Deepa Chandrashekar

PORTFOLIO PERFORMANCEEVALUATION- LITERATURE

REVIEW

1[Date]

Table of Contents

1. Introduction........................................................................................................................................... 2

2. Portfolio Returns Calculation................................................................................................................4

2.1. Time weighted rate of return......................................................................................................... 5

2.2. Value weighted rate of return........................................................................................................ 6

2.3. Internal rate of return ....................................................................................................................6

3. Literature Review.................................................................................................................................. 7

3.1. Sharpe ratio ................................................................................................................................... 9

3.2. Sortino ratio ................................................................................................................................10

3.3. Treynor ratio ...............................................................................................................................12

3.4. Jensen ratio..................................................................................................................................13

Weaknesses of traditional performance measures ......................................................................................15

Conclusion ..................................................................................................................................................18

Bibliography ...............................................................................................................................................20

2[Date]

1. Introduction

In this paper, we conduct an in-depth literature review on the topic “portfolio performance

evaluation”. A portfolio is a collection of different investments, which is owned by an individual

or an institution on which an investor bets to make a profit without compromising the principal

investment. A portfolio can include assets such as real estate and gold; however, most investment

portfolios are made up on securities such as stocks, bonds, mutual funds, exchange traded funds

and cash. Portfolio evaluation refers to assessing the performance of an investment portfolio. It

is a process of comparing the returns earned on one portfolio with the returns earned on another

investment portfolio by measuring and evaluating the performance. Performance measurement

measures the returns earned and performance evaluation discloses additional issues such as out-

performance or under-performance of the investment and the reasons behind such performance i.e.

due to investment managers’ skills, market, economic, political circumstances or sheer luck. It is

all about differentiating those investment managers who truly add value through active

management from those who do not. (Lehmann & Timmermann, 2007)

In the 1950s performance of a portfolio was measured only based on returns earned. Although

investment managers realized that risk was a crucial element to consider while measuring the

performance, there was no easy approach to factor that in. Subsequent to this period, Harry Max

Markowitz, an American economist conceptualized a Modern Portfolio Theory (MPT). The

theory provides a framework to construct and select portfolios based on the expected performance

of the investment and the risk appetite of the investor. (Fabozzi, Gupta, & Markowitz, 2002) In

later years many more financially sophisticated tools and concepts were established to assess the

3[Date]

performance of a portfolio allowing investment managers to better serve the needs of their clients.

For instance, in 1960, the Capital Asset Pricing Model (CAPM) was introduced by William Sharpe

and John Litner, which marked the birth of asset pricing theory, a tool that allowed one to connect

expected return and risk. Jack L. Treynor in 1965 and William Jenson in 1968 introduced similar

tools that combined risk and return performance into a single value.

Markowitz, in 1952, stated that an efficient portfolio is one where no other portfolio has higher

return for the same risk and lower risk for the same return. He used the following assumptions to

develop a computer algorithm that would track all efficient portfolio from a given set of stock i.e.

the efficient frontier: (Henriksen & Hansen, 2013)

All investors seek to maximise the expected return of total wealth

All investors have the same expected single period investment horizon

All investors are risk adverse, that they will only accept greater risk if they are compensated

with a higher expected return

All investors base their investment decisions on expected return and risk

All markets are perfectly efficient (e.g. no taxes or transaction costs)

In this paper, we will limit our discussions to conducting a literature review on some of the

traditional portfolio performance evaluation tools introduced by William Sharpe, Frank Sortino,

Treynor and Jenson along with critically examining their theories. The paper will start with a brief

introduction of some basic formulas used for calculating the return on a portfolio, followed by a

detailed literature review on the four stated performance measurement theories as conceptualized

by Sharpe, Sortino, Treynor and Jenson. The document will conclude with the author’s remarks

on the four models.

4[Date]

2. Portfolio Returns Calculation

This sections provides a brief introduction on some of the basic formulas investment managers use

to calculate the returns earned on a portfolio for a given period. Calculating returns on an asset or

an individual portfolio is straightforward; however, it gets more complex when calculating returns

on mutual funds, which involves variable capital.

The simplest formula used by investment managers to calculate a return on portfolio for a given

period of time is obtained by measuring the sum of cash received (dividends) and the change in

the portfolio’s market value (market value less initial investment) divided by the initial investment.

