LIST OF TABLES 3
LIST OF FIGURES 4
1. INTRODUCTION 6
2. LITERATURE SURVEY 9
2.1 MUTUAL FUND 9
2.1.1 Definition of Mutual Fund 9
2.1.2 Investment Principals of Mutual Funds 10
2.1.3 Manager’s Role in Fund Management 12
2.1.4 Issues Effecting Fund Performance 12
2.1.5 Advantages Provided by Mutual Funds 16
2.1.6 Certificate of Participation 17
2.1.7 Brief Legal Process of Fund Establishment in Turkey 19
2.1.8 The Assets That Turkish Mutual Funds Can Invest 19
2.1.9 The Classification of Mutual Funds 20
2.1.10 Taxation of Mutual Funds 21
2.2 CAPM (Capital Asset Pricing Model) 22
2.2.1 The Theory Of Capital Asset Prices Under Conditions
Of Risk 22
2.2.2 Properties of CAPM 25
2.3 TREYNOR INDEX 26
2.4 SHARPE INDEX 35
2.4.1 Definition 35
2.4.2 Critique of Treynor Index From Sharpe’s Point of View 41
2.5 JENSEN INDEX 44
2.5.1 Definition 44
2.5.2 The Foundations of Jensen Model 45
2.5.3 The Measurement of Fund Performance 49
2.6 GRAHAM HARVEY METHOD 54
2.6.1 Definition 54
2.6.2 GH1 Index 54
2.6.3 GH2 Index 55
2.6.4 An Illustration of G&H Method 57
3. COMMENTS ON FOUR SINGLE INDEX METHODS 58
4. RESEARCH METHODOLOGY 60
4.1 DATA 60
4.2 METHODOLOGY 64
5. RESEARCH FINDINGS 66
5.1. RISK PREMIUMS 66
5.2 PORTFOLİO RANKINGS 67
5.3 COMPARISON OF SHARPE INDICES 68
6. QUESTIONS AND SUGGESTIONS 73
7. CONCLUSION 75
8. TABLES AND INDICES 78
8.1 PERFORMANCE MEASUREMENT RESULTS 78
8.1.1 A TYPE FUNDS 78
8.1.2 B TYPE FUNDS 84
8.2. MUTUAL FUNDS INDEX RANK RESULTS 92
8.2.1 A TYPE FUNDS 92
8.2.2 B TYPE FUNDS 95
8.3 MEAN/STANDARD DEVIATION/SLOPE VALUES 99
8.3.1 A TYPE FUNDS 99
8.3.2 B TYPE FUNDS 102
8.4 GRAPHS OF STAN. DEV. AND BETA VALUES OF FUNDS 106
8.4.1 A TYPE FUNDS 106
8.4.2 B TYPE FUNDS 107
LIST OF TABLES
Table 1 : Sample of Weekly Cahnge Data of ISE-100 62
Table 2 : Sample of Data Calculation For Weekly Risk Free Rate 64
Table 3 : Results of “Spearman r” correlation analysis 67
Table 4 : Average Sharpe Index (%) 69
Table 5 : Number of A Type Funds That Beat The Market 70
Table 6 : Number of B Type Funds That Beat The Market 70
Table 7 : Z Test Results For The Significance of Mean Differences 71
All other tables are presented under Section 6 “Tables and Indices”
LIST OF FIGURES
Figure 1: Treynor and Mazuy Results 14
Figure 2 : Characteristic Line 29
Figure 3: Effects of management change to fund expected return/risk line 31
Figure 4 : Illustration of Treynor method for a single asset. 34
Figure 5: Average return and variability 38
Figure 6 : Theoretical/Empirical CAPM line 53
Figure 7 : GH1 Index 54
Figure 8 : GH2 Index 56
Figure 9 : Risk Premiums 66
Figure 10 : Sharpe Coefficients 68
T-Bills : Government Treasury Bills
ISE-100 : Istanbul Stock Exchange Index
CAPM : Capital Asset Pricing Model
CMB : Capital Market Board
NYSE : New York Stock Exchange
GH : Graham Harvey
G&H : Graham Harvey
This thesis aims at measuring portfolio performance of type-A and type-B investment funds in Turkey with alternate indices over the period of January 1, 1998 – June 30, 2000, testing whether alternative evaluation criteria give similar results, comparing portfolio performances of Type A and Type B with those of T-Bills, and ISE-100 in order to see the significance of the differences and validity of capital market theory in Turkey over the same period.
Before 1960, investors evaluated portfolio performance almost entirely on the rate of return, first studies related to the performance of a fund was done by Cowles in 1933. But this study and the related studies were no more than comparing returns of a group of fund to a passive portfolio.
Although they knew that risk was a very important variable in determining investment success. The reason for ignoring risk was the lack of knowledge of how to measure and quantify it. After the pathbreaking contributions of Markowitz and development of portfolio theory in early 60s, and CAPM in subsequent years, risk, measured as either by standard deviation or beta, was included in evaluation process. However since there was not a single measure combining both return and risk, two factors were to be considered separately: Researchers grouped portfolios into similar risk classes and compared rates of return of portfolios in the same risk class.
In the mid 1960’s the issue of “How to evaluate the performance of mutual funds?” was widely discussed throughout the finance world and literature. The main reason for bothering with performance was mainly based on the doubts about the differences between the theory and the fact. If the market is perfectly effficient, performance is expected to be random over time, which means we can not rely on past data to predict the future. However the relation of past performance and future expectations is worth to note at this point. In order to draw characteristic line of a fund, portfolio-possibility line or criticize the performance pattern of a fund we look at the past and rely on the ex-post data for further commands. We do this when we consider about our portfolio preferences related to the future. The reason for relying on the past performance to predict the future performance depends on the idea that “a good historical performance pattern is one, which, if continued into the future, would cause investors to prefer the specific asset to others”. After all, if past performance is unrelated to future performance, then performance evaluation is of no help when selecting a fund or manager. On the other hand if superior management exists, then this performance will either be reflected in higher fees or there would be persistance in performance. However the fact of fees as a percent of assets tend to be lower for good performing funds mean that, if superior management exists, it should be rreflected in persistance performance. This suggest that performance of mutual funds shows that there is persistence. This suggests at least some managers have superior information and this can be beneficial to the investors if can be theorized.
