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

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.

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.

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

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

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.

Chapter 2

PORTFOLIO MEAN AND VARIANCE

2.1 Mean Return of a Portfolio

Suppose that there are nassets with rates of returnr₁,r₂,…,r and these have expected _n values E(r₁)= r ,₁ E(r₂)= r , ….. ,₂ E(r_n) = r_n . We form a portfolio of these nassets using the weightsw_i, i = 1,2,…,n. The rate of return in terms of the individual return of the portfolio is

r = w₁r₁+ w₂r₂+ … + w_nr_n. (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)

) (r

E = w₁E(r₁) + w₂E(r₂) + … + w_nE(r_n). (2)

2.2 Variance of Portfolio Return

The variance of the return of asset i is denoted byσ_i², the variance of the return of the portfolio byσ², and the covariance of the asset i with asset j byσ_ij. We can perform the following calculation,

σ2 = ^E

[

]

= 



 





∑

∑

= =

n

i

n

i i i i

ir w r

w E

1 1

)2

( ,

= 



 





∑

∑

= =

n

i

n

j

j j j i

i

i r r w r r

w E

1 1

) (

))(

(

( ,

= 



 





∑

= n

j i

j j i i j

iw r r r r

w E

1 ,

) )(

( ,

σ² =

∑

= n

j i

ij j iw w

1 ,

σ . (3)

Equation (3) represents the Variance of the return on the portfolio.

Example 2.1 Consider there are two assets with expected values r₁ = 0.22, r₂ = 0.55 and the variance are σ₁ = 0.80, σ₂= 0.88 and σ₁₂ = 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) + w₂E(r₂)

= 0.25(0.22) + 0.65(0.55)

= 0.4125.

Variance of the portfolio σ2 =

∑

= n

j i

ij j iw w

1 ,

σ

= w₁²σ₁² + w₂²σ₂² + w₁w₂σ₁₂ + w₂w₁σ₂₁

= (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,..., r_n and their covariances are σ i j , for i = j = 1,2,…,n. The portfolio is defined as a set of n weightsw_i, 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

∑

j i

ij j iw w

1

2 ,

min1 σ ,

subject to

∑

= n =

i i

ir r

w

1

,

∑

= n =

i

wi 1

1.

Solution of Markowitz Problem

Lagrange Multipliers λ and µ 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.

f =

∑

= n

j i

ij j iw w

1

2 ,

1 σ ,

∑

= n −

i i

ir r

w

1

= 0,

∑

= n −

i

wi 1

1 = 0.

Then the each left hand side is multiplied to its Lagrange Multiplier and subtracted from the objective function.

Lagrangian function

L (w ,_i w_j) =

∑

= n

j i

ij j iw w

1

2 ,

1 σ - λ(

∑

= n −

i i

ir r

w

1

) - µ(

∑

= n −

i

wi 1

1 ).

Differentiate the Lagrangian with respect tow and _i w and put equal to zero. If the type _j 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

L (w ,_i w_j) =

∑

= 2

1

2 ,

1

j i

ij j iw

w σ - λ(

∑

= 2 −

1 i

i

ir r

w ) - µ(

∑

= 2 −

1

1

i

wi ) or

L (w ,_i w_j) = 2

1(w₁²σ₁² + w₁w₂σ₁₂ + w₂w₁σ₂₁+ w₂²σ₂² ) - λ(r₁w₁+ r₂w₂- r ) - µ( w₁ + w₂- 1 ).

Differentiate the above equation with respect tow₁ andw₂, we obtain

w1

L

∂

∂ = 2

1(2w₁σ₁²+ w₂σ₁₂+σ₂₁w₂) - λ r - ₁ µ , and

w2

L

∂

∂ = 2

1(σ₁₂w₁+ w₁σ₂₁+ 2w₂σ₂² ) - λ r - ₂ µ.

Now putting w1

L

∂

∂ = 0, w2

L

∂

∂ = 0, and using the fact σ₁₂=σ₂₁, we get

1 2 1w

σ + σ₁₂w₂+ λ r - ₁ µ = 0, and

1 21w

σ + σ₂²w₂ - λ r - ₂ µ = 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 µ.

Equation for efficient set

The efficient portfolio for two Lagrange Multipliers λ and µ and the portfolio weight w for i = 1, 2… n having the mean rate of return r satisfy i

∑

j i ijw

1

σ - λ r_i - µ = 0, for i = 1, 2… n (1)

∑

= n

i i ir w

1

= r , (2)

and

∑

= n

i

wi 1

= 1. (3)

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.w_i' , s λ andµ. 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.