Arithmetically put,

Return of a portfolio = ( )

The above stated formula works well to calculate the returns on a static portfolio. In the case of a

variable portfolio such as a mutual fund when investors add or withdraw capital at varying periods,

the formula must be adapted to take these movements into account. There are three popular

methods that are used to calculate returns on such portfolios.

1. Time weighted rate of return

2. Value weighted rate of return

3. Internal rate of return

5[Date]

2.1. Time weighted rate of return

In a time weighted rate of return the actual returns to an investor are determined by the returns of

the funds they hold, timing and magnitude of their cash flows into and out of these funds.

(Schneider, 2007) The principle behind this method is to break down the period into elementary

sub-periods. The return for the complete period is then calculated by using the geometric mean

of the returns calculated for the sub-periods. This calculation assumes that the distributed cash

flows, such as dividends, are reinvested in the portfolio. (Le Sourd, 2007)

The return for the sub-periods are written as follows:

= − ( )+The return for the whole period is then given by the following formula:

= [∏ (1 + )]1/t -1

Although the time weighted return measures the performance of the fund manager, there is a small

reservation when applying this method. The formula ignores the month-to-month variation in

assets under management. (Schneider, 2007)

6[Date]

2.2. Value weighted rate of return

Value weighted rate of return takes into account the month-to-month variation in assets under

management that is ignored by the time-weighted method. The formula used to calculate the value-

weighted rate of return (r) is given by:

Wt denotes withdrawal at time t and

Di denotes deposit at time t

M denotes the number of withdrawals

T denotes the length of time in years

N denotes the number of deposits during the period. (Shahid, 2007)

2.3. Internal rate of return

Commonly known as the IRR, this method is based on actuarial calculation. The IRR is a discount

rate that is used to make the final value of the portfolio equal to the sum of its initial value and the

capital flows that occurred during the period. The cash flow is the net cash flow for each sub-

period (incoming cash flow such as reinvestment of dividends or client contributions less outgoing

cash flows resulting from payments to clients). The formula used to determine IRR R1 is as stated

below:

7[Date]

Where:

T denotes the length of the period in years

ti denotes the cash flow dates, expressed in years over the period

V0 denotes the initial value of the portfolio

VT denotes the final value of the portfolio

Ctj denotes the cash flow on date ti. (Le Sourd, 2007)

The IRR method provides for more accurate results than the other two methods discussed above

when there are a significant number of capital flows of different sizes.

There are several methods of calculating portfolio returns. Each one provides different results

with one common idiom, which states that a return value should always be accompanied by more

information. (Le Sourd, 2007)

3. Literature Review

In this section, we conduct an elaborate literature review on some of the performance evaluations

tools. Literature suggests and discusses several different performance evaluation tools. Several

other academics and researchers have proposed various methods to measure performance. For

instance, in 1972 Fama proposed a useful decomposition of performance between timing and

8[Date]

selection abilities. Treynor and Mazuy, Henriksson and Merton designed performance measures

that aimed at measuring market-timing abilities. (Hubner, 2005) In 2002 Martine Lettau and

Harald Uhlig examined the effect of preferences on the market price for risk using a lognormal

framework whilst, in 2006 Russ Wermers, used portfolio holdings to evaluate the performance of

an asset manager. (Kolbadi & Ahmadinia, 2011) Annaert, Van Osselaer and Verstraete (2009)

evaluated the performance of the stop-loss, synthetic put and constant proportion portfolio

insurance techniques based on block-bootstrap simulation. They considered both, traditional

performance measures and recently developed measures to capture the non-normality of the return

distribution i.e. value-at-risk, expected shortfall and the Omega measures. In 1997, Modigliani and

Modigliani suggested an alternative measure of risk that uses the volatility of returns in the context

of the capital asset pricing model. (Hubner, 2005) As one can see, many methods have been

developed and used for measuring performance. Not all evaluation tools can be included in this

study. Therefore, we limit our discussions to the four risk adjusted ratio is as conceptualized by

Sharpe (1966), Sortino (1980), Treynor (1966) and Jenson (1968). As return and volatility cannot

be considered in isolation to evaluate the attractiveness of a portfolio, risk-adjusted measures

provide a more efficient perspective to the performance of a portfolio. We are well aware of the

capital asset pricing theory (CAPM), a model that describes the relationship between risk and

expected return. According to the model, the expected return of a stock equals the risk-free rate

and the portfolio’s beta multiplied by the excess return of the market portfolio (Risk premium).