From then on several methods have been suggested for evaluation of mutual fund performance. However they are mainly based on three methods generated in the second half of 1960’s. Those three major methods which are used widely by fund managers and investors in order to evaluate the performance of mutual funds are Sharpe, Treynor and Jensen indexes.
Today we may include one more method to the above mentioned methods: That is Graham-Harvey method. However reliability of the results of these four methods is being widely discussed. All four methods do not display similar results while evaluating mutual funds.
The main purpose of this paper is to measure if there is significant difference between the results of these four evaluation methods and to display the correlation between each method.
In order to do it, these four methods are applied to the same initial set of mutual funds weekly returns that are traded in Capital Market Board. The thesis can be divided into four main parts.
a) Presentation of the four methods and related topics.
b) The data set and the research methodology.
c) Evaluation results and ranking of mutual funds.
d) Analysis of the correlation between the ranking results of the four measurement methods.
2. LITERATURE SURVEY
2.1 MUTUAL FUND
2.1.1 Definition of Mutual Fund
The word portfolio refers to all of the assets that an individual or corporation holds. And we can describe funds as a special type of portfolio. However fund is legal body. It is a type of portfolio that is managed by professional managers based on some legal regulations. These points are all explained in detail below. Fund portfolios can be formed of either money market instruments or valuable deposits (minerals such as gold).
Technician, portfolio analyst, security analyst work in cooperation for the best possible portfolio, that promises the greatest expected return for any given degree of risk. The security analyst provides information to predict future performance of the security. Portfolio performance uses that information to predict the future performance of portfolio possibilities and select the efficient ones. And finally it is the individual investor who will decide which one of the efficient portfolios he or she will invest based on his or her choice of risk and expected return.
The processes, which are related each of the groups mentioned above other than final investor aim to find incorrectly priced securities for a possible premium. And the process of finding incorrectly priced securities is related to portfolio analysis theory. However the last step – individual investor’s role related to the issue of mutual fund cannot be clearly defined and explained by the portfolio analysis theory. This is actually more like a subject of the field of Behavioural Finance. “A mutual fund cannot practically determine the preference patterns of its investors directly” (Sharpe 1966: p 120). So what usually done is to go simply backwards. First mutual fund manager sets a management policy. This management policy is basically based on choosing a policy of risk and expected return. And then the manager convinces the investors with the similar risk/return desire to invest in the fund.
“Mutual funds allege that they can provide two types of service to their clients. First, they minimize the amount of the unsystematic risk an investor must face. This is done through efficient diversification in the face of the transaction costs. Second, they may be able to use their professional expertise to earn abnormal returns through successful prediction of future security prices. This second claim is contradictory to the semistrong form of capital market efficiency unless, for some reason, mutual fund managers can consistently obtain information that is not publicly available.” (Fabozzi/Francis,1979: 1245)
2.1.2 Investment Principals of Mutual Funds
Large fund portfolios have the possibility for a better diversification of the risk to the levels that is hard to be achieved by individuals.
Suppose an individual have capital of $5000. As long as his investment is limited by that certain amount he can hardly divide it among couple of assets. As the number of the assets that are invested decrease, chance of achieving a successful diversification also increases. Since less diversified a portfolio more riskier it is, a decrease in the prices of two stocks has more probability to effect the individual’s portfolio that contains them than a large portfolio which has invested in 25 different stocks including those two. This is certainly a question of weights. And individuals face more risk of losing a large and unexpected amount of capital. Mutual funds have the chance of creating large pools of liquid investment portfolios which provide a chance for a better diversification.. They can easily invest in many stocks, as well as government bonds and other investment instruments.
This is also true for a whole decrease in the market, because large portfolios usually invest in stocks of different industries or tools other than stock market. The effects of September 17th decrease of NYSE on an individual investor who has invested half of his portfolio to airline companies and on a diversified portfolio worth to compare this point.
Besides the diversification issue to limit the risk of the market, it is a fact that, financial markets display somewhat a complex technical structure, which requires knowledge and close observation. It is obviously hard for individuals to keep close attention on the markets and respond effectively to rapid changes in the market. Mutual funds have the financial opportunities to hire professionals for this purpose. These professional portfolio managers can study the market continuously and respond to any change immediately.
As an illustration to this issue following example can be simple case. This act of a single investor can also give us an idea about the management of fund. Managing a portfolio invested in fund with different investment tools carries importance when the investor seeks to change the volatility of the portfolio. (We can easily say that a personal portfolio can be considered as acting as a fund). “Furthermore, if the investor desired a risk different from that offered by the fund, he or she would modify the risk by lending and/or borrowing. The relevant definition of performance may change if the problem is examined from the point of view of the fund manager. This leads directly to our second measure of performance.” (Elton/Gruber: p 652)
As an example of modifying the risk by lending and/or borrowing, suppose the investor has a portfolio of 10.000 dollar and he invests all of it to Fund A. Fund A has a volatility of 2%. Suppose the investor is pessimistic and tries to decrease the volatility his portfolio from 2% to 1%. What the investor simply does is investing in a risk free tool with 5000 dollar of his portfolio and leave rest of his fund invested in the fund that has a volatility of 2%. Total risk of his portfolio will be.
(0.02*5000) / 10000 = 0.01
In an opposite case suppose an optimistic investor invests his portfolio to a fund with a volatility of 0.5% wants to increase the volatility of his portfolio from 0.5% to 1%.
(0.005 * X) / 10000 = 0.01
Solving for X we get 20000. The investor should borrow 10000 dollar and invest in the fund.
In both cases investor’s portfolio has the volatility of 1% but invested in tools with different volatilities. Investors should follow different strategies according to the fund they invest just like the fund-manager follows to change the risk of the fund. Also the expected return of the fund. This policy change is discussed more by Sharpe.
2.1.3 Manager’s Role in Fund Management
The manager of the mutual fund is considered as the only legal body for the management of the fund. By definition of the fund, investors agree to deliver all their rights on any investment exercise of their capital through the fund, to the manager. Therefore manager of a fund is expected to behave with respect to the ethical rules. For example “Money collected from individuals for the establishment of fund can not be used by the management for any purpose other than investing for the benefit of the fund.” is part of the rules and regulation for the establishment of a fund.
Management of the fund can be limited in some way facing a certain amount of agency cost by the establishing company. For that purpose mutual fund managers may be required to invest in a certain instrument to a limited extent, in order to keep manager of the fund away from manipulating the portfolio depending on his personal interest and desire.
