonds), slope of the term structure (difference between long and short run government bond yields) Ferson, W. and R.W.Schadt (1996) Measuring Fund Strategy and Performance in Changing Economic Conditions, Journal of Finance, vol. 51,

15 Questions: 1) A USD based investor has invested $100 million in a portfolio of US, Asian and European stocks. On December 31 the portfolio is invested in 500 IBM shares (NYSE listed), 200 Sony shares (Tokyo SE listed) and 50 BMW shares (Frankfurt listed). Her portfolio has appreciated 4.065% while her benchmark rose only 0.735%. The benchmark has a weighting of 50% US, 25% Japan, 25% Europe. Calculate the contribution of currency, asset allocation and stock selection to her portfolio s and to the world index s return. The Table below summarises stock price and currency values at December 31 and March 31. Asset December 31 March 31 IBM USD 100 USD 105 SONY JPY 10,000 JPY 11,000 BMW EUR 600 EUR 600 US Equity Index (LC) Japan Equity Index (LC) Euro Equity Index (LC) JPY/USD EUR/USD ) You are managing a portfolio worth GBP 200K at the beginning of the year. After 6 months the portfolio s value has risen to 220, but the client withdraws 100. The end of year value of the portfolio turns out to be 110. Calculate the TWR and NWR rates of return. Comment. 3) Given the following data calculate for the portfolio Jensen, Treynor, Sharpe and Modigliani risk adjusted performance measures and the Trenor-Mazuy timing coefficient. An analysis of a fund s portfolio shows that an appropriate benchmark would be 0.4 LV, 0.2 LG, 0.4 S, calculate the fund s information ratio using the Market Index and a Style adjusted benchmark. Comment. Period Fund Market T bill LV LG S ) Use the data for fund, market and t bill from question 3 but assume the manager sets his beta at the beginning of each period according to the formula beta = 1.0 (t bill/10) Calculate his beta at the beginning of each period. Does he exhibit market timing? 15

16 Comprehension Check 1. Define money weighted and time weighted rates of return. 2. Define attribution analysis. 3. What are the components of attribution analysis? 4. Define Jensen Treynor, Sharpe performance measures. 5. Explain the idea behind the M2 performance measure. 6. What is market timing? what is the Treynor-Mazuy measure? 7. What is the information ratio and how does it differ from the Sharpe Ratio? 16