The four ratios that we will discuss now are based on the CAPM principles. Sharpe, Sortino and

Treynor ratios are based on the ratio of the return to risk and Jensen ratio is a measure of the

relative performance based on the security market. (Shahid, 2007) In his research Hubner (2003),

9[Date]

notes that Sharpe and Sortino ratios use the Capital Market Line as the risk-return referential, that

uses standard and target downside deviation of the portfolio returns as the measure of risk, whereas

Treynor and Jensen’s alpha model directly related to the beta of the portfolio using the Security

Market Line. (Hubner, 2005)

3.1. Sharpe ratio

William Sharpe introduced the Sharpe Ratio in 1966. It is a widely used portfolio evaluation tool.

It is defined as the measure of excess return over the risk-free rate per unit of risk. It combines

both the average return on the asset and the volatility of the return for a particular period. (Aftab,

Jungwirth, Sedliacik, & Virk, 2008) The following formula defines Sharpe ratio in mathematical

terms:

= −Where,

rp denotes average return on asset p

rf denotes average return of the risk free rate

σp is the volatility of the return on assets i.e. standard deviation of the return of p

All other characteristics of two portfolios being equal, rational investors would prefer a portfolio

with a greater Sharpe ratio than one with a lesser Sharpe ratio. (Henriksen & Hansen, 2013) This

ratio measures the performance of portfolios that are not well. Veronique Le Sourd (2007) states

10[Date]

that the Sharpe ratio offers significant possibilities for evaluating portfolio performance, whist

remaining simple to calculate.

3.2. Sortino ratio

Dr. Frank Sortino undertook a research in 1980s to come up with an improved measure for risk –

adjusted returns. It is a modification of the Sharpe ratio as it uses the downside deviation that

only considers the negative deviations from the mean or a minimum return threshold. (Weisinger,

2010) Sortino’s model was even recognized by Markowitz when he developed the Modern

Portfolio Theory in 1959. He acknowledged that only downside deviation is relevant to investors

and using it to measure risk is more appropriate than using standard deviation. Duda and Batyuk

(2009) also concur in their research work that a usual investor is more concerned about the

downside effect of volatility on assets, since the upside, is a potential profit. They suggested using

downside risk rather than total risk.

The Sortino ratio is defined as:

Where,

R denotes average period return

MAR denotes minimum acceptable return

DR is the target downside deviation

The target downside deviation is calculated mathematically as shown below (Rollinger &Hoffman, 2013):

11[Date]

Where,

Xi denotes return

N denotes the total number of returns

T denotes target return.

The above stated formula is very similar to the definition of standard deviation, which is a measure

of dispersion of data around its mean. However, target downside deviation significantly differs

from standard deviation and is a measure of dispersion of data below some user-selectable target

return with all above target returns treated as underperformance of zero. According to Rollinger

and Hoffman (2013), large return swings mean volatility and risk. However, they state that if a

portfolio consistently produces strong upward swings with lower downward swings, it should not

be punished for those strong moves in its favour.

12[Date]

In 2008, Dr. Ashraf Chaudhry and Dr. Helen L. Johnson investigated the suitability of various

performance measures under the assumption of a clearly defined benchmark. Their study indicated

that Sortino ratio was a superior performance measure that exhibited more power and less bias

than the Sharpe ratio when the distribution of excess returns are skewed. (Kolbadi & Ahmadinia,

2011)

3.3. Treynor ratio

In 1965, Jack Treynor introduced the Treynor ratio that was computed using the systematic risk of

the portfolio. The model requires the selection of good reference index (a.k.a Treynor index) as

the denominator is heavily reliant on the selected benchmark. (Cogneau & Huber, 2009) This

index is a ratio of return generated by a fund over and above the risk free return for a given period.

