Measuring portfolio performance
Muhammad Shahid U.U.D.M. Project Report 2007:19
Examensarbete i matematik, 20 poäng Handledare och examinator: Johan Tysk Juni 2007
Department of Mathematics Uppsala University
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Dedications I dedicate this thesis to my beloved parents and teacher (Prof. Abdul Hafeez Sh), who played the most vital role in my upbringing and grooming. Today what ever I am is due to virtue of their nurture and prays. My Degree (Master in Financial Mathematics) would not been completed without their support and encouragement. May Allah bless them.
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Acknowledgement All prays to Allah Almighty who induced the man with intelligence, knowledge and wisdom. It is He who gave me ability, perseverance and determination to complete this thesis. Teachers are lighthouses spreading the light of knowledge and wisdom everywhere and guiding the new generation so that they can cruise safely towards their destination. They are really lamps that are kindling the candles of knowledge in the heart of young generation. They are performing the job, which Allah himself acknowledge as the noblest to all jobs; the job of teaching. They will get its reward not only from Allah but also in the form of immense respect that every student carries for them in the core of his heart. I offer my sincerest thanks and deepest gratitude to my research supervisor Prof. Johan Tysk for his inspiring and valuable guidance, encouraging attitude and enlightening discussions enabling me to pursue my work with dedication. I would like to say a big thanks to all the teachers who taught me in the entire program. They did not only teach me how to learn, they also taught me how to teach, and their excellence has always inspired me. I also wish to express my feeling of gratitude to my parents, sisters, brothers and friends, who prayed for my health and brilliant future. A very special thanks and appreciation goes to my dearest teacher Prof. Abdul Hafeez Sh, though you were for away, your persistent telephone calls and the thought of you gave me the enthusiasm to carry on with my academic work.
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Table of Contents 1 Introduction 1 2 Portfolio Mean and Variance 3 2.1 Mean Return of a Portfolio 3 2.2 Variance of Portfolio Return 3 2.3 The Markowitz Problem 4 2.4 The Capital Market Line 8 3 The Capital Asset Pricing Model 11 3.1 History of CAPM 11 3.1.1 Systematic Risk 11 3.1.2 Unsystematic Risk 11 3.2 Assumption of Capital Asset Pricing Model 14 3.3 The Security Market Line 17 3.4 CAPM as a Pricing Formula 17 3.4.1 Linearity of Pricing and Certainty Equivalent Form 19 4 Measuring Portfolio Performance 20 4.1 Measuring the Rate of Return of a Portfolio 20 4.1.1 Time Weighted Rate of Return 20 4.1.2 Value Weighted Rate of Return 20 4.2 Risk Adjusted Performance Measure 21 4.2.1 Public Information 21 4.2.2 Private Information 21 4.3 Risk Adjusted Performance Indices 23 4.3.1 The Jensen Index 23 4.3.2 The Treynor Index 26 4.3.3 The Sharpe Index 29 4.4 Comparison of Three Indices 36 4.5 Conclusion 39 Reference 40
Measuring Portfolio Performance
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Chapter 1
INTRODUCTION Measuring of portfolio performance has become an essential topic in the financial markets for the portfolio managers, investors and almost all that have something to do in the field of finance and it plays a very important role in the financial market almost all around the world. Earlier then 1950, portfolio managers and investors measured the portfolio performance almost on the rate of return basis. During that time, they knew that risk was a very important variable in determining investment success but they had no simple or clear way of measure it. In 1952 Markowitz created the idea of Modern Portfolio Theory and proposed that investors expected to be compensated for additional risk and provided a framework for measuring risk. In early 1960, after the development of portfolio theory and capital asset pricing model in subsequence years, risk was included in the evaluation process. The capital asset pricing model of William Sharpe and John Litner marks the birth of asset pricing theory. The attraction of capital asset pricing model was that it offered power predictions about how to measure risk and the relation between expected return and risk. Treynor (1965) was the first researcher developing a composite measure of portfolio performance. He measured portfolio risk with beta and calculated portfolio market risk premium and later on in 1966 Sharpe developed a composite index which is similar to the Treynor measure, the only difference being the use of standard deviation instead of beta. In 1967 Sharpe index evaluated funds performance based on both rate of return and diversification but for a completely diversified portfolio Treynor and Sharpe indices would give identical ranking. Jensen in 1968, on the other hand, attempted to construct a measure based on the security market line and he showed the difference between the expected rate of return of the portfolio and expected return of a benchmark portfolio that would be positioned on the security market line. According to Prof. K. Spremann, “ Portfolio measurement has not only the goal to inform about the quality of a portfolio performance__ but and that’s even more important__ to decompose and analyze the success factors of a portfolio ”. This thesis is organized as fellows. In Chapter 2, we explore the concept of Mean Variance portfolio theory with example. We also describe the Markowitz problem, solution of the Markowitz problem and the concept of capital market line. In Chapter 3, we describe the capital asset pricing model and prove its theorem along with assumptions. CAPM as a pricing formula and linearity of pricing and certainty equivalent Measuring Portfolio Performance 2 form are also explained in this chapter. Finally, in Chapter 4, we explore the concepts of measuring portfolio performance, including definitions of measuring the rate of return of a portfolio, time weighted and value weighted rate of returns. Finally, we discuss the risk adjusted performance measure based on capital asset pricing model. Based on three risk adjusted performance indices (Jensen, Sharpe and Treynor) we calculate the performance of different portfolio and compare these indices. The main references of this thesis are [1], [2], [4], [5], [9], [10] and [11]. In Chapter 2, we refer to [3], [7], [8] and [16]. In Chapter 3, we also refer to [6], [12], [13] and [15] frequently. Some data also refer to web pages listed at the end of reference section.
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Chapter 2
PORTFOLIO MEAN AND VARIANCE
2.1 Mean Return of a Portfolio
Suppose that there are n assets with rates of return r1 , r2 ,…, rn and these have expected values E(r1 ) = r1 , E(r2 ) = r2 , ….. , E(rn ) = rn . We form a portfolio of these n assets using the weights wi , i = 1,2,…,n. The rate of return in terms of the individual return of the portfolio is
r = w1r1 + w2 r2 + … + wn rn . (1) We find the expected rate of return by taking the weighted sum of individual expected rates of return. Using the property of linearity, we take the expected values on the both sides of equation (1)
E(r) = w1E(r1 ) + w2 E(r2 ) + … + wn E(rn ) . (2)
2.2 Variance of Portfolio Return
2 The variance of the return of asset i is denoted by σ i , the variance of the return of the 2 portfolio by σ , and the covariance of the asset i with asset j by σ ij . We can perform the following calculation, 2 2 σ = E[(r − r) ], n n 2 = E(∑wi ri − ∑ wi ri ) , i=1i = 1 n n = E(∑wi (ri − ri ))( ∑ w j (rj − rj ) , i=1j = 1 n = E∑ wi w j (ri − ri )(rj − rj ) , i, j=1 n 2 σ = ∑ wi w jσ ij . (3) i, j=1 Equation (3) represents the Variance of the return on the portfolio.