2.1.4 Issues Effecting Fund Performance
Diversification of a fund portfolio is the biggest issue on the fund’s performance without any doubt. Diversification is usually measured by the percentage of total risk that can depend on market movements. However researches show that by investing in a fund investors bears some nondiversifiable risk. And most mutual funds do not earn sufficient return to justify incurring this extra risk and there is no evidence for the opposite. Since diversification of the portfolio is the direct and major effect of management, the portion of performance related to the diversification is needed to be studied.
On the other hand the portion of a fund risk that depends on the diversification is fairly low. According to the study of Sharpe “88% of the total variance of return of mutual funds was due to the market movements, leaving 12% diversifiable risk. Merrill Lynch, in their analysis of funds using their service, found that funds diversified away all but about 10% of the risk. Elton and Gruber show that for an equally weighted randomly selected portfolio to be 90% diversified, 48 securities are required” (Elton/Gruber; 1991; p 676)
Empirical studies, such as Friend, Blume and Crockett’s suggest that funds are not well managed ,in term of diversification, compared to the market. In their study they create equally weighted empirical portfolios with different risk groups based on beta. Later they compare mutual funds of the same beta group with the empirical portfolios and they show that funds do not beat the market. Besides they also show that since most of the manager move together almost all of the funds come up with similar result and end up not to be able to beat the market.
Besides the limited role of diversification, there are some other issues that affect the performance of a fund portfolio. For example in his study Sharpe compares his data set of 34 funds with Dow Jones portfolio and finds that 19 of the performed better than Dow Jones during the time period of 1954-1963. Statistically he concludes with .99 percent probability that the average mutual fund did as well as the Dow Jones portfolio. One point that has to be criticized about this analysis. Sharpe ignores any transfer cost and other issue which may has some impact on the performance of the fund, compared to again his result that only 12%of the fund performance depends on diversification.
Another aspect is timing. How well the managers react to changes in the market. The first way managers follow is to change the percentage of bonds and stocks in the market. If the manger expects the market to rise he increases the amount of security in portfolio. If he expects the market to decrease he increases the amount of the bond (riskless asset) in his portfolio. Another way managers attempt is they change the beta of the fund. They switch between low beta funds and high beta funds following the market movements. The simplest way to measure the timing performance is to draw a line graph of beta, bond market returns and stock market returns on a time scale. Another way is suggested by Treynor and Mazuy.
As we tell from Figure (1a), the line that passes through the origin will remain the same. However we need to draw a curve shaped line through the points in Figure (1b) where managers manage well at timing and keep the fund above the line.
. . . . . . .
. . . . . . . .
. . … . Rm . . Rm
. . . . . .
Figure 1a Figure 1b
Figure 1: Treynor and Mazuy Results
Source : ELTON E., GRUBER M.; “Modern Portfolio and Investment Analysis”; 666
Treynor and Mazuy concludes from that point if the case is such as the second situation the regression equation will result such;
Another issue is transaction costs. Transaction cost is considered as the main reason for individual’s investment in a fund. Besides in many studies transaction costs are omitted which may be a reason for bias in the results.
There are three types of transaction cost. Although the percentage of them changes from market to market, they exist in almost all of them. First on is the cost that mutual funds incur when they buy and sell securities for the purpose of changing the combination of the portfolio. Second type of transaction cost is the yearly management fee that mutual fund charges. This is an optional type of cost based on the regulation and the policy of the market. However Jensen claims that it is not optional and he divides it in two parts. Analysis and bookkeeping cost and management fee itself. And the third type is an initial fee that is charged from the purchaser as a percentage of the purchase. The fund, which charges the initial fee is called as “load fund” and the fund without the charge of the initial fee is named as “no-load”. Although researches related to this topic display the result of no-load funds are tend to perform better overall than the load funds. By ignoring this type of cost the performance of load funds are overestimated and no-load funds are underestimated.
Ratio of expenses to asset is used as an indicator at this point. Reducing the expenses can also be explained as amount of expenses per unit invested (per TL. , per Euro etc.)
E = E / P (16)
E is Expense
P is total size of the fund portfolio
e is amount spent for per unit of investment in fund
if Ai (net) = Ai (actual) – E then we can conclude that bigger the size of the fund more probable smaller “e” will be. On the other hand one can claim that a larger fund would require a larger amount of investment. And studies tend to show results supporting the idea that “funds with greater expense have superior performance”. The question is once again if additional cost is covered by additional profit.
Fund size is also claimed to affect the performance of a fund. However this issue has two point of view. “Large funds have an advantage in that they have more to spend for information and analysis. On the other hand, large funds may have more impact on the market when they engage in purchase or sales.” (Elton/Gruber: p 678)
Any policy change of the fund management that would lead it to another risk group than stated before will be also source of a performance change. This issue will be detailed further again.
Turnover is another indicator that affects the fund performance. Different studies depict different results. While Friend, Blume and Crockett’s study show that funds with higher turnover outperform the funds with lower turnover, Sharpe’s study show the opposite. Sharpe shows that the transaction costs of the larger turn-over are larger than any additional performance.
2.1.5 Advantages Provided by Mutual Funds
- Investor’s capital is managed by professional and trustworthy individuals
- Since the whole portfolio is diversified between exchange, stocks and bonds, risk is considered to be diversified better than the individuals may achieve.
- Since the payment such as dividend and interest of the assets that fund invests will be received by the fund managers on a programmed base.
- Opportunity for making investments in assets with a high premium potential which are hard for small investors to invest due to their large amount of capital requirements.
- Provide partial liquidity option and reflection of daily returns to whole portfolio.
- Fund saves time and money during transactions since it always exercises with large capitals.
- Some funds give option to write cheque through share of participation .
For an investor with limited capital, very large transaction costs are required to obtain the same degree of diversification. Thus, for small investors, mutual funds, even with slightly poor performance, still provide a reasonable alternative to direct purchase.
All of the explanations related to funds that are given above apply to the funds in Turkish market. However there are certain points that distinguish Turkish Fund Market from other market.
2.1.6 Certificate of Participation
Certificate of participation is the legal note, which is a proof for an individual investor to claim on the rights and benefits of a certain mutual fund.