The systematic risk associated with it is measured by Beta. (Bansal, Garg, & Saini, 2012)

Mathematical representation of the Treynor ratio is stated below:

(Tp) = Rp-Rf) /βp

Where Rp = portfolio average return, Rf = Risk free rate of return and βp = slope of the characteristic

line. The above formula is then benchmarked against the following formula:

(Rm – Rf)/βm

Where Rm is market average return and βm is the beta of the marker portfolio. (Bansal, Garg, &

Saini, 2012)

13[Date]

Bansal, Garg and Saini (2012) conclude in their research that a high and positive Treynor’s index

demonstrates superior risk-adjusted performance while a low and negative index indicates

unfavourable performance. In 2002, Carnahan noted that market risk as measured by beta cannot

be diversified away through investing in many different funds and hence, deserves to be penalized.

(Alenius, N.D.)

Hubner (2005) identified a gap in using Treynor ratio. He noted a lack of a multi-index counterpart

of the Treynor ratio, which if resolved, will allow relating the level of abnormal returns to the

systematic risk taken by the portfolio manager in order to achieve it. He proposed a generalized

measure of the Treynor ratio. The proposed solution was a simple measure that filled the gap

whilst still keeping the original economic interpretation of the model intact. Dr. Kucuksille and

Acar (2011), note that Treynor ratio differs from the Sharpe ratio only through the choice of the

beta factor.

3.4. Jensen ratio

In 1968, Michael Jensen developed a composite portfolio evaluation technique that considered

returns adjusted for risk difference. (Bansal, Garg, & Saini, 2012) Also known as, Jensen’s alpha,

this ratio measures the extra return that the portfolio earns after adjusting for its “beta” risk.

Jensen’s index is used to determine the required (excess) return of a stock, security or portfolio by

the capital asset pricing model. According to Jensen, an asset with a positive alpha has higher

return than the risk adjusted return estimated by the CAPM. (Shahid, 2007) This model allows

the flexibility to adjust the level of beta risk so that riskier securities can expect higher returns. It

also allows the investor to statistically test whether portfolio produced an abnormal return relative

14[Date]

to the overall capital market. (Shahid, 2007) Shahid (2007) states that Jensen’s alpha calculates

relative returns after considering the systematic risk of the portfolio in the CAPM framework,

whilst Sharpe ratio simply uses the return and variance of the portfolio itself. (Hwang & Salmon,

2001)

Jensen’s alpha calculates the performance of a portfolio by measuring the deviation of a portfolio’s

return from the securities market lines

Where,

rpt denotes the portfolio’s return at time t

rf denotes the risk free rate

rmt denotes to the market return at time t

βp refers to the systematic risk of the portfolio

Jensen’s alpha is the expected excess return of the portfolio less the product of the expected excess

return of the market portfolio and the portfolio’s beta. (Hwang & Salmon, 2001)

Nielsen and Vassalou (2004) proposed modifications to Jensen’s alpha that are consistent with the

expected utility maximization in a continuous-time model. The modifications considered that

investors might change the split of their wealth between the fund and the riskless asset over time.

(Nielsen & Vassalou, 2004)

15[Date]

Weaknesses of traditional performance measures

All the above stated portfolio measurements tools have been subject to criticisms. This section

will elaborate on the gaps identified in each of them as observed through various studies. One of

the main problems with traditional performance measures is the usage of a benchmark, especially

when it comes to estimating the securities market line. According to Aldrian (2000), whenever a

security market line is incorrectly estimated, it projects an inefficient market index which has a

ripple effect on the models proposed by Treynor and Jensen. Aldrian identifies two reasons for

such incorrect positioning of the security market lines, 1) The true risk free return is different from

the risk free return used in these models and 2) a non-optimized market index has been employed

i.e., an index whose expected returns differ from the expected return of the optimized index which

is appropriate for the true risk-free return. (Aldrian, 2000)

A growing number of literatures have identified problems concerning the application of Sharpe

Ratio as a performance measure. Kidd (2011) states that the Sharpe ratio measures only one

dimension of risk i.e., the variance and is only designed to be applied to investment strategies that

have a normal expected return distributions and not for measuring investments that provide an

asymmetric returns. According to Dybvig and Ingersoll (Dybyig & Ingersoll, 1982), non-linear

pay offs limit the utility of the Sharpe ratio in evaluating the performance of a portfolio. The main

problem with Sharpe ratio is that although the returns are definite and quantifiable, risk is not. For

instance, the standard deviation can be calculated from any time series of return data; however, its

meaning will not be the same for all such series. Bernardo and Ledoit (2000) demonstrated that

the Sharpe ratio returns inconsistent results when the payoffs are far from normal. To address this