Using the equations,

∑

j i ijw

1

σ - λ r_i - µ = 0, for i = 1, 2… n (1)

∑

= n

i i ir w

1

= r , (2)

and

∑

= n

i

wi 1

= 1. (3)

we have

σ₁² = σ₂²= σ₃²= 1 and σ₁₂= σ₂₃= σ₃₁ = 0.

Using all known values in equations (1) – (3), we get the following five equations.

w1 - λ - µ = 0, (4)

w2- 2λ - µ = 0, (5)

w - 33 λ - µ = 0. (6)

And w₁ + 2w₂ + 3w = r , ₃ (7)

w₁ + w₂ + w = 1. ₃ (8)

We have to find the values of λ andµ, substituting the values ofw₁,w₂ and w from ₃ equations 4, 5 and 6 in equations 7 and 8, we obtain

(λ + µ) + 2(2λ +µ) + 3(3λ +µ) = r ,

14λ + 6µ = r , (9)

and

(λ + µ) + (2λ +µ) + (3λ +µ) = 1,

6λ + 3µ = 1. (10)

Solving equation (9) and (10) simultaneously, we get λ and µ as

λ = 2 r - 1,

µ = 2(

3

1) - r .

Substituting λ and µ in equations (4) to (6), we get the values ofw₁, w₂and w ₃

w1 = 3 4 - (

2 r ),

w2 = 3 1, and

w = ( 3

2 r ) - (

3 2).

The standard deviation is given by σ = w₁² +w₂² +w₃²

σ =

2 2 3

7 r²

− r +

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 µ_pon the y-axis and risk σ_pon 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

increases then corresponding expected rate of return must also increase. “M” is the market portfolio and r is the risk free rate of return. _f

The capital market line states that

r = p r + _{f p}

M f

M r

r σ

σ

− .

Here

r p = expected return of an efficient portfolio r f = risk free rate of return

r M = expected return of the market portfolio σp = standard deviation of the efficient portfolio σM = standard deviation of the market portfolio.

The slope of the capital market line is given by

K =

M f

M r

r σ

− .

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

r = 15% = .15 , f r = 23% = .23 _M σM = 17% = .17 , σ_p = 45% = .45

From the capital market line, we know that r = _p r + _{f p}

M f

M r

r σ

σ

− ,

= (.15) + (.23 .15 (.45) .17

− ),

= (.15) + (0.47) (.45),

= 36%.

Actual expected rate of return is

r = ( 2000 1750 ) – 1, = .14,

= 14%.

After comparing the both values it is clear that the oil venture lies well below the capital market line.

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 r_i of any asset i satisfies the relationship.

ri - r = _f β_i( r_M − ), r_f where

r f = risk free rate βi = beta of the security r M = expected market return ( r_M − ) r_f = equity market premium and

βi = ₂

M iM

σ σ .

Proof Suppose for anyα , 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

r_α = α r_i + (1 - α ) r _M (1) Standard deviation of the rate of return is

σ_α = [α²σ_i²+ 2α (1 - α ) σ_iM+ (1 - α ) ² σ_M² ]^1/2 (2) The values of α are traced out as shown in the diagram below.

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 toα, we get

α

α

d r

d = r_i - r . _M Furthermore

α σ_α d d =

2

1[α²σ_i²+ 2α (1 - α )σ_iM+ (1 - α )² σ_M² ]⁻¹²× [2ασ_i+ 2(1 - α )σ_iM + 2(1 - α )(-1) σ_M² ],

M

= 2 1

1/2 2 M 2 iM

2 i 2

2

] ) - (1 + ) - (1 2 + [

) 1 ( 2 )

1 ( 2 2

σ α σ

α α σ

α

σ α σ

α

ασ_i + − _iM + − _M

= _{2 1/2}

M 2 iM

2 i 2

2

] ) - (1 + ) - (1 2 + [

) 1 ( )

1 (

σ α σ

α α σ

α

σ α σ

α

ασ_i + − _iM + − _M

.

At α = 0, we get

=0 α α

α σ d

d = _{2 1/2}

M 2 iM

2 i 2

2

] ) 0 - (1 + 0) - (1 2 + ) [(0

) 1 0 ( )

0 1 ( ) 0 (

σ σ

α σ

σ σ

σ_i + − _iM + − _M

=

M 2

] [σ

σ σ_iM − _M

=

M M iM

σ σ σ − ²

.

Using the relation

α α

σ d

r

d =

α σ

α

α α

d d

d r

d ,

at point α = 0, we get

=0 α α

α σ d

d =

M m iM

M

i r

r σ

σ σ − ²

−

= ( ) ₂

M iM

M M

i r

r σ σ

σ

−

− .

This slope is also equal to the slope of the capital market line.