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Example 2.1 Consider there are two assets with expected values r1 = 0.22, r2 = 0.55 and the variance are σ 1 = 0.80, σ 2 = 0.88 and σ 12 = 0.55 respectively. A portfolio with weights w1 = 0.25 and w2 = 0.65 is formed. Calculate the mean and variance of the portfolio? Solution
Mean of the portfolio r = w1E(r1 ) + w2 E(r2 ) = 0.25(0.22) + 0.65(0.55) = 0.4125. Variance of the portfolio n 2 σ = ∑ wi w jσ ij i, j=1 2 2 2 2 = w1 σ 1 + w2σ 2 + w1w2σ 12 + w2 w1σ 21 = (0.25) 2(0.80) 2 + (0.65) 2(0.88) 2+ (0.25)(0.65)(0.01)+(0.65)(0.25)(0.01) = 0.040 + 0.327 + 0.002 + 0.002 = 0.371. 2.3 The Markowitz Problem The Markowitz problem explicitly addresses the tradeoff between expected rate of return of a portfolio and variance of the rate of return of a portfolio. This problem is mainly used when the risk free assets as well as risky assets are available. The Markowitz problem can be solved numerically. When we solve the problem numerically then we get a numerical solution.
Consider that there are n assets and their expected rates of return are r1 , r2 ,..., rn and their covariances are σ i j , for i = j = 1,2,…, n . The portfolio is defined as a set of n weights wi , for i = 1, 2… n , that its sum equal to 1. In order to find a minimum variance of a portfolio, some arbitrary value r is assign to the mean value. Hence the problem can be formulated as
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1 n min ∑ wi w jσ ij , 2 i, j=1 subject to n ∑ wi ri = r , i=1 n ∑ wi = 1. i=1
Solution of Markowitz Problem Lagrange Multipliers λ and are used to solve the problem. In the Lagrangian, first we convert all the constraints to one with zero on the right hand side as shown below. 1 n f = ∑ wi w jσ ij , 2 i, j=1 n ∑ wi ri − r = 0, i=1 n ∑ wi −1 = 0. i=1 Then the each left hand side is multiplied to its Lagrange Multiplier and subtracted from the objective function. Lagrangian function 1 n n n L ( wi , w j ) = ∑ wi w jσ ij - λ ( ∑ wi ri − r ) - ( ∑ wi −1 ). 2 i, j=1 i=1 i=1
Differentiate the Lagrangian with respect to wi and w j and put equal to zero. If the type structure is unfamiliar then it is difficult to differentiate it. Here we consider the case of only two variables and it is easy to generalize it to n variable. Functions of two variables 1 2 2 2 L ( wi , w j ) = ∑ wi w jσ ij - λ ( ∑ wi ri − r ) - ( ∑ wi −1 ) 2 i, j=1 i=1 i=1 or 1 L ( w , w ) = ( w2σ 2 + w w σ + w w σ + w2σ 2 ) i j 2 1 1 1 2 12 2 1 21 2 2
- λ ( r1w1 + r2 w2 - r ) - ( w1 + w2 - 1 ).
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Differentiate the above equation with respect to w1 and w2 , we obtain
∂L 1 2 = (2 w1σ 1 + w2σ 12 +σ 21w2 ) - λ r1 - , ∂w1 2 and
∂L 1 2 = (σ 12 w1 + w1σ 21 + 2 w2σ 2 ) - λ r2 - . ∂w2 2 ∂L ∂L Now putting = 0, = 0, and using the fact σ 12 =σ 21 , we get ∂w1 ∂w2 2 σ 1 w1 + σ 12 w2 + λ r1 - = 0, and 2 σ 21w1 + σ 2 w2 - λ r2 - = 0 . Here we have four equations, two as above and two from the constraints. These equations can be solved for the four unknown w1 , w2 , λ and . Equation for efficient set The efficient portfolio for two Lagrange Multipliers λ and and the portfolio weight wi for i = 1, 2… n having the mean rate of return r satisfy n ∑σ ij wi - λ ri - = 0, for i = 1, 2… n (1) j=1 n ∑ wi ri = r , (2) i=1 and n ∑ wi = 1. (3) i=1 From equation (1), we have n equations and two equations of the constraints. Now we have total n + 2 linear equations, with n + 2 unknown i.e. wi 's , λ and . Using the linear algebra method these equations can be solved easily. Example 2.2 Consider we have three uncorrelated assets. Each has variance 1 and the mean values are 1, 2 and 3, respectively, there is a bit of simplicity and symmetry in this situation, which makes it relatively easy to find an explicit solution.
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Using the equations, n ∑σ ij wi - λ ri - = 0, for i = 1, 2… n (1) j=1 n ∑ wi ri = r , (2) i=1 and n ∑ wi = 1. (3) i=1 we have 2 2 2 σ 1 = σ 2 = σ 3 = 1 and σ 12 = σ 23 = σ 31 = 0. Using all known values in equations (1) – (3), we get the following five equations.
w1 - λ - = 0, (4)
w2 - 2 λ - = 0, (5)
w3 - 3 λ - = 0. (6)
And w1 + 2 w2 + 3 w3 = r , (7)
w1 + w2 + w3 = 1. (8)
We have to find the values of λ and , substituting the values of w1 , w2 and w3 from equations 4, 5 and 6 in equations 7 and 8, we obtain ( λ + ) + 2(2 λ + ) + 3(3 λ + ) = r , 14 λ + 6 = r , (9) and ( λ + ) + (2 λ + ) + (3 λ + ) = 1, 6 λ + 3 = 1. (10) Solving equation (9) and (10) simultaneously, we get λ and as r λ = - 1, 2 1 = 2( ) - r . 3
Substituting λ and in equations (4) to (6), we get the values of w1 , w2 and w3 Measuring Portfolio Performance 8
4 r w = - ( ), 1 3 2 1 w = , 2 3 and r 2 w = ( ) - ( ). 3 2 3 The standard deviation is given by 2 2 2 σ = w1 + w2 + w3 2 7 r σ = − 2r + 3 2 The minimum variance point is, by symmetry, at r = 2, with σ = /3 3.