16[Date]

problem they proposed a semi parametric alternative gain-loss ratio. In 2001 Spurgin showed that

annualized standard deviation of returns tends to be higher for shorter periods such as daily returns

over weekly and monthly returns. Hence, Sharpe ratio can be easily manipulated by lengthening

the measurement period. (Kidd, 2011)

Cerny (2004) also showed why the Sharpe ratio is not a good reward-for-risk measure. According

to Cerny (2004), the Sharpe ratio is based on the assumption that investors have a quadratic utility

function and this utility has a bliss point, beyond which one is penalized for achieving more wealth.

This is demonstrated in the below stated example:

Consider the two assets A and B in the table below:

Table 1:Probability 1/6 ½ 1/3 Sharpe Ratio

Return of Asset A -1% 1% 2% 1.0

Return of Asset B -1% 1% 11% 0.8

As one can see, although asset B has a higher return than asset A, but because its optimal wealth

has extended beyond the bliss point, it achieves a lower Sharpe ratio and, thereby, makes it a less

attractive investment. This is because high outlier returns can have the effect of increasing the

value of the denominator more than the value of the numerator, thereby, lowering the value of the

ratio. (Rollinger & Hoffman, 2013) Cerny’s (2004) solution to this problem was to make the

utility non-decreasing after the bliss point by using truncated quadratic utility function called the

Arbitrage Adjusted Sharpe ratio. He further developed a generalized version that is consistent

with a wider class of utility functions. (Sheridan & Poti, 2007)

17[Date]

Much of the criticism of the Sharpe ratio has been focussed on the differential return distribution

not being symmetric. Henriksson (2005) states that use of standard deviation as a risk of measure

may significantly overstate and understate the riskiness of positively and negatively skewed return

distributions respectively. Qamruzzaman (2014) in his study of performance evaluation of mutual

funds has noted that Sharpe ratio is superior to the Treynor ratio as the former considers the point

whether investors are reasonably rewarded for the total risk in comparison to the market. Dr.

Kucuksille and Acar (2011), note that both Sharpe and Treynor measures share the disadvantage

that they do not provide any guidance for analyzing return differentials. Scholz and Wilkens noted

in 2005 that investors who are not familiar with capital market theory and regression analysis

would find the Treynor ratio difficult to interpret. (Kucuksille & Acar, 2011) Further, the Treynor

ratio has also been criticized for its inability to be used in multi-index asset pricing models, which

is not sufficient to keep track of the systematic sources of portfolio returns in excess of the risk

free rate. (Hubner, 2005)

As with Sharpe ratio, the Sortino ratio has also been subject to criticism. Henriksson (2005) states

that Sortino ratio can be manipulated with derivatives to achieve artificially high values. Another

problem identified by Donald Martin, a financial advisor at Mayflower Capital is that both Sharpe

and Sortino ratios use the stock price as a key element of their data. Stock prices are subject to the

behaviours of irrational investors and, hence, Martin considers it an inappropriate tool to assess

risk. By contrast, he considers the earning stability of a company to be a better judge of risk.

(Martin, 2011) Jensen’s alpha has also been considered difficult to use in practice. It has

predominantly been criticized by fund managers since it can ascribe a negative performance to

market timer because it is based on an upwardly biased estimate of systematic risk for a market-

timing investment strategy. (Hwang & Salmon, 2001) Besides, it has been noted that Jensen’s

18[Date]

alpha does not control non-systematic sources of risk that could matter to investors. (Aragon &

Ferson, 2006)

Conclusion

Having gone through the extensive literature review on the four models, one can conclude that

there are several schools of thought when it comes to evaluating a portfolio’s performance. As

mentioned in the AIMR performance presentation, “the use of a variety of measures with an

understanding of their shortcomings will provide the most valuable information because no one

statistic can consistently capture all elements of risk of an asset class or a style of management”.

