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MEASURING INVESTMENT PERFORMANCE Index construction — the issues • Index construction — the different methods in practice • Other problems in Index Construction Comparing fund returns and index returns • Money Weighted Return (MWR) • Time Weighted Return (TWR) • Choosing between MWR and TWR Risk-adjusted performance measures • Treynor's measure • Jensen's Measure • Sharpe's measure An illustration of risk-adjusted performance measures Conclusion 1 Learning Objectives • The different ways in which stock market indices may be constructed, and why certain types of index are preferred for performance appraisal purposes • The different ways in which the return on a fund can be calculated, and the underlying logic of time-weighted-return (TWR) and money- weighted-return (MWR) calculations • How to make appropriate comparisons between the return on a fund and a stock market index so as to identify underperformance and superior performance 2 Introduction • There is a great deal of money invested on stock exchanges. In 1998 the capitalisation of the London stock market was about one trillion pounds. The stock markets in the US are about four times larger. • Some of the shares are held directly by private individuals and they do not normally carry out sophisticated calculations to assess how well they have made their investment decisions. Some private investors put their money into funds managed by professional investment firms. These funds are often cleverly marketed, but the private investors are not given complex and detailed measures of the funding performance – and if they were they would probably not understand them. 3 Index construction the issues There are three main decisions to be made when creating an index. • (i) Sampling Which shares should be included in the index? It is not practical to include them all. Should only the largest companies be included? This can produce bias if small firms behave differently, in stock market terms, from large ones. In the UK, the large firms are generally international in the scope of their activities. Smaller firms are much more closely linked to the UK economy. If the firms in the index include both large and small firms, it is possible that the shares of small firms are not traded every day and there may be no up-to-date price to include in the index. The shares of large companies listed on major markets, however, have very frequent transactions – perhaps one every few minutes – so this problem does not arise. 4 Weighting (ii)Weighting • The simplest system is to give equal weight to all the shares in the index. Alternatively, the shares of larger companies can be given more influence on the index by weighting shares in proportion to their market capitalisations. We shall consider in detail the effects of these two methods. • Other systems can be used. Dow Jones originally constructed his index by adding up the prices of the 30 shares in his sample and then dividing by 30. So the index was, quite simply, an average price. The Dow-Jones is calculated in a similar way today. This system gives more importance to companies with heavier share prices. If a share with a price of $100 goes up 10%, it has twice as much effect on the index as if a share priced at $50 goes up 10%. If a company in the index had a stock split – so that exactly the same business was divided into a larger number of shares each with a lower price – the weight of the company in the index would be reduced. It is a thoroughly illogical system. This has not prevented the Dow Jones from being the best known index in the world for more than a century. But it is not used by professional investors. 5 Averaging (iii) Averaging • The arithmetic average of a set of n numbers, v1, v2, ... , Zn is simply: (v1+v2+... +v)/n • But this is not the only way of averaging. Consider the following example. If your investment has fallen by 10% one year and risen by 60% the next, your 'wealth relatives' have been 0.90 and 1.60. • Over the two-year period, your wealth has grown by a factor of 0.90 x 1.60 = 1.44. What has been your annual compound growth rate over the two year period? 6 • We calculate that: x/0.90 x 1.60 = 1.20 and so the average annual compound rate of return has been 20%. 7 Index construction - the different methods in practice • We need some illustrative numbers to show how the different methods work. We can do this with a sample of just 2 shares on three trading days. Table 5.1 shows the prices of shares A and B over this period: Table 5.1 Illustrative share prices for index construction Share Day 1 Day 2 Day 3 A 80 160 80 B 120 60 120 Index 100 ? ? Notice that the share prices on Day 3 are exactly the same as they were on Day 1. We shall see how well our different methods of index calculation reflect this fact. 8 Equal weights; arithmetic average • Between day 1 and 2, share A goes up by 100% and share B goes down by 50%. We take an equally weighted arithmetic average of these numbers: • +100% — 50%/2 = +25% • Between Day 1 and Day 2, therefore, we increase the value of our index by this percentage, from 100 to 125. • Between day 2 and 3, A falls by 50% and B rises by 100%, so the change in the index is: • -50% + 100%/2 = +25% and the index rises by a further 25% to 156.25 since: 125 x 1.25 = 156.25 • Do you notice something strange? Although the prices of both A and B are exactly the same on Day 3 as they were on Day 1, the index has gone up by 56.25% Clearly the index is not accurately representing the movements of the underlying shares. So something is wrong with our calculation method. Let's try again. 