2.4 The Capital Market Line The linear efficient set of capital asset pricing model (CAPM) is known as capital market line. It is also stated as “The efficient set consisting of a single straight line, from the risk free point and which is passing through the market portfolio, that line is known as capital market line”.
Figure 1: The Capital market line
The capital market line is illustrated above, with return p on the y axis and risk σ p on x axis. The line shows the relationship between the expected rate of return and the risk of return for efficient portfolios of assets. It is also referred to pricing line and if the risk
Measuring Portfolio Performance 9 increases then corresponding expected rate of return must also increase. “M” is the market portfolio and rf is the risk free rate of return. The capital market line states that
rM − rf rp = rf + σ p . σ M Here
rp = expected return of an efficient portfolio
rf = risk free rate of return
rM = expected return of the market portfolio
σ p = standard deviation of the efficient portfolio
σ M = standard deviation of the market portfolio. The slope of the capital market line is given by r − r K = M f . σ M It is also called the price of risk. It tells us how the expected rate of return of a portfolio must be increase if the standard deviation of that rate is increase by one unit.
Example 2.3 Consider an oil drilling venture. The price of a share of this venture is $1750. After one year, it is expected to yield the equivalent of $2000. The standard deviation of the return is σ = 45%. Currently, the risk free rate is 15%. The expected rate of return on the market portfolio is 23% and the standard deviation of this rate is17%. Compare this oil venture with the asset on the capital market line.
Solution Here
rf = 15% = .15 , rM = 23% = .23
σ M = 17% = .17 , σ p = 45% = .45 From the capital market line, we know that
rM − rf rp = rf + σ p , σ M .23− .15 = (.15) + ( (.45) ), .17 Measuring Portfolio Performance 10
= (.15) + (0.47) (.45), = 36%. Actual expected rate of return is 2000 r = ( ) – 1, 1750 = .14, = 14%. After comparing the both values it is clear that the oil venture lies well below the capital market line.
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Chapter 3
THE CAPITAL ASSET PRICING MODEL (CAPM) 3.1 History of CAPM The capital asset pricing model (CAPM) was introduced by Jack Treynor (1961) while parallel work was also performed by William Sharp (1964) and Lintner (1965). In 1990, Sharp received the Nobel Memorial Prize in Economics with Harry Markowitz and Merton Miller in the field of financial economics. The Capital Asset Pricing Model is an economic model which is used for valuing the securities, stocks and assets by relating risk and expected rate of return. In the capital market line, the expected rate of return of an efficient portfolio relates to its standard deviation but cannot show how the expected rate of return of an individual asset relates to its individual risk. This relation is expressed by the capital asset pricing model (CAPM). The CAPM help us to calculate investment risk and what is the return on the investment. This investment contains two types of risk. • Systematic Risk • Unsystematic Risk
3.1.1 Systematic Risk Systematic risks are market risks that cannot be diversified away. For example, wars and interest rates are good examples of the systematic risk.
3.1.2 Unsystematic Risk Unsystematic risk is specific to each individual stocks and it can be diversified away as the investor increases the number of stocks in portfolio. It is also known as “specific risk”.
Theorem Suppose that market portfolio M is efficient, the expected return ri of any asset i satisfies the relationship.
ri - rf = βi ( rM − rf ), where
rf = risk free rate
βi = beta of the security
rM = expected market return
( rM − rf ) = equity market premium and Measuring Portfolio Performance 12
σ iM βi = 2 . σ M
Proof Suppose for any α , the portfolio consisting of a portion α invested in the asset i and the remaining potion 1 α invested in the market portfolio M. The expected rate of return of this portfolio is
rα = α ri + (1 - α ) rM (1)
Standard deviation of the rate of return is 2 2 2 2 1/2 σ α = [α σ i + 2 α (1 - α ) σ iM + (1 - α ) σ M ] (2) The values of α are traced out as shown in the diagram below.
M
In particular α = 0 corresponding to the market portfolio M. This curve cannot cross the capital market line. If it crosses the capital market line then it would violate the definition of the capital market line. The curve must be tangent to capital market line. At the point M the slope of the curve is equal to the slope of the capital market line. Differentiating equations (1) and (2) with respect to α , we get d r α = r - r . dα i M Furthermore dσ 1 α = [α 2σ 2 + 2 α (1 - α ) σ + (1 - α ) 2 σ 2 ] −1 2 × dα 2 i iM M 2 [2 ασ i + 2(1 - α ) σ iM + 2(1 - α )(-1) σ M ],
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2 1 2ασ i + 1(2 −α)σ iM + (2 α − )1 σ M = 2 2 2 2 1/2 2 [α σ i 2 + α (1 α ) σ iM + (1 α ) σ M ]
2 ασ i + 1( −α)σ iM + (α − )1 σ M = 2 2 2 2 1/2 . [α σ i 2 + α (1 α ) σ iM + (1 α ) σ M ] At α = 0, we get 2 dσ α )0( σ i + 1( − )0 σ iM + 0( − )1 σ M = 2 2 2 2 1/2 dα α =0 [(0) σ i 2 + α (1 0)σ iM + (1 ) 0 σ M ]
2 σ iM −σ M = 2 1/2 [σ M ] σ −σ 2 = iM M . σ M Using the relation d r d r dα α = α , dσ α dσ α dα at point α = 0, we get
dσ α ri − rM = 2 dα α =0 σ iM −σ m
σ M
(ri − rM )σ M = 2 . σ iM − σ M This slope is also equal to the slope of the capital market line.
(ri − rM )σ M rM − rf 2 = , σ iM − σ M σ M and therefore,
2 2 2 riσ M − rM σ M = (rM − rf ) (σ iM −σ M ),
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2 2 2 2 riσ M − rM σ M = rM σ iM − rM σ M −σ iM rf + σ M rf . Thus 2 2 riσ M − rf σ M = (rM − rf )σ iM , and
(rM − rf )σ iM ri = rf + 2 . (3) σ M Now let
σ iM βi = 2 . σ M
Putting the value of βi in equation 3, we get
ri = rf + βi (rM − rf ). Thus the theorem is proved.