(Hwang & Salmon, 2001)

Subsequent to the criticism, each of the four traditional models was subject to, several modified

theories and models were developed and adapted by portfolio managers globally. For example,

since the advent of Sharpe ratio, there were other close analogues developed such as the

information ratio, the squared Sharpe ratio, the M-squared Sharpe ratio, the Generalized Sharpe

ratio, instantaneous Sharpe ratio and so on. Hubner came up with a Generalized Treynor ratio to

fill the gaps identified in the original Treynor ratio. Then there were other alternative measures of

performance evaluation developed such as the Arbitrage pricing theory by Ross in 1976, Fama and

French’s three and five factor APT model, Value-at risk, Appraisal ratio that is a modified version

the Jensen’s alpha model adjusted for specific risks, Grinblatt and Titman’s no benchmark model

etc. Several empirical studies have been conducted by various academics and researchers who

have compared the results of a specific portfolio performance such as hedge funds, mutual funds,

derivatives, asset pricing in incomplete markets, commodities trading etc., using multiple methods

and ratios. The results of each study differed significantly. For instance, a performance evaluation

19[Date]

study on Danish mutual and hedge funds conducted by Duda and Batyuk (2009) concluded that

by using multi-factor model to evaluate performance, hedge funds were seen to outperform mutual

funds when compared to passive benchmark but the same when evaluated using Sortino ratio,

showed no difference in performance. A study conducted by Shahid (2007) of 15 international

mutual funds indicated that the ranking of these funds differed when using Sharpe, Treynor and

Jensen ratio’s, whereas a similar study conducted by Bansal, Garg and Saini (2012) using Sharpe

and Treynor’s ratio provided similar rankings of the mutual funds in India. Aldrian (2000) notes

that traditional measures of performance considered risk, disregarded the style of managers and,

hence, revealed very limited information on the returns generated in a series of Australian funds.

Chen and Knez (1996), in their study conclude that any portfolio performance measure should

satisfy four conditions, it should assign zero performance to each reference portfolio, and it is

linear, continuous and non-trivial. The author is not biased against suggesting the superiority of

one model over the other given that no empirical studies have been personally conducted to arrive

at such a conclusion.

With such varied opinions, what is the best way to then quantify and evaluate a portfolio’s

performance? The answer is open to debate until date. Performance evaluation is very arbitrary.

Ultimately, as stated in the framework of the capital asset pricing model, the investors seek returns

that provide superior performance on their investment compared to that which they could obtain

by combining a risk-free borrowing or lending market portfolio. This area is subject to vast

changes in the upcoming years. More research and empirical studies are being undertaken even

today. Although older measures continue to be used, new models are being proposed on a

continuous basis as market dynamics change.

20[Date]

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Omisore, I., Yusuf, M., & Christopher, N. (2012, March). The modern portfolio theory as an investmentdecision tool. Journal of Accounting and Taxation, 4(2), 19-28.

Qamruzzaman, M. (2014). Comparative Study on Performance Evaluation of Mutual Fund Schemes inBangladesh: An analysis of Monthly Returns. Journal of Bbusiness Studies Quarterly, 5(4), 190-210.

Rollinger, T., & Hoffman, S. (2013, February). Sortino ratio: A better measure of risk. Retrieved fromFutures Magazine: http://www.futuresmag.com/2013/02/01/sortino-ratio-a-better-measure-of-risk

Schneider, L. (2007, July). Are UK Fund Investors Achieving Fund Rates of Return. Retrieved May 20,2015, fromhttp://www.ebisgroup.org/uploaded_images/docs/Thesis%20Lukas%20Schneider1.pdf

Shahid, M. (2007, June). Measuring portfolio performance. Retrieved 05 20, 2015, from ilehttp://www.diva-portal.org/smash/get/diva2:304183/FULLTEXT01.pdf

Sheridan, C., & Poti, V. (2007, May 23). Hedge Fund Performance and Generalized Sharpe Ratios or WhyRisk Averse Investors Love Hedge Funds. Retrieved May 15, 2015, fromhttp://webpages.dcu.ie/~potiv/Main.pdf

Venkataramani, C. (2003). Random Walk Hypotheses and Profitability of Momentum Based TradingRules. Retrieved May 20, 2015, from http://www.e-m-h.org/Venk03.pdf

Weisinger, A. (2010, May). Risk-Adjusted Performance Measurement - State of the Art. Retrieved May15, 2015, from http://www.unisg.ch/~/~/media/0D77AF25603347E795C77DDA1DD83676.ashx