A certificate of participation of a fund works similar to the share of a corporation. The owner of a certificate of participation claims right on the benefits of the fund like the owner of share of a corporation claims right to receive any issued dividend. Investor can do it in two ways. He can buy or sell the certificate of participation any time and keep any premium he makes out of it or he can keep the certificate of participation and receive any dividend payments of the shares that are issued by the companies whose stocks are included in the fund portfolio.
Certificate of participation does not give any right to the holder of it on the management of the fund like the share of a corporation gives right to its holder on the corporation’s management.
Certificate of participation can be either kept by the investor or by the management of the fund upon investor’s request.
Certificate of participation can be purchased from the authorized banks or institutions announced by the management of the fund if it is being issued for the first time. Once the issuing is terminated they can be traded in the secondary market similar to stocks.
Investors may invest a fund by purchasing certificate of participation. They keep the right to deliver certificate of participation any time to the fund in return of cash. In some cases investor may be required to announce fund management the return of certificate of participation certain time before the exercise. This return policy had to be set and announced by the management committee of the fund during the establishment.
The price of the certificate of participation is calculated by the management of the fund at the end of each day and announced to the market as purchasing price at the beginning of the following day.
· Fund portfolio value is calculated with respect to the market prices of the assets that the portfolio contains.
· Fund’s total value is calculated by adding receivables and subtracting debts (including any transaction and operating cost) of the fund portfolio from total value of fund portfolio
· By dividing fund’s total value to the number of share of participation, price of per certificate of participation is calculated.
Other gains of the assets included in the portfolio such as interest or dividend payments gained from the securities that are included in the portfolio, premiums or losses from change of the market value of all assets held in the portfolio are also reflected to the price of the mutual fund. Individual investors receive from these gains with respect to the proportion of participation in the whole portfolio. Value of the per certificate of participation of the fund is announced on daily base.
2.1.7 Brief Legal Process of Fund Establishment in Turkey
Banks, dealers, security companies, retirement organizations etc. which meet the requirements of Capital Markets Board of Turkey are declared as eligible for fund establishment. Establisher of the fund is responsible of the fund’s charter with respect to the ethical principles that are mentioned above. Responsibility belongs to the establisher although the fund can be managed by a hired professional manager. Establisher has the right to make any change in the field portfolio within the rules set by the fund’s charter.
Any eligible individual who fits the legal requirements of the Capital Markets Board can be chosen as manager of a fund by the establisher of the fund.
2.1.8 The Assets That Turkish Mutual Funds Can Invest
- Stocks and bonds of Turkish corporations, and government bonds.
- Foreign government bonds and corporation stocks and bonds.
- Gold and other valuable metal traded in domestic and international markets and stocks in secondary markets.
- Other financial market tools that are declared as tradable by Capital Markets Board of Turkey.
Mutual Funds are formed by investing the portfolio on one or more investment tools listed above. This is declared in the fund’s charter.
Risk and return of a mutual fund depends on the assets that the fund includes. Therefore investor should decide which to invest after studying each fund in detail based on this desired risk and return.
2.1.9 The Classification of Turkish Mutual Funds
Mutual Funds can be established in two types in Turkish market. A Type and B Type. A Type mutual fund portfolios must form at least 25% of their portfolio by purchasing shares of companies of Turkish origin. However B Type funds do not have such a requirement.
Type of the fund carries importance during taxation process. A Type benefits some taxation advantages that will be explained briefly below.
Other than the classification according to their type, mutual funds are classified in order to the assets that they contain. Purpose of such a classification is to offer the investor various types of investment opportunities. There are 11 different types of mutual funds according to the assets they contain. Bond fund, Stock fund, Sector fund, Ownership Interest fund, Group fund, Foreign Security fund, Gold and Valuable Metal Deposit fund, Variable fund, Liquid fund, Index fund.
Fund types give the investor an idea about taxation of the fund and the structure of the assets that are included in the portfolio of the fund.
If at least 51% of the portfolio is constantly invested in;
- government and private bonds, then the fund is named as “Bond fund”.
- stocks of domestic corporations, then the fund is named as “Stock fund”.
- stocks of corporations in a specific sector, then the fund is called as “Sector fund”.
- the stocks of corporation that are owned by the establisher of the fund, then the fund is named as “Ownership Interest fund”.
- the stocks and bonds of corporations owned by a group or holding (such as Koc, Sabancı etc.), then the fund is named as “Group fund”.
- foreign stocks and bonds, then the fund is named as “Foreign Security fund”.
- gold and other valuable deposits and related money market instruments, then the fund is named as “Gold and Valuable Deposit fund”.
If the whole portfolio is invested in;
-At least two of the instruments out of stock, bond, gold and valuable deposit and related instruments where the proportion of each of the two is not less than 20% then the fund is called as mixed fund.
- Money market instrument such as bonds and T-Bills with a maturity less than 90 days then the fund is called as liquid fund.
If the fund cannot be fit to any of these criteria then it is named as the “Variable fund”.
If at least 80% of the fund portfolio is always invested in all of the stocks or just a group of stocks that are used in the calculation of the market index and the correlation between the fund’s per certificate of participation and stock market index value always be above 90%, then the fund is named as Index fund.
2.1.10 Taxation of Mutual Funds
Any premium from the transaction of the portfolio is subject to a 10 % taxation for B Type funds. However A Type funds are not subject to any kind of premium taxation.
Dividend payments are subject to taxations as explained below.
Since dividend payments of stocks in a certain portfolio are subject to income tax, according to Article 4444, item 55 ; between the dates of 01.01.1999-31.12.2000 no other taxation would be applied.
Dividend payments of the assets in the fund portfolios are subject to 30% taxation for both A Type and B Type as long as they are held by a fund management corporation.
2.2 CAPM (Capital Asset Pricing Model)
The reason for giving brief information about the CAPM theory in this thesis is because of the reason that each four of the performance measurement criteria in someway link their theory to it.
Till 1960, investors evaluated portfolio performance almost entirely on the rate of return, although they knew that risk was a very important variable in determining investment success. The reason for omitting risk was the lack of knowledge how to measure and quantify it. The development of portfolio theory in early 60s, and CAPM in subsequent years made risk possible to be measured as either by standard deviation or beta. However, since there was not a single measure combining both return and risk, two factors were to be considered separately: Researchers grouped portfolios into similar risk classes and compared rates of return of portfolios in the same risk class.