The ri − rf is expected excess rate of return of asset i , it is defined as the amount by which the rate of return is expected to exceed the risk free rate. Similarly, rM − rf is the expected excess rate of return of the market portfolio. The capital asset pricing model (CAPM) tells us that the expected excess rate of return of an asset is proportional to the expected excess rate of return of the market portfolio. 3.2 Assumptions of Capital Asset Pricing Model (CAPM) The capital asset pricing model (CAPM) is valid within a special set of assumption. These assumptions are • All investors have homogenous expectations about the assets. • Investor may borrow and lend unlimited amount of risk free asset. • The risk free borrowing and lending rates are equal. • The quantity of assets is fixed. • Perfectly efficient capital markets. • No market imperfections such like taxes and regulation and no change in the level of interest rate exists. • There are no arbitrage opportunities. • There is a separation of production and financial stocks. • Returns (assets) are distributed by normal distribution.
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Example 3.1 Suppose the rate of return of the market has an expected value 14% and a standard deviation of 15%, let the risk free rate be 10%. Using the capital asset pricing model formula calculate an expected rate of return. Solution Consider an asset has covariance of .045 with the market; first, we have to find the value of the β (beta).
σ iM β = 2 , σ M .0 045 = , ().0 15 2 = 2.
Here rf = 10% = .10 , rM = 14% = .14 The capital asset pricing model formula is
ri = rf + βi ( rM − rf ), r = (.10) + 2(.14 .10), = (.10) + (.08), = .18, = 18%.
Example 3.2 Assume that the risk free rate is 8% and the expected market return is 12%. Find the expected rate of return when (a) β = 0 (b) β = 2.
Solution rf = .08, rM = .12 Case (a) When β = 0
ri = rf + βi ( rM − rf ), = .08 + 0 (.12 .08), = .08. Measuring Portfolio Performance 16
Case (b) When β = 2
ri = rf + βi ( rM − rf ), = .08 + 2(.12 .08), = .08 + .08 = .16. This example shows that the higher the degree of the systematic risk β , the higher the return on a given security demanded by investors.
Example3.3 Consider that you have $30,000 in the following 4 stocks. Security Amount Beta Xi R = r + βi (Rm − r) Stock A 5,000 0.75 5/30 0.1225 Stock B 10,000 1.10 10/30 0.1610 Stock C 8,000 1.36 8/30 0.1896 Stock D 7,000 1.88 7/30 0.2468 The risk free rate is 4% and the expected return on the market portfolio is 15%. Using the capital asset pricing market, what is the expected return on the above portfolio? Solution
Here rf = .04 , rM = .15.
Here βi denotes the beta coefficient of the stock i . We calculate the beta coefficient βi for the portfolio and get the expected return on the portfolio from the capital asset pricing model equation.
Here βi = x A β A + xB β B + xC β C + xD β D = (5/30)(0.75) + (10/30)(1.10) + (8/30)(1.36) + (7/30)(1.88) = 1.29 Capital asset pricing model equation is
ri = rf + βi ( rM − rf ), = .04 + (1.29) (.15 .04), = .04 + 0.15, = 0.19.
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3.3 The Security Market Line The security market line is the graphical representation of the capital asset pricing model. The capital asset pricing model equation describes a linear relationship between risk and return. This linear relationship is termed as the security market line.
The graph shows the relation in the form of beta. In this case, the market portfolio to the point beta is equal to one. According to the capital asset pricing model this line expresses the risk reward structure of assets. The expected rate of return increases linearly as beta increases. It is a useful tool in determining if an asset being considered for a portfolio offers a reasonable expected return for risk on the security market line. We plotted individual securities, if the security’s risk versus expected return is plotted above the security market line, then it is undervalued and the investor can expect a higher return for the inherent risk. If a security’s risk versus expected return is plotted below the security market line, then it is overvalued and the investor would be accepting less return for the amount of risk assumed. 3.4 CAPM as a Pricing Formula CAPM is a pricing model. It only contains the expected rate of return but cannot contain price explicitly. We want to see why the CAPM is called a pricing model. Consider an asset is being purchased at price P and after some time it is sold at price Q . (Q − P) Then r = is the rate of return, Here P is known and Q is random (unknown), P the CAPM formula is
r = rf + βi (rM − rf ). (1) Putting the value of r in equation (1), give us
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(Q − P) = r + β (r − r ), P f M f
Q − P = P ( rf + β (rM − rf ) ),
Q = P + P ( rf + β (rM − rf ) ),
= P (1 + ( rf + β (rM − rf ) ),
Q P = . (2) 1( + (rf + β (rM − rf ))) It is the price of the asset according to the CAPM. Here β is the beta of the asset. Example 3.4 Let us consider an oil drilling venture. The possibility of investing in a certain oil share that produces a payoff, it is random because of the uncertainty in future oil price. The expected payoff is $1200 and standard deviation of return is 40%.The β of the asset is 0.8 that is relatively low. The risk free rate is 20% and the expected return on the market portfolio is 70%. What is the value of this share of the oil venture using the CAPM? Solution We know that Q P = (1) 1( + (rf + β (rM − rf ))) here Q = $1200 , β = 0.8
rf = 20% = 0.20 , rM = 70% = 0.70 putting this value in above equation (1) 1200 P = , 1+ .0 20 + .0(8.0 70 − .0 20) 1200 P = , 1.6 P = $750.
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3.4.1 Linearity of Pricing and the Certainty Equivalent Form Linearity of the pricing formula is a very important property. Its mean that the price of the sum of two assets is the sum of their prices, similarly, the price of a multiple of an asset is also the same multiple of the price. The formula does not look linear, in the case of sums. Consider the example Suppose
Q1 Q2 P1 = , P2 = . 1+ rf + β1 (rM − rf ) 1+ rf + β 2 (rM − rf )
Adding P1 and P2 , we get
Q1 Q2 P1 + P2 = + 1+ rf + β1 (rM − rf ) 1+ rf + β 2 (rM − rf )
Q + Q = 1 2 . 1+ rf + β1+2 (rM − rf )
It is the sum of assets 1 and 2, here β1+2 is the beta value of the new asset.