9 Equal weights: geometric average • Can geometric averaging solve the problem? From Day 1 to Day 2, the 'price relative' for A is 160/80 = 2.00. • For B it is 60/120 = 0.50. • The 'price relative' for the index is the geometric mean of these num-bers: • √2.00 x 0.50 = √1.00 = 1.00 • And the value of the index on Day 2 is 100 x 1.00 = 100. • From Day 2 to Day 3 the price relative for A is 0.50 and for B it is 2.00. So the same calculation shows that the index on Day 3 remains at 100. This Day 3 value is the number that we would expect from a properly constructed index. So the geometric averaging method has passed its first test. But geometric averaging has another characteristic which is less accept-able. Suppose company B had gone bankrupt on Day 3 and its share price went to zero. Its price relative would be zero, and the index on Day 3 will be the Day 2 index multiplied by √2.00x0.00=√0.00=0.00 So the index on Day 3 goes to zero. 10 Comparing fund returns and index returns • Some investment funds are 'closed end' funds. Investment Trusts in the UK have this characteristic. An investor cannot get his money out of the fund; if he wants cash he has to sell his shares in the Trust to another investor. • But most investment funds have cash inflows or outflows as money is put in or taken out. A company pension fund or an insurance company fund will vary in size. If the sales force is successful in selling insurance policies, the fund will rise. If policyholders decide to terminate their policies, they can take money out. • Inflows and outflows of cash create problems for performance mea-surement. 11 Money Weighted Return (MWR) • In this case we compare the return achieved on the fund with the return that would have been achieved by investing in a way that just matched the stock market index. This would often be called 'investing in the index' although, of course, investors cannot actually buy the index. They have to buy a portfolio of shares carefully constructed to-perform just like the index. • The MWR method is often called the Notional Fund method, because it compares the performance achieved by the fund manager with the performance of a notional fund, with the same cash flows, which is invested in the index. 12 Time Weighted Return (TWR) • A pure measure of the manager's stock- selection skills, independent of how much money he is working with. 13 Choosing between MWR and TWR • If the fund manager has no control over the timing of inflows and outflows from his fund, then it is surely right to give equal weight to each annual investment performance whether the fund was large or small. This means using TWR. An additional advantage is TWR can compare a lot of different fund managers with each other as well as with the index. MWR can only be used to compare one fund and the index. • So TWR is usually the preferred method. But remember that MWR gives the direct answer to the question "What rate of return has this fund made on the money entrusted to it?" 14 Risk-adjusted performance measures • We know that expected return is related to market-related risk. It would not be fair to expect a fund manager with a low risk fund to achieve the same return as one with high risk. So we need to include some risk-adjustment in our performance measures. There are three ways of doing this, and they are appropriate in different circumstances. 15 Treynor's measure • A fund has a characteristic line, just as an individual share does. If we have a well diversified portfolio, then the characteristic line gives us all the information we need to determine the attractiveness of a fund. • Treynor solved the problem in the following way. Investors can change the characteristic line of funds by 'gearing up' or 'gearing down'. Suppose an investor puts half his assets into fund A and half into a risk-free investment. Compared with fund A held on its own, this combination will have lower β and a flatter characteristic line. The characteristic line will pivot. 16 Jensen's Measure • Jensen's measure is the difference between: the return actually achieved by a fund A the return that would have been achieved, for the same level of market- related risk, Q, by investing in a combination of the market portfolio M and the risk-free asset F. 17 Sharpe's measure • The formula is: • SB = RB - RF /σB • It looks very much like Treynor's measure, except that the total (his-toric) standard deviation of fund return, σ B, is substituted for βB which only measures market- related risk. • Visually, Sharpe's measure can be seen in risk-return space. 18 Conclusion • The investment world is a very competitive place; the investment markets are very large; and measuring the performance of funds is, in itself, a major business activity. • We looked first at the construction of appropriate stock market indexes against which performance could be measured. We looked at several alternatives, and concluded that market-value weights and arithmetic averaging formed the only acceptable index construction method. • We have also considered how to measure fund return. When the fund experiences inflows and outflows of money, there are two methods-of measurement – Time-weighted Return (TWR) and Money-weighted Return (MWR). The time-weighted return of a fund (which gives equal importance to the fund's performance in each sub-period whether the fund is large or small) can be directly compared to the return on the index. The money- weighted return can only validly be compared to an adjusted index return – the return on a so-called 'notional fund'. The TWR method is more widely used – but it can give misleading results when the fund manager can control the inflows or outflows of cash from his fund. 19