“CAPM is basically involved with the pricing of individual risk in equilibrium. Sharp, Mossin and Lintner have developed the model. It will be shown that in equilibrium the rates of return of each assets are a function of their covariance between the market portfolio. In proof of CAPM the market portfolio must be an efficient portfolio. Therefore, the efficiency of the market portfolio and the capital asset pricing are two indispensible hypothesis in this context.” (Dybvig/Ross, 1985: 385)
2.2.1 The Theory of Capital-Asset Prices Under Conditions of Risk
CAPM is basicly a portfolio analysis theory for all perfectly investors, using all the information they have, that is based on the empirical result of “market responds very rapidly to new information affecting the value of securities”.
The predicted performance of a portfolio can described by two measures.
1. The expected rate of return ()
2. The predicted variability of risk, expressed as the standard deviation of return of the asset ().
The main assumptions are;
1. There is a risk free rate in the market which all investors can invest. This assumption includes that investors can borrow funds at the same rate, and invest in them at least to the desired extent.
2. Since the market is efficient and all information is available to everybody, all investors are assumed to share the same future predictions for the future performance of securities.
We can easily extend this assumotion from a single security to a whole portfolio because portfolio is a combination of different proportions of securities. And if we remember that a fund itself is a portfolio, we can say that the assumptions are also true for funds. Under these assumptions we can conclude that any efficient portfolio will satisfy the equation of;
: Expected return of the security (portfolio etc.)
: Riskfree interest rate.
b : risk premium
σ : standard deviation of the asset.
“Since investors are assumed to be risk-averse, b will be positive.” (Sharpe 1965: 418)
Sharpe also notes that this definition suggests the result that for inefficient
“If an investor can borrow or lend at some riskless interest rate p and/or invest in a portfolio with predicted performance (), then by allocating his funds between the portfolio and borrowing or lending he can attain any point on the line
Any portfolio will thus will give rise to a complete (linear) boundary of E,s combinations. The best portfolio will be the one giving the best boundary; clearly it is the one which is the greatest. If more than one portfolio is to be efficient, all must lie along a common line and give identical values of this ratio.” (Sharpe, 1966 : p-122)
value of the market is known. and are predicted by the individual investor. So we can conclude that is a constant, and say that E and σ has a linear relationship and will lie on a linear boundry. One of the main purposes of the capital-market model is to predict future performance. But since future results can not be known exactly from today, the capital model can not be tested with a 100% certainty. Instead of that ex-post values are used-the average rate of return of a portfolio must be substituted for its expected rate of return and the actual standard deviation of its rate of return for its predicted risk. Expected rate of return can be denoted and the risk can be denoted as . and have a positive linear relationship. That means they should lie on a line with a positive slope. But due to the unpredictible risk in the stock market, points will not lie precisely along a line. They will be a little above or below the line. However the results are significant enough for a relationship to be present, visible and statistically significant.
The implications of this model for mutual fund performance are relatively straightforward. “If all funds hold properly diversified portfolios and spend the appropriate amount for analysis and administration, they should provide rates of return giving , values lying generally along a straight line . Points that diverge from the underlying relationship should reflect only transitory effects and not persistent differences in performance. On the other hand, if some funds fail to diversify properly, or spend too much on research and/or administration, they will persistently give rates of return yielding inferior, values. Their performance will be poorer and can be expected to remain so.”(Sharpe, 1966: 122)
2.2.2 Properties of CAPM
- Every asset must be priced so that its risk adjusted required rate of return falls exactly on the SML
- Individual can diversify away all the risk except the risk of overall economy which is unavoidable.
- Total risk can be partioned into systematic and unsystematic risk.
Total risk = systematic + unsystematic risk
- Measure of a risk is for an individual assets is linearly additive when the assets are combined into portfolios.
- The benchmark for the market beta is 1 for the purpose of portfolio selection. In other words, if beta is greater that 1 then the portfolio will be more aggressive then average.
(Weston/Brigham, 1993, 154)
2.3 TREYNOR INDEX
Treynor (1965) was the first researcher developing a composite measure of portfolio performance. He measures portfolio risk with beta, and calculates portfolio’s market risk premium relative to its beta:
T = ( Rp - Rf ) / βp (6)
Ti = Treynor’s performance index
Rp = Portfolio’s actual return during a specified time period
Rf = Risk-free rate of return during the same period
βp = beta of the portfolio
We can interpret Treynor index as folows. If both Rp> Rf and βp > 0 than the portfolio wit a bigger Treynor index is consired to perform better for all ivestors regardless of their individual risk preferences. There are two possibilities for Treynor index to be negative. Either Rp < Rf or βp < 0. If the negativity comes from the numerator, then we can say that portfolio has a return less than risk-free asset, which we can consider the fund performance very poor. If the negativity comes from the denominator than we consider that, despite negative beta fund performs better than market. This is called as a superb fund performance. If both the numerator Rp- Rf and the denominator βp are negative, which result as a positive Treynor index, in order to qualify the fund’s performance one should see if Rp is above the security market line (if fund performed better than ).
This approach is assuming that the market may not fit the CAPM theory which is the case in Turkish market. Normally rreturn of a portfolio acan not be above the return of market if it is already below risk free rate of returrn.
“Consider portfolios in expected return-Beta space. (Beta on the x axis and expected return on the y axis) It is easy to show that all combinations of a riskless asset and a risky portfolio lie on a straight line connecting them. Furthermore, the slope of the line connecting the risky asset A and the risk free rate is .” (Elton/Gruber, 1991: p 657). There, we can say that the portfolio with the most counterclockwise (highest slope) will bring the most return for a certain level of beta.
Treynor bases his analysis on basically two types of the risks. Market risk itself and the risk that is based on the portfolio manager decisions. Investment manager has a variety of tools to invest and variety of policies to follow and the risk that depends on the decisions of the manager is free from movements in the market so free from risk related to market movements. This risk can be measured separately from the market risk and makes performance comparisons possible.
The risk that is created by the decisions of the investment manager has two kinds within itself. ‘There is a risk produced by general market fluctuations-the volatility of the stock market. There is also a risk resulting from fluctuations in the particular securities held by the fund.’ (Treynor, 1965, 63)
The first kind of the risk affects the performance during big market fluctuations. More volatile the fund that investment manager generates, more sensitive it is to the changes in the market. The second kind of risk derives from the diversification policy of the investment-manager. Not every investment-manager follows the same diversification policy. The need for diversification differs from the each investor’s point of view.