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Chapter 4
MEASURING PORTFOLIO PERFORMANCE 4.1 Measuring the rate of return to a portfolio The rate of return of a portfolio is measured as the sum of cash received (dividend) and the change in the portfolio’s market value (capital gain or loss) divided by the market value of the portfolio at the beginning of the portfolio, mathematically, Cash( Dividend )+ Capital ( gainor loss ) Return of a portfolio = Market valueof a portfolio( purchase price ) The rate of return is the most important outcome from any investment. It works well for static portfolio. Managed portfolios receive additional amount to be invested in the period (a month or a quarter) and their investors can also withdraw fund from the portfolio. Suppose that the market value of a portfolio $ 1 million invested for the period of a quarter and $1 million is added at the end of the first month and then $1.5 million is withdrawn at the end of 2 nd month. How is the return to be calculated for the quarter? There are two methods to calculate this return. • Time Weighted rate of return • Value Weighted rate of return 4.1.1 Time Weighted rate of return The first method is called time weighted rate of return. The time weighted rate of return measures the performance of the portfolio manager. The amount of funds invested is neutralized in the calculation of time weighted return because the funds have deposits and withdrawals by the investors are not under the control of the fund manager but their return are computed on the basis of cash distributions and the changes in the market value of a single share in the fund but the time weighted return is calculated by dividing the beginning value of a share into the cash distribution and the change in the value of a share during the period. However, to calculate the time weighted rate of return, divide the portfolio into shares and compute the return to a single share in the portfolio across the period. In the same way we can calculate the rate of return of a mutual fund. 4.1.2 Value Weighted rate of return The second method is called value weighted rate of return. The time weighted method ignored the deposits and withdrawal to and from the portfolio during the period over which return to be measured but the value weighted method takes deposits and withdrawal into its account. Suppose that wT is a withdrawal at time T and Dt is a
Measuring Portfolio Performance 21 deposit at time t and further assumed that cash (dividend) to the portfolio are received at the end of the period. The value weighted rate of return r is found by solving the following particular equation. n D m w Beginning portfolio Value = t + T ∑ t ∑ t t=1 (1+ r ) t=1 (1+ r ) Total ending valueof portfolio + . (1+ r ) t Here m is number of withdrawals, t is the length of time in years and n is number of deposits during the period. 4.2 Risk Adjusted performance Measure (Based on Capital Asset pricing model) Suppose that for a set of information relevant to any given stock, we can divide this information into two major types. 4.2.1 Public information it is also called open end information. These pieces of information are available to everyone and the manager can offer new shares at any time. 4.2.2 Private information This information is available for selected individuals only. Suppose that if we were to estimate the expected returns, variance and covariance based on the analysis of the public available information alone, we would see the market portfolio positioned on the capital market line shown in the Figure 4.1 and every stocks and portfolio would be positioned on the security market line shown in the Figure 4.2. Figure 4.1 E(r) 0´ 14 Jensen Index 12 0 SML 0 M 10 0 A´ 8 A 0 rf Expected return β 0 .50 1.00 1.50
0 Beta Measuring Portfolio Performance 22
Figure 4.2
E(r) 0´ CML M 0 E( Tm ) A` A 0 rf Expected return Standard deviation σ(r) 0 σ(rm) Standard deviation Consider two professionally managed portfolio, Alpha fund and Omega fund. In the Alpha fund, the managers have private information relating to a single company and this private information is favorable in the sense that expected rate of return to the stock is higher then we think about the public information alone. Suppose that the managers of the Alpha fund invest 100% of money in their portfolio in single stock of this company. If we plot the portfolio position based on the public information alone. Its point label A in figures 4.1 and 4.2 as shown above. Alpha fund plot at point A / in the above two figures, based on the both public and private information and it is above the security market line. With the 2% additional increment in its expected rate of return, its still position inside the efficient set. In the Omega fund, the managers are more skillful because they have able to acquire private information on many other companies. Suppose in this case, the private information affects only in the estimate of expected return but not in the estimate of risk. In the Figures 4.1 and 4.2 the Omega is point at O and O / based on public information alone and both public and private information respectively. Omega does not look like very special to those of us who only have public information to make one estimates. The typical structure of a risk adjusted performance measure is Risk adjusted performance = performance / Risk
Measuring Portfolio Performance 23
4.3 Risk-Adjusted Performance Indices There are three indices available for measuring the risk adjusted performance. • The Jensen Index (Jensen, 1968) • The Sharp Index (Sharp, 1966) • The Treynor Index (Treynor, 1965) All three indices are based on the capital asset pricing model and they are in widespread use. The Jensen Index is a measure of relative performance based on the security market line, whereas the Treynor and Sharp indices are based on the ratio of the return to risk. It is generally assumed in the Jensen and Treynor Indices that stocks are priced according to the capital asset pricing model. We know that capital asset pricing model theory proposes that the expected return on a risky investment is composed of the risk free rate and a risk premium, where the risk premium is the excess market return over the risk free rate multiplied by beta. The Jensen and Treynor indices deal with risk adjusted performance stickle based within the framework of capital asset pricing model and both are bounded by capital asset pricing model assumptions. We have already discussed these assumptions in Chapter 3. 4.3.1 The Jensen Index An index that uses the capital asset pricing model (CAPM) to determine whether a money manager outperformed a market index. In finance, Jensen’s index is used to determine the required (excess) return of a stock, security or portfolio by the capital asset pricing model. Jensen index utilizes the security market line as a benchmark. In 1970’s, this measure was first used in the evaluation of mutual fund managers. This model is used to adjust the level of beta risk, so that riskier securities are expected to have higher returns. It allows the investor to statistically test whether portfolio produced an abnormal return relative to the overall capital market. An important issue regarding the use of Jensen Index is the choice of the market index, because the portfolio performance will be compared with the market portfolio. According to capital asset pricing model (CAPM), in an equilibrium risk return model (Levy and Sarnat, 1984) the expected rate of return on an asset or portfolio is expressed as
Erpp= r +( Er mfp − r ) β . (1) Here
Er p = expected return of an asset or portfolio
rf = risk free rate of return
Er m = expected return on the market portfolio
Measuring Portfolio Performance 24
β p = beta or systematic risk of the asset or portfolio. We want to obtain the Jensen Index, a time series regression of the security’s return
(rp− r f ) is regressed against the market portfolio excess return (rm− r f ) . Now
(rrpf−=+−) α pmfpp( rr ) β + ε . (2) Here
rp = return on the portfolio
rf = risk free rate of return
α p = Jensen Index measure of the performance of the portfolio
β p = beta or systematic risk of the portfolio
rm = return of the market portfolio
ε p = portfolio random error term. Now by taking mean on the both sides of equation (2), we obtain
(rrpf−) =α pmfp +( rr − ) β . (3)
By Levy and Sarnat 1984, the average error term ε p is always zero. So equation (3) become
αpp=−r( r f +( rr mf − ) β p ) . (4)
In the framework of capital asset pricing model (CAPM), α p should be zero. It means that the stock has performed exactly same as the market expected based on its systematic risk.