( ) ₂

M iM

M M

i r

r σ σ

σ

−

− =

M f

M r

r σ

− ,

and therefore,

2 2

M M M

i r

rσ − σ =

(

) (

)

M M M

i r

rσ − σ = r_Mσ_iM −r_Mσ_M −σ_iMr_f +σ_Mr_f . Thus

2 2

M f M

i r

rσ − σ =

(

)

ri = r + _f

2

) (

M iM f

M r

r σ

σ

− . (3)

Now let

β_i = ₂

M iM

σ σ .

Putting the value of β_i in equation 3, we get

r_i = r + _f β_i

(

)

The r −_i r_f 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, r −_M r_fis 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.

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 theβ (beta).

β = ₂

M iM

σ

σ ,

=

(

)

045 .

0 ,

= 2.

Here r = 10% = .10 , _f r = 14% = .14 _M The capital asset pricing model formula is

ri = r + _f β_i( r −_M r_f), 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 r = .08, _f r = .12 _M Case (a)

When β = 0

ri = r + _f β_i( r −_M r_f ), = .08 + 0 (.12-.08), = .08.

Case (b)

When β = 2

ri = r + _f β_i( r −_M r_f ), = .08 + 2(.12-.08), = .08 + .08 = .16.

This example shows that the higher the degree of the systematic riskβ, 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+β_i(R_m −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 r = .04 , _f r = .15. _M

Hereβ_i denotes the beta coefficient of the stock i . We calculate the beta coefficient β_i for the portfolio and get the expected return on the portfolio from the capital asset pricing model equation.

Here β_i = x_Aβ_A + x_Bβ_B + x_Cβ_C + x_Dβ_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

r_i = r + _f β_i( r −_M r_f ), = .04 + (1.29) (.15-.04),

= .04 + 0.15, = 0.19.

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 . Then r =

P P

Q )

( − is the rate of return, Here P is known and Q is random (unknown), the CAPM formula is

r = r_f + β_i

(

)

P P

Q )

( − = r + _f β

(

)

Q − = P ( r + _f β

(

)

(

)

= P (1 + ( r + _f β

(

)

P =

))) (

( 1

( r_f r_M r_f Q

− +

+ β ^{. (2)}

It is the price of the asset according to the CAPM. Hereβ 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%.Theβ 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 P =

))) (

( 1

( r_f r_M r_f Q

− +

+ β ⁽¹⁾

here Q = $1200 , β = 0.8

rf = 20% = 0.20 , r_M = 70% = 0.70 putting this value in above equation (1)

P =

) 20 . 0 70 . 0 ( 8 . 0 20 . 0 1

1200

− +

+ ,

P = 6 . 1 1200,

P = $750.

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 P1 =

) (

1 ₁

1 f M

f r r

r Q

− +

+ β ^, P² =

) (

1 ₂

2

f M

f r r

r Q

− +

+ β ^.

Adding P₁ andP₂, we get

P1 + P₂ =

) (

1 ₁

1

f M

f r r

r Q

− +

+ β ⁺ 1 ₂( )

2

f M

f r r

r Q

− +

+ β

=

) (

1 _{1 2}

2 1

f M

f r r

r

Q Q

− +

+

+

β+ ^.

It is the sum of assets 1 and 2, here β₁₊₂ is the beta value of the new asset.

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,

Return of a portfolio = ^{( ) ( )}

( )

Cash Dividend Capital gain or loss Market value of 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 w_T is a withdrawal at time T and D_tis a

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.

Beginning portfolio Value =

1(1 )

n t

t t

D

= +r

∑

1 (1 )

m T

t t

w

= +r

∑

+ (1 )^t

Total ending value of portfolio

+r .

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

⁰

8 10 12 14 E(r)

1.00 1.50

.50 A

0 A´ M SML

0

0 0´

Jensen Index

0 Beta

Expected return

0 rf

β

Figure 4.2

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

0

Standard deviation Standard deviation E(r)

Expected return

0 rf

σ(r)

CML

E(Tm)

M

A A`

σ(rm)

0 0´

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

( )

p p m f p

Er =r + Er −r β . (1)

Here

Erp= expected return of an asset or portfolio rf = risk free rate of return

Erm= expected return on the market portfolio

βp = 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

(

)

(

)

Now

(

)

(

)

Here

rp = return on the portfolio rf = risk free rate of return

αp= Jensen Index measure of the performance of the portfolio βp= beta or systematic risk of the portfolio

rm = return of the market portfolio εp = portfolio random error term.

Now by taking mean on the both sides of equation (2), we obtain

(

)

(

)

By Levy and Sarnat 1984, the average error term ε_pis always zero.