While measuring the performance of a fund under the type of risks explained above Treynor points out two important points that the measurement criteria that would be generated has to consider about “It should tend to remain constant so long as management performance is constant – even in the face of severe market fluctuations. Also, it should take into account the aversion of individual shareholders or beneficiaries to investment risk’ (Treynor 1965; 64)
In order to relate performance measure of a fund to the rate of return of a suitable market average a device as a reference point is generated by Treynor which is named as the characteristic line by him.
Treynor forms the characteristic line specific for each fund. He uses market rate of return and fund rate of return for the same time period. On the horizontal line lies the market rates of return and on the vertical axis lies the fund rates of return. Each fund displays a characteristic line with a different slope. The slope of the characteristic line obviously represents the reaction of the asset to the market. It represents the change in the return of the fund for per change in the market. The rate of the change (slope of the line) gives us idea about the volatility of the fund. Although Treynor observes that the funds exhibit wide swing in the rate of return over the time interval of observation, they more or less fall a little below or above a specific line.
Since the slope of the characteristic line which also represents the volatility of the fund creates a basis for a comparison. “A volatility of two means that a 1% increase (or decrease) in the rate of return demonstrated by the Dow-Jones Average is accompanied, on the average, by a 2% increase (or decrease) in the rate of return demonstrated by the particular fund in question” (Treynor 1965 p66)
“The slope of the characteristic line obviously provides a more refined measure of a fund’s volatility than the usual categories of “balanced fund”, “stock fund” or “growth fund”. (Treynor 1965, p66)
For a comparison; numerical values for the rate of each fund is needed to be obtained. Figure 2 shows us how this numerical value is achieved. A random selected risk free rate of return line is drawn on the horizontal axis intersects characteristic line of the fund at point T. As we can see on the figure the slope of the characteristic line is “B”. For a particular market rate of return, which is the value of “D” in our example, expected fund rate of return is “μ”. The distance between the fixed income securities line and horizontal axis (market rate of return line) α is shown for the calculation of the value of r.
Using the geometry, we have for volatility (β)
Solving for r we obtain
From this equation we can conclude that, for any given level of market rate of D, r is uniquely related to ranking fraction. Substituting for μ Treynor comes up with the result;
However this value of r is negative of the Treynor Index I used in the thesis. Treynor Index is accepted as the –1*r after it was criticized by Sharpe. Further explanation is given under the title of Sharpe Index.
Figure 2 : Characteristic Line
Source : TREYNOR J. L., “How To Rate Management of Investment Funds”
Grouping the funds and comparing their volatility within the group even presents more reliable results than comparing funds from different groups of fund types. For any risk-averse investor, the result of Treynor index is surely worth measuring the effects of management policy on the performance of the fund.
If the characteristic line of a fund is studied further, it will be observed that not all data points lay exactly on the characteristic line but rather a little below or above the line. This suggests the conclusion that all of the risk of the fund in question can be explained by the fluctuation in the general market level. This conclusion agrees with the previous explanations of Treynor. Investment risk of a diversified fund has two parts. First part of the risk is the general market fluctuations. General market fluctuations covers most of the risk of the fund and gives most of its shape to the path that characteristic line follows. Second group of the risk comes from fluctuations peculiar to the particular investment tools held by the fund. It is the second part of the risk that depends on the performance of the fund manager. We can simply say that it’s the management policy the fund manager follows which gives its final level to the fund’s performance and volatility of the characteristic line. “If the management of a fund attempts to maintain a constant degree of volatility, then the slope of the characteristic line will tend to measure the volatility” (Treynor 1965, p 66)
Any observation of excessive deviation from the characteristic line leads us to two conclusions:
- The fund is not efficiently diversified.
- The fund manager speculates on fluctuations in the general market by changing the volatility of the fund. He increases the volatility on purpose when he expects a positive return from an optimistic point of view. He decreases it purposely when he expects a high decline on the market of the investment tools which the fund portfolio holds.
It is simply a problem of timing for the manager. Manager achieves this goal simply switching between low and high beta funds according to his expectations related to the movement of the fund. This detailed before from the Treynor’s perspective.
As an example; If I manage Fund A which is simply a copy of the general market portfolio, what I simply do is just sticking with the market. This is absolutely one of the possible policies. However remembering the main reason of establishing a fund is beating the market, it is waste of capital for the effort just to copy the market. Starting from the moment of diversifying my portfolio, I face the second part of the risk. I am no longer unaware of the possible future changes. If I expect the government policy succeed in decreasing the interest rates by the end of 2001, I may include riskier stock in my portfolio believing the stock market will rise by the decrease of investment rates or vice versa.
Another possibility is a shift of characteristic line without any change in the volatility. This is usually possible by a managerial change. Let there be two imaginary funds. Fund A and Fund B in Figure (3a) We observe that although both of the funds’ characteristic line have the same slope (they both have the same volatility) Fund A’s characteristic line is above Fund B’s. This tells us that for a certain risk level Fund A has more returns than Fund B. Fund A performs all the time better that Fund B. We can reach the conclusion that Fund A management’s performance is better than Fund B.
Figure 3(a) Figure 3(b)
Figure 3: Effects of management change to fund expected return/risk line
In order to interpret the performance of the fund management and to understand if the performance change is not temporary, Treynor suggests drawing a check-line above and below the characteristic line of the fund and observe if the plots constantly fall outside the lines drawn.
“In summary, therefore, the graphical method provides a simple test of:
- The extent to which a fund has adhered, purposely or not, to a single characteristic line.
- The degree of volatility associated with the fund.
- The success of fund management in maintaining a high rate of return under a variety of market conditions.” (Treynor 1965, p67)
Of course we do not always compare funds with the same volatility as in Figure (3b) and comparing fund performances is not so simple when volatility differs. We compare two funds with the same volatility (reaction against risk) according to their places on the diagram. In order to compare two funds with different volatilities (means different slopes) using Treynor method we need to define the tool of indifference curve. Indifference curves are the curves that the investor is indifferent to risk/return combinations (portfolios) laying on a particular indifference curve. The horizontal axis of the diagram of indifference curves represents risk and the vertical axis represents expected returns.