The Jensen Index ( α p ) for a particular portfolio is identified by the vertical intercept of the regression model described in equation (4), from the equation (4) it is clear that the higher the vertical intercept ( α p ), the greater the abnormal return achieved by the portfolio in the excess of the market return. Here we discussed three scenarios of super market performance along with the diagrams. In all the scenarios, the excess returns on the fund are plotted against the excess returns on the market.
Measuring Portfolio Performance 25
First Scenario
The excess returns on the fund are plotted against the excess returns on the market as shown above. The regression line in the first scenario has a positive (+ve) intercept. This is the abnormal performance.
Second Scenario
The second scenario shows what is known as market timing. If the portfolio manager knows when the stock market is going up, he will shift into high beta stocks. If the portfolio manager knows the market is going down, he will switch into low beta market. In the high beta stocks, these stocks will go up even further then the market and in the case on low beta stocks, these stocks will go down less then the market. Here we notice that the Jensen measure is positive signaling superior performance.
Measuring Portfolio Performance 26
Third Scenario
The third scenario shows market timing, suppose the manager is so good that there are no negative returns. The managers know the good market timing abilities. Suppose that the market goes up. In this case, the fund goes up by more than the market, which is indicating that it shifts into high beta stocks. It is important to notice that the Jensen measure in this case is negative. Even though the manager has exhibited strong market timing abilities, the performance evaluation criteria tells that he is not doing a superior job. It is a major problem in the Jensen measure. 4.3.2 The Treynor Index
In 1965, Treynor’s was the first researcher who computed measure of the portfolio performance. A measure of a portfolio excess return per unit of risk is equal to the portfolio rate of return minus the risk free rate of return, dividing by the portfolio beta. This is useful for assessing the excess return, evaluating investors to evaluate how the structure of the portfolio to different levels of systematic risk will affect the return.
Symbolically, the Treynor Index ( Tp ) is presented as
rp− r f Tp = . β p Here
rp = portfolio rate of return
rf = risk free rate of return
β p = portfolio beta.
When rp> r f and β p > 0 , we get a larger Treynor value. It means a better portfolio for all the investors regarding of their individual risk performance. Measuring Portfolio Performance 27
We discuss two cases, in which we may have a negative Treynor Value.
• When rp< r f The Treynor is negative because rp< r f , we judge the portfolio performance very poor.
• When β p < 0 The negativity becomes from beta, the funds performance is superb.
There is another very important case, suppose that when rp− r f and β p are both negative, then Treynor will become positive but in order to qualify the funds performance as good or bad we should see whether rp lies above or below the security market line. The Treynor index uses the security market line as a benchmark. This index has a geometric interpretation which is similar to the sharp index. It measures the slope of a line that starts at the risk free rate and connects with the point that marks the fund beta and expected return. All risk averse investors would like to maximize this, while a high and positive (+ve) Treynor index shows a superior risk adjusted performance of a fund, while a low and negative ( ve) Treynor Index shows an unfavorable risk adjusted performance of a fund.
The excess returns on the fund are plotted against the beta. The security market line is drawn with excess returns on the vertical axis. The security market line is the dashed line that starts from zero in the excess return axis. Notice that the mutual funds distributed randomly above and below the security market line.
Demonstration with example
As we discuss above, when rp− r f and β p are both negative, then Treynor will be positive (+ve). In order to find the fund performance as good or bad we should see whether rp above or below the market line. Consider the following example.
Measuring Portfolio Performance 28
Assume that we have the following data for three funds namely, ABC, DEF and GHI, with their rate of return and beta. The risk free rate is 12%. The risk for market (M) is 1.0 and the rate of return for the market (M) is 18%. Manager Rate of Return Beta Market 18% 1.00 ABC 16% 0.90 DEF 20% 1.05 GHI 22% 1.20 Now by using the Treynor index equation, we can calculate the value of each manager
For Market We know that
rp− r f Tmarket = (1) βM
Here rp = 18%, rf = 12%, βM = 1.0 Putting these values in equation (1) (0.18− 0.12) T = market 1.0
Tmarket = 0.06 Manager ABC (0.16− 0.12) T = ABC 0.90
TABC = 0.044
Manager DEF (0.20− 0.12) T = DEF 1.05
TDEF = 0.076
Manager GHI (0.22− 0.12) T = GHI 1.20
TGHI = 0.083
Measuring Portfolio Performance 29
Values of each Manager
• Tmarket = 0.06
• TABC = 0.044
• TDEF = 0.076
• TGHI = 0.083
Treynors SML
0.2 GHI 0.15 DE 0.1 F
Return 0.05 ABC 0 0 1 2 3 Beta Securities Market line It can be calculated as 0.12 + (0.06 * Value of beta) Manager ABC = 0.12 + (0.06 * 0.90) = 0.174 Manager DEF = 0.12 + (0.06 * 1.05) = 0.183 Manager GHI = 0.12 + (0.06 * 1.20) = 0.192 These results show that GHI had the best performance and ABC did not beat the market and DEF also beat the market as shown in the above figure.