So equation (3) become

( )

( )

p rp rf rm rf p

α = − + − β . (4)

In the framework of capital asset pricing model (CAPM), α_pshould 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.

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.

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 ( T ) is presented as _p

p f

p

p

r r

T β

= − .

Here

r = p portfolio rate of return r = f risk free rate of return βp= portfolio beta.

When r_p >r and_f β_p > 0, we get a larger Treynor value. It means a better portfolio for all the investors regarding of their individual risk performance.

We discuss two cases, in which we may have a negative Treynor Value.

• When r_p <r The Treynor is negative because_f r_p < r , we judge the portfolio _f 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 r_p− and r_f β_pare both negative, then Treynor will become positive but in order to qualify the funds performance as good or bad we should see whether r lies above or below the security market line. _p

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 r_p −r_f and β_pare both negative, then Treynor will be positive (+ve). In order to find the fund performance as good or bad we should see whether r above or below the market line. Consider the following example. _p

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

p f

market

M

r r

T β

= − (1)

Here r = _p 18%, r = _f 12%, β_M = 1.0 Putting these values in equation (1)

(0.18 0.12)

market 1.0

T −

=

market 0.06

T =

Manager ABC

(0.16 0.12)

ABC 0.90

T −

= 0.044 TABC = Manager DEF

(0.20 0.12)

DEF 1.05

T −

= 0.076 TDEF = Manager GHI

(0.22 0.12)

GHI 1.20

T −

= 0.083 TGHI =

Values of each Manager

• T_market =0.06

• T_ABC =0.044

• T_DEF =0.076

• T_GHI =0.083

Treynors SML

0 0.05 0.1 0.15 0.2

0 1 2 3

Beta

Return

GHI DE

F ABC

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.

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

p f

P

p

r r

S σ

= − .

Here

r = p portfolio rate of return r = f risk free rate of return σp= 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

p f

P

p

r r

S σ

= −

For Market

p f

market

M

r r

S σ

= −

r = _p 0.18, r = _f 0.12, σ_p = 2.0

Putting in above equation

(0.18 0.12)

market 0.20

S −

=

0.300

market

S =

Portfolio UV

(0.17 0.12)

UV 0.18

S −

=

0.278 SUV = Portfolio WX

(0.21 0.12)

WX 0.22

S −

=

0.409 SWX = Portfolio YZ

(0.20 0.12)

YZ 0.23

S −

=

0.348 SYZ = Values of each Portfolio

• S_market =0.300

• S_UV =0.278

• S_WX =0.409

• S_YZ =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.

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

p f

P

p

r r

S σ

= −

Here r = _p 0.15, r = _f 0.07, σ_p = 0.3 Putting in above equation

(0.15 0.07)

p 0.3

S −

= 0.267 S =p

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

( )

( )

p rp rf rm rf p

α = − + − β (1)

Here

r = p 0.18 , β_p = 1.8 r = f 0.07 , r = 0.14 _m putting all values in above equation

( )

0.18 0.07 0.14 0.07 1.8 αp = − + − 

0.18 0.196 αp = −

0.016 αp = −

or α_p = −1.6%

Portfolio 2 Here

r = p 0.12 , β_p = 0.7 r = f 0.07 , r = 0.14 _m again putting this values in equation (1)

( )

0.12 0.07 0.14 0.07 0.7 αp ^{= −} ^{+ −} 

0.12 0.119 αp = −

0.001 αp =

or α_p =0.1%

Part (b) Treynor Index Portfolio 1

We know that

p f

p

p

r r

T β

= − (2)

Here r = _p 18%, r = _f 7%, β_p = 1.8 putting these values in equation (2).

(0.18 0.07)

p 1.8

T −

=

p 6.11 T =

Portfolio 2

Here r = _p 12%, r = _f 7%, β_p = 0.7 Again putting these values in equation (2)

(0.12 0.07)

p 0.7

T −

=

p 7.14 T =

Treynor index for the market

m f

m

r r

T β

= −

Here r = 0.14, _m r = _f 0.07, β_m = 1.25 putting these values in above equation.

(0.14 0.07) T 1.25−

= 5.6 T = Part (C) Sharpe Index Portfolio 1

Standard deviation for portfolio 1 is given by the following equation.

2 2 2 2

p p m p

σ =β σ +σ

2 2 2 1/ 2

[ ]

p p m p

σ = β σ +σ

2 2 1/ 2

(1.8) (0.02) (0.04) σp =^ + ^

0.0538 σp = = 5.38%

Sharpe index for portfolio 1

p f

P

p

r r

S σ

= − (3)

Here r = _p 0.18, r = _f 0.07, σ_p = 0.0538