We generally have two types of portfolios that we will draw the characteristic line of, passing through the indifference curves, on Treynor’s approach
“1. Money fixed claims, such as checking deposits; government, municipal and corporate bonds.
2. Equity assets, including equity in personal business and partnership and corporate common stocks” (Treynor 65, p 68)”
The first type of portfolio is subject to two different risks. They are interest rate and price level. Returning to Figure (3b) one can see that the indifference curve “IA” which Fund A is tangent is located higher than the indifference curve “IB” which Fund B is tangent to. The point where Fund A risk/return line is tangent to IA curve is equilibrium point for Fund A risk/return fraction. We can say the same for the point where IB is tangent to Fund B line. We can conclude than Fund A displays a better performance than Fund B and has more return for the same amount of risk “The steepness of the portfolio-possibility line associated with a given fund is thus a direct measure of the desirability of the fund to the risk-averse investor” (Treynor 65, p 69).
As a final step of studying the theory of the rating of the fund management, Treynor draws a diagram of risk and expected return. He places characteristic line of 20 funds (as his data group) on the chart. Slope of each fund’s characteristic line gives us the risk level of that certain fund for any certain level of market risk. Treynor takes fund risk levels for 10% and 30% market risk and places them on separate two dimension coordinate systems. Expected rate of return changes for each risk level, however their rank of performance remains the same. “Although the absolute position of funds on a risk-return chart may vary with the level of market return assumes, the ranking of funds with respect to each other does not” (Treynor 1965, p 70)
Treynor’s method carries importance for some points.(See Figure 4)
- Any market rate of return there is always a level of risk. And a level of return for that risk.
- Slope of the characteristic line can be a criteria as a ranking index for funds.
- Performance ranks of the funds are independent from market fluctuations.
0 * * *
-10 0 10 20 30 40 50%
Figure 4 :Illustration of Treynor method for a single asset.
Source : TREYNOR J. L., “How To Rate Management of Investment Funds”
2.4 SHARPE INDEX
Sharpe (1966) developed a composite index which is very similar to the Treynor measure, the only difference being the use of standard deviation, instead of beta, to measure the portfolio risk:
Si = ( Rp - Rf ) / σp (10)
Si = Sharpe performance index
σp = Portfolio standard deviation
This formula suggests that Sharpe prefers to compare portfolios to the capital market line rather than the security market line (SML). Sharpe index, therefore, evaluates funds performance based on both rate of return and diversification. For a completely diversified portfolio Treynor and Sharpe indices would give identical rankings.
Sharpe’s method is based on the empirical test of Treynor’s Index predictability and combining it with the studies that are done after Treynor’s study.
Sharpe considers the theory of portfolio analysis on two different bases. One is the analysis of an individual security. Second is the analysis of the portfolio of a group of securities. In either way, it is the analyst’s emphasis on both expected return and risk, because in the very end it’s up to the investor’s decision of which security would be included in the portfolio.
Two common ways to set the relation between risk and expected return of the fund are:
- The fund attempts to make a description of entire pattern of indifference curves (the structure of indifference curves is explained under title of Treynor Index) where on each the investment will get different amount of returns for a different level of risk but still be on the same indifference curve.
- The second way and maybe more common one is; fund manager makes a description of the general degree of risk planned for the fund’s portfolio. The next step will then be just selecting the most efficient portfolio for the desired amount of risk. And that portfolio will be the one with the greatest expected return.
Unfortunately portfolio analysis theory cannot explain the pattern of security prices or the skill of investment managers. One cannot compare the performance of different mutual funds using portfolio analysis. However it is a fact that funds perform from one another. Possible reasons of the performance differences that are suggested by Sharpe are;
- Different funds could exhibit different degrees of variability in return, due either to conscious selection of different degrees of risk or due to mistakes made during the predictions of the risk
- Because of choosing incorrectly priced securities or mistakes of effective diversification, funds with similar variability may end up with completely different average returns.
As we can see most of the funds contain stock exchange within their portfolio. Therefore the main question for the manager of the fund is whether they are successful to find undervalued stocks.
If one can analyze the stocks in order to find undervalued ones is a further point to discuss. According to random walk theorem we cannot because past behavior of security’s price carries no value in predicting its future price. If so, then any cost for fund management will be a loss for its investors.
Under these conditions we can say security analysis is directed more toward evaluating the interrelationships among securities. Portfolio analyses is concerned primarily with diversification and the selection of a portfolio of the risk desired. If the market is perfect which means any properly diversified portfolio will be efficient then its manager’s primary job will be choosing among properly diversified portfolios, the one with the appropriate degree of risk.
Risk classes can be a reason for differences in variability of return. Besides as Sharpe suggests funds usually are diversified well enough (This suggestion maybe needs further discussion). That leaves only one thing as the one affecting the performance of the fund portfolio from Sharpe’s point of view, “continued expenditure of large amounts of a fund’s assets on the relatively fruitless search for incorrectly valued securities” (Sharpe 1966, 121).
The theory of capital asset pricing method deserves more attention for further proofs and explanations of Sharpe. So we briefly will give only the major features here. The predicted performance of a portfolio has two measures. One is expected rate of return (Ei), second is the predicted risk which can also be expressed as the standard deviation of return (σi). Risk free interest rate (p) and future expectations for all efficient portfolios.
Ei = p + b σi (11)
b represents the risk premium.
Any point on the line of efficient portfolios will satisfy the equation.
If we assume that any investor can invest at a predicted performance (Ei, σi)
The best portfolio will be the one with the greatest value of (Ei-p)/ σi. And for all efficient portfolios with a risk of Vi (as Sharpe uses) and expected return of Ai, we can say they lie on a straight line which satisfies
At>Az ═> Vt>Vz
It is assumed here that all funds hold properly diversified portfolios. Enough time and administrative effort is put through them and any divergence from this underlying relationship reflects the transitory effects but not any difference in the performance of the portfolio. Not being well diversified or spending too much resource on administration can also be considered as another reason for poor performance.