4.3.3 Sharpe Index
In 1966 Sharpe developed a composite measurement of portfolio performance which is very similar to the Treynor measure. The only difference being the use of standard deviation instead of beta. The Sharpe index is a measure in which we may measure the performance of our portfolio in a given period of time. Measuring Portfolio Performance 30
In Sharpe index, we must know three things, the portfolio return, and the risk free rate of return and the standard deviation of the portfolio. Another thing is that for the risk free rate of return, we may use the average return (over the given period of time). The standard deviation of the portfolio is measure the systematic risk of the portfolio. The Sharpe index is computed by dividing the risk premium of the portfolio by its standard deviation or total risk. Symbolically, the Sharpe index is presented as
rp− r f SP = . σ p Here
rp = portfolio rate of return
rf = risk free rate of return
σ p = standard deviation. The Sharpe index uses the capital market line as a benchmark. Suppose that mutual fund is positioned on the capital market line then the fund has natural performance. This makes sense under capital asset pricing model, because on the basis of the public information only, any investor can construct a portfolio that is positioned on the capital market line. The higher the Sharpe measure indicates a better performance because each unit of total risk (standard deviation) is rewarded with greater excess return. Demonstration with example Rate of return and standard deviation for three portfolios are given below, the risk free rate is 0.12. The systematic risk for the market (M) is1.0 and the rate of return for market (M) is 18%. Portfolio Rate of Return SDEV Market 18% 2.00 UV 17% 0.18 WX 21% 0.22 YZ 20% 0.23 Sharpe Measure
The Sharpe index equation is
rp− r f SP = σ p For Market
rp− r f Smarket = σ M
rp = 0.18, rf = 0.12, σ p = 2.0
Measuring Portfolio Performance 31
Putting in above equation (0.18− 0.12) S = market 0.20
Smarket = 0.300
Portfolio UV (0.17− 0.12) S = UV 0.18
SUV = 0.278
Portfolio WX (0.21− 0.12) S = WX 0.22
SWX = 0.409 Portfolio YZ (0.20− 0.12) S = YZ 0.23
SYZ = 0.348
Values of each Portfolio
• Smarket = 0.300
• SUV = 0.278
• SWX = 0.409
• SYZ = 0.348
Capital Market line It can be calculated as 0.12 + (0.30 * SDEV) Portfolio UV = 0.12 + (0.30 * 0.18) = 0.174 Portfolio WX = 0.12 + (0.30 * 0.22) = 0.186 Portfolio YZ = 0.12 + (0.30 * 0.23) = 0.189 Thus, the portfolio YZ did the best performance and UV failed to beat the market and WX also beat the market.
Measuring Portfolio Performance 32
Example 4.1 Suppose a portfolio manager achieved a return of 15% his portfolio has standard deviation of 0.3 and a market achieved a return of 14.6%, and a risk free rate of return of 7%. Calculate the Sharpe Index. Solution The Sharpe index equation is
rp− r f SP = σ p
Here rp = 0.15, rf = 0.07, σ p = 0.3 Putting in above equation (0.15− 0.07) S = p 0.3
S p = 0.267 Example 4.2 Suppose we have to ask to analyze two portfolios having the following characteristics. Portfolio Observed r Beta Residual Variance 1 0.18 1.8 0.04 2 0.12 0.7 0.00 • The return on the market portfolio is 0.14. • The risk free rate is 0.07. • The standard deviation of the market portfolio is 0.02. Compute
a) The Jensen Index for portfolios 1 and 2. b) The Treynor Index for portfolios 1 and 2 and the market portfolio. c) The sharp Index for portfolios 1 and 2 and the market portfolio. Solution Part (a) Jensen index
Portfolio 1 We know that
αpp=−r( r f +( rr mf − ) β p ) (1)
Measuring Portfolio Performance 33
Here
rp = 0.18 , β p = 1.8
rf = 0.07 , rm = 0.14 putting all values in above equation
α p =0.18 − 0.07 +( 0.14 − 0.07) 1.8
α p =0.18 − 0.196
α p = − 0.016 or α p = − 1.6% Portfolio 2
Here
rp = 0.12 , β p = 0.7
rf = 0.07 , rm = 0.14 again putting this values in equation (1)
α p =0.12 − 0.07 +( 0.14 − 0.07) 0.7
α p =0.12 − 0.119
α p = 0.001 or α p = 0.1% Part (b) Treynor Index Portfolio 1 We know that
rp− r f Tp = (2) β p
Here rp = 18%, rf = 7%, β p = 1.8 putting these values in equation (2). (0.18− 0.07) T = p 1.8
Tp = 6.11
Measuring Portfolio Performance 34
Portfolio 2
Here rp = 12%, rf = 7%, β p = 0.7 Again putting these values in equation (2) (0.12− 0.07) T = p 0.7
Tp = 7.14 Treynor index for the market r− r T = m f βm
Here rm = 0.14, rf = 0.07, βm = 1.25 putting these values in above equation. (0.14− 0.07) T = 1.25 T = 5.6 Part (C) Sharpe Index
Portfolio 1 Standard deviation for portfolio 1 is given by the following equation. 2 22 2 σp= βσ pm + σ p 2 2 21/2 σp=[ βσ pm + σ p ] 1/ 2 2 2 σ p =(1.8) (0.02) + (0.04)
σ p = 0.0538 = 5.38% Sharpe index for portfolio 1
rp− r f SP = (3) σ p
Here rp = 0.18, rf = 0.07, σ p = 0.0538
Measuring Portfolio Performance 35
Putting values in equation (3) (0.18− 0.07) S = p 0.0538
S p = 2.044 Portfolio 2 Standard deviation for portfolio 1, we know that 2 22 2 σp= βσ pm + σ p 2 2 21/2 σp=[ βσ pm + σ p ] 1/ 2 2 σ p =(0.7) (0.02 + 0.00)
σ p = 0.0989 = 9.89%
Here rp = 0.12, rf = 0.07, σ p = 0.0989 Putting values in equation (3) (0.12− 0.07) S = p 0.0989
S p = 0.505 Sharpe Index for Market
r− r S = m f σ m (0.14− 0.07) S = 0.02 S = 3.5
Measuring Portfolio Performance 36
