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

         Learning Objectives
• The different ways in which stock market indices
  may be constructed, and why certain types of
  index are preferred for performance appraisal
• 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
• 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.

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.

•      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
•      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.

(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?
• We calculate that:
x/0.90 x 1.60 = 1.20
and so the average annual compound rate
  of return has been 20%.

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
     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.
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.

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
• √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.

    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.
 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

Time Weighted Return (TWR)
• A pure measure of the manager's stock-
  selection skills, independent of how much
  money he is working with.

    Choosing between MWR and
• 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?"

    Risk-adjusted performance
• 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
           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.

         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.

         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.
•   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.


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