Sharpe’s data set also confirms the relationship between risk and return. Funds with more risk display more return and the relation is approximately linear and significant. Approximately linear means some funds have greater value of expected return (Ai) for smaller value of risk (Vi)
Sharpe derives the following equation for ex-post measures (A and V) using ex-ante measures that are previously plugged in ex-ante measures (E and σ)
Z T M
Figure 5: Average return and variability
Source : SHARPE WILLIAM, “Mutual Fund Performance”,Journal of Business, January 1966, p 124
If we say that in Figure 5 point Z is the point where there is no risks (like the government bonds theoretically) we can say a fund at point V brings |ZT| much return for |VT| much risk and a fund at the point of Q brings |ZM| much return for |QM| much risk. We can easily say Fund V performs better then Fund Q just by taking the cotangent of the angle each line makes with the horizontal Average return axis.
|VT| / |ZT| > |ZM| / |QM|
|ZT| shows the difference between the average annual return and riskfree (pure) interest rate. We can consider this as a reward for bearing |VT| much risk. “The denominator measures the standard deviation of the annual rate of return; it shows the amount of risk actually borne. The ratio is thus the reward per unit of variability” (Sharpe ’66, p 123) This is the main point where Sharpe differs from Treynor. Risk is measured by standard deviation. All further study of Sharpe depends on this approach.
The argument at this point is between those who hypothesize the market is perfect, managers diversify perfectly and claim that any difference is either transitory or due to excessive expenditures and others who hypothesize that differences are related to management skills.
Sharpe discusses this argument and tries to find a reliable measurement technique for the second hypothesis if it is proved to be true. The problem Sharpe faces is setting a reliable riskfree (pure) interest rate. This is a problem, which I also faced for the thesis. However, since Sharpe’s study is not a comparison but rather a single calculation of E.
And any rf value would not change the rank he chooses yield on a 10 years bond. For thesis this point turned out to be an unfortunate problem because of high default risk of Turkish Government Bonds and T-Bills. This issue will be explained later.
Sharpe shows that any fund that is ranked in the high or low performance group has a tendency to remain in the same group. He uses Spearman’s rank correlation test coefficients to prove it. Both for the actual values and fund ranks he comes up with coefficients that are greater than +0.3. “These results show that differences in performance can be predicted, although imperfectly. However, they do not indicate the sources of the differences. Equally important, there is no assurance that past performance is the best predictor of future performance.” (Sharpe ’66, 127). This last part is also the base for his argument about Treynor index, which will be discussed about later.
Sharpe’s results for the term he studies suggest just the possibility of a correlation between size and return or expense and return. Funds with bigger size or less expense tend to display better performance. However the results are not statistically meaningful for deriving a theorem.
According to the calculations we see that Sharpe concentrates on relationship between the reward and risk of the fund rather than the risk of individual stock exchanges. That’s all right as long as the investor can follow a policy to set the desired risk for a certain fund. Because holders (investors) of the fund will have difficulty to arrange their over-all risk and return policy if manager do not stick with a decided risk in order to obtain a consistency over time. The best way for the investor in this situation is maybe having some idea about the variability of the fund, so that investor may forecast the results of possible policy changes of fund management.
Sharpe tests the question using his data set and comes up with the result that there are times when risk of a certain fund makes high switches. But in the end “One might reasonably argue that the data show that mutual fund managers fulfill remarkably well the obligation to stay within their selected risk classes” (Sharpe ’66, p 135)
Overall Sharpe shows that performance of a fund can be theoretically measured by average return and risk. We can expect a fund to have greater average return when it’s riskier. Differences can be explained by expense ratios as long as the market is assumed to be highly efficient “Good managers concentrate on evaluating risk and providing diversification, spending little effort (and money) on the search for incorrectly priced securities (Sharpe ’66, p 138)
2.4.2 Critique of Treynor Index From Sharpe’s Point of View
Sharpe’s first critique about Treynor method is about the sample set and how this sample set manipulates the results. This is a fact that all true diversified portfolios will move with the over-all market. Also the sample group of Treynor does. “During the period 1954-63, almost 90 percent of the variance of the return on the typical fund in our sample was due to its co-movement with the return on the thirty securities used to compute the Dow-Jones Industrial Average moreover, the percentage was quite similar for most of the thirty-four funds. Treynor has taken advantage of this relationship by using the volatility of a fund as a measure of its risk instead of the total variability used the risk/return ratio” (Sharpe ’66, p 127)
For a well diversified portfolio, any change in the market will be reflected to its value. Fund’s rate of return can stand as a good measure of the total variability of the fund’s return over time. Sharpe agrees with Treynor at this point that, for a well diversified portfolio observing ex post data, we can come up with an estimate of volatility (the change in the rate of return of the fund for 1% change in the market). Sharpe names this estimate as Bi of ith fund and rewrites the Treynor Index as the following;
TI = (Ai – p) / Bi (15)
Where Ai is actual fund return and p is risk free rate. “And the extent of the contribution of volatility to over-all variability makes the ranking of the funds on the basis of the Treynor Index very close to that based on the R/V ratio” (Sharpe’66, p 127)
As long as they both study well diversified portfolios Treynor and Sharpe come up with similar results. However Sharpe points out that Treynor Index will result different than Sharpe “Since Treynor index cannot capture the portion of variability that is due to lack of diversification, it is an inferior measure of past performance. But for the same reason it may be superior measure for predicting future performance” (Sharpe ’66, p128)
We also can say that for well diversified portfolio any major discrepancies in the variability of it’s return and the portion related to changes in the market are likely to be due to transitory effect. But by focusing on the systematic part of the fund’s variability which is its volatility, one can avoid transitory effect and can come up with a more permanent relationship. At this point we can say that (as Sharpe also agrees) better the portfolio is diversified, better it is to use Treynor Index rather than Sharpe Index.
The Treynor Index (TI) as used by Sharpe here is the negative of the Treynor’s result.
Slope angle = (-1) * TI
The way Treynor rewrites it is more beneficial to use because the equation of (Ai – p) / Bi
also requires Treynor index to be rewritten as Sharpe does it for a comfortable comparison.
Before clarifying his approach, Sharpe discusses possible reasons for performance differences between funds. For the use of Treynor Index, past performance can be a tool for predicting future performance. Assuming the market is efficient Sharpe points out the fact high correlation among mutual funds mean one thing. Funds are diversified well. So the differences in the performance can be a result of the following reasons :
- Differences in ability to find incorrectly priced securities.
- Differences in expense ratio.
Any manager who ;
- Shows better skills finding incorrectly priced securities.
- Spend least.
- Gain enough to offset the expenses.