4.4 Comparison of three indices While studying the composite performance measurement of the funds, we see that Sharpe uses Standard deviation as a measurement of risk; on the other hand, Treynor uses Beta. If we are examining a well diversified portfolio, the ranking should be similar for all these indices. For my analysis, I have chosen the data from the Paper “Performance Evaluation of the Mutual Funds, by Hewad Walasmel”, but he took the data from the JP Morgan Global index bond as the risk free rate. In his paper, he has chosen around about 80 international mutual funds, but I take only 30 and compare of these. All of the funds have been in the market at least 10 year period. In the first table mentioned below, we show the values of Mean, standard deviation and beta value against each fund. Table 4.1 Fund Mean SDEV Beta ABBE 0.001 0.015 0.460 ABFA 0.001 0.016 0.140 ASNA 0.001 0.016 0.760 BHUM 0.003 0.029 1.020 CISE 0.002 0.015 0.530 CITY 0.001 0.020 0.140 COMM 0.001 0.022 0.630 FPSI 0.001 0.011 0.370 FPSE 0.002 0.013 0.400 KBCE 0.003 0.020 0.770 NPIP 0.004 0.016 0.610 ROBU 0.001 0.025 0.140 TSB 0.003 0.020 0.750 VARL 0.004 0.028 0.950 VGRN 0.001 0.031 0.410 CAVE 0.001 0.019 0.120 INGG 0.004 0.025 1.010 POST 0.004 0.024 0.920 SEBA 0.004 0.029 1.010 WAAA 0.002 0.030 1.020 HSBC 0.005 0.024 0.900 CONS 0.002 0.015 0.080 DRGE 0.001 0.015 0.110 LLOY 0.004 0.023 0.730 WALS 0.003 0.023 0.810 CERA 0.002 0.034 1.240 UBSM 0.002 0.020 0.460 LAKE 0.003 0.018 0.730 SEBG 0.002 0.022 0.920 DNBR 0.001 0.032 0.980
Measuring Portfolio Performance 37
The first step of analysis is to measure the performance of all listed above 30 funds according to the performance measurement of Sharpe, Treynor and Jensen. We know the formulas how to calculate the Sharpe, Treynor and Jensen indices. The table 4.2 shown below provides an analysis of the performance of the given funds. The first, second and third columns report the Sharpe, Treynor and Jensen respectively. Table 4.2 Performance of the given funds
Fund Sharpe Treynor Jensen ABBE 0.088 0.0028 0.0002 ABFA 0.097 0.0112 0.0012 ASNA 0.097 0.0020 0.0032 BHUM 0.110 0.0031 0.0008 CISE 0.174 0.0047 0.0012 CITY 0.071 0.0099 0.0011 COMM 0.057 0.0020 0.0003 FPSI 0.158 0.0046 0.0005 FPSE 0.189 0.0060 0.0011 KBCE 0.176 0.0044 0.0015 NPIP 0.243 0.0063 0.0024 ROBU 0.049 0.0087 0.0009 TSB 0.191 0.0050 0.0020 VARL 0.140 0.0041 0.0017 VGRN 0.010 0.0008 0.0012 CAVE 0.046 0.0074 0.0006 INGG 0.168 0.0042 0.0018 POST 0.174 0.0045 0.0019 SEBA 0.137 0.0039 0.0015 WAAA 0.093 0.0027 0.0003 HSBC 0.201 0.0054 0.0027 CONS 0.156 0.0320 0.0022 DRGE 0.108 0.0141 0.0013 LLOY 0.169 0.0053 0.0021 WALS 0.163 0.0047 0.0018 CERA 0.053 0.0015 0.0048 UBSE 0.114 0.0046 0.0001 LAKE 0.206 0.0052 0.0020 SEBG 0.093 0.0023 0.0001 DNBR 0.004 0.0001 0.0025
Measuring Portfolio Performance 38
After measuring the ratio of these funds, we separated top 15 funds according to each measure. It is shown in table 4.3.1, 4.3.2 and 4.3.3. This shows whether there is identical ranking for these three ratios. Table 4.3.1 Top 15 ranking of funds using Sharpe index
Ranking Sharpe Value 1 NPIP 0.243 2 LAKE 0.206 3 HSBC 0.201 4 TSB 0.191 5 FPSE 0.189 6 KBCE 0.176 7 POST 0.174 8 CISE 0.174 9 LLOY 0.169 10 INGG 0.168 11 WALS 0.163 12 FPSI 0.158 13 CONS 0.156 14 SEBA 0.137 15 UBSE 0.114 Table 4.3.2 Top 15 ranking of funds using Treynor index
Ranking Treynor Value 1 CONS 0.0320 2 DRGE 0.0141 3 ABFA 0.0112 4 CITY 0.0099 5 ROBU 0.0087 6 CAVE 0.0074 7 NPIP 0.0063 8 FPSE 0.0060 9 HSBC 0.0054 10 LLOY 0.0053 11 LAKE 0.0052 12 TSB 0.0050 13 CISE 0.0047 14 WALS 0.0047 15 FPSI 0.0046
Measuring Portfolio Performance 39
Table 4.3.3 Top 15 ranking of funds using Jensen index
Ranking Jensen Value 1 ASNA 0.0032 2 HSBC 0.0027 3 NPIP 0.0024 4 CONS 0.0022 5 LLOY 0.0021 6 LAKE 0.0020 7 TSB 0.0020 8 POST 0.0019 9 WALS 0.0018 10 INGG 0.0018 11 VARL 0.0017 12 KBCE 0.0015 13 SEBA 0.0015 14 DRGE 0.0013 15 UBSM 0.0001
Tables 4.3.1, 4.3.2 and 4.3.3 show that each fund in the above 15 ranking has a different rank according to the different performance measurements. We find that none of the funds are fully diversified.
4.5 Conclusion From the above tables we conclude that there is no identical ranking of the three measurements for any funds. This also shows that these funds are not completely diversified because we know that completely diversified funds have the similar ranking for the composite performance measurement of Sharpe, Treynor and Jensen. It means that there is still some degree of unsystematic risk that any manager can remove by diversification.
Measuring Portfolio Performance 40
References [1] C. T. Gurroy and Y. Omer, 2001, “Evaluation of portfolio performance”, Dogus University [2] Dr. K .Spremann and Dr. Pascal, 2000, “Approaches to Modern Performance Measurement”. Journal of portfolio management. [3] Droms W G and D. A, Walker 1996 “Mutual funds investment performance” Quarterly Review of economics. [4] Hewad Walasmel, 2005, “Performance evaluation of Mutual funds”, Journal of finance 0509023 [5] Jensen, M 1968 “The Performance of Mutual Funds in the period 1945 1964”, Journal of Finance. [6] Luenberger, D.G. (1997) Investment Science. Oxford University. Press Inc, New York. [7] Markowitz, H. M, “Portfolio Section”, Journal of finance, 1952, V7(1), 77 91 [8] Markowitz, H. M, “Portfolio Section: Efficient Diversification of Investment”, John Wiley & Sons, Inc., 1959 [9] Robert A. Haugan, Modern Investment Theory, Third edition University of California. [10] Sharpe, William F “Mutual fund performance” Journal of business, January 1966 [11] Treynor, Jack L 1965 “How to Rate Management of Investment Funds”, Harvard Business Review [12] http://www.effisols.com/basics/mno.htm [13] http://www.investopedia.com/university/concepts/concepts8.asp [14] http://en.wikipedia.org/wiki/capital asset pricing model [15] http://www.mutualfundsindia.com/perf.asp [16] http://www.duke.edu/~charvey/Classes/ba350/perfeval/perfeval.htm
Measuring Portfolio Performance