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1

1 Comparison between different methods of calculating return

1.1 The money-weighted rate of return (MWRR)
1.2 Time-weighted rate of return (TWRR)
. 1.4  Linked internal rate of return (LIRR) 
The LIRR for a fund over a given year is found using the following process:

2

The money-weighted rate of return (MWRR)

The money-weighted rate of return (MWRR) is useful as an absolute measure of the achieved return. It can be compared with the actuarial assumptions underlying the fund to see whether the achieved return is higher or lower than that expected.

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The standard formula for calculating the MWRR is:

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4

Cashflow v investment income

Cashflow in the above formula represents the excess of contributions received over claims and expenses paid. It is the money moving into (or out of) the fund.

5

Tax and expenses

For rates of return net of tax and expenses, VT and all disinvestments should be net of tax and expenses.

6

Basic problem with the MWRR

The money-weighted rate of return is not a good basis for comparing two different fund managers. The main reason for this is that the rate of return can be heavily influenced by the timing and size of cashflows. The payments into and out of a particular fund are not usually within the control of the investment manager, so rates of return influenced by cashflows are not very useful when comparing investment managers.

7

Fund Managers A and B are both given 10 million to invest. Both managers invest the whole 10 million in equities. Over the next six months the equity market is very sluggish, and the funds for both managers are still standing at 10 million. (There are no contributions to and no payments from either fund during the first six months. )
Then Manager A is asked to return 5 million to the trustees of the fund who need to make a large payment out. Meanwhile Manager B is given another 5 million to invest because the trustees of Fund B happen to have received a large payment into the fund. In the next six months, there is a strong bull market in equities and both Managers achieve a 50% return on their equity investments over the six-month period.
Calculate the annual money-weighted rate of return for each manager over the year described above. Comment on the results.

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8

Will money-weighted returns always give figures that are unsuitable for comparing different fund managers? Under what circumstances is the money-weighted rate of return least appropriate for comparison purposes?

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9

Time-weighted rate of return (TWRR)

The time-weighted rate of return overcomes the basic problem associated with money-weighted rates of return. Theoretically, it is usable as a basis for comparing different investment managers because the timing and size of cashflows will not distort the rates calculated.

It is calculated by
1. assessing the fund value at each time there is a cashflow in or out 

2. calculating the return achieved for each period between cashflows 

3. linking these returns together to give the time-weighted rate of return. 

Note that the amount of money invested at each time will not affect the result.

10

Give a formula for the TWRR.

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11

Comparison of time-weighted return and money-weighted returns 


The time-weighted rate of return and the money-weighted rate of return will be very similar when either the:
  cashflows during the valuation period are small relative to the funds involved or 

  the rate of return is stable over the period. 
When neither of these conditions hold, the two rates of return can be very different. 


12

Impracticality of time-weighted rate of return

The problem with using the time-weighted rate of return in practice is the amount of data that is required: fund values are needed for every occasion on which there is a cashflow. This is often impracticable in practice. 
A practical compromise solution is to use the linked internal rate of return as an approximation for the time-weighted rate of return; this is often done using quarterly sub-intervals.

13

 Linked internal rate of return (LIRR)

The LIRR for a fund over a given year is found using the following process:
1. Determine the value of the fund at various dates throughout the year (eg at monthly or quarterly intervals). 

2. For each inter-valuation period, calculate the money-weighted rate of return. 

3. Link the inter-valuation MWRRs together to get the linked internal rate of return for the year.

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Under what conditions will the linked internal rate of return provide a very good approximation for the time-weighted rate of return?

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15

For each of the money-weighted rate of return, time-weighted rate of return and linked internal rate of return give one sentence summarising the main use and one sentence summarising the main weakness.

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16

How to do the assessment Comparison of portfolio performance with an index two basic ways in which to compare the performance of a portfolio with an index:

There are two basic ways in which to compare the performance of a portfolio with an index:
1. By comparing the actual value of the portfolio at the end of a defined period with the value that would have been achieved had the initial value of the portfolio and subsequent net new money been invested in the same way as the index. 

2. By comparing the time-weighted return from each (or the linked internal rate of return as an approximation to the time-weighted rate of return). 

In practice the method chosen and the type of return calculated will depend partly on the data available.

17

An investor has total assets at the start of a year of $47 million. At the end of the year the fund has assets of $57 million. Over the same year, the index rose from 2,000 to 2,090.
What can you say about the performance of the portfolio? What further information would you need to assess the performance of the investor’s portfolio over the year compared with the index?

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18

Calculate the internal rate of return for the portfolio and the notional index tracking fund.

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Although we have carried out a reasonable assessment of the portfolio against the index based on an internal rate of return, the calculation is not ideal because the calculations are not robust enough to be used for other portfolios over the period. This is because the internal rate of return for the index will vary when we look at portfolios that have different cashflows. The solution is to use the time-weighted rate of return, or the linked internal rate of return as an approximation.

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20

Example assessment

To the initial data in the question above, we will add the missing data and compare the fund against the index.
  There was just one cashflow in the year: an amount of $4 million paid in on 1 July. There was also investment income of $0.6 million (net of tax) at the end of each quarter. (The $57 million is quoted after all of the cashflows. ) 

  The investor pays tax at 20% on all income, but no capital gains tax. At the end of the year, there is no outstanding tax due. 
We now know that the increase in value of $10 million was made up of:
  $4 million in new money 

  $2.4 million in investment income (net of tax) 

  $3.6 million capital gains (as the balance to make the total $10 million). 
It is often useful to do this summary very quickly as it gives you a feel for what is going on. For example, we know from this summary that the net rate of return achieved is approximately 12% – ie $51 million growing into $57 million in one year. In fact, the return is better than this because the $4 million new money was invested for 6 months only. Knowing this will help us understand whether our answer looks right. This is important because examiners are very unimpressed by candidates who fail to spot silly answers. 


21

Assessment based on internal rate of return

If we wanted to measure the out-performance in relative terms (eg with a % figure), then we could also use the time-weighted rate of return. However, in this example, we do not have the market value of the portfolio on 1 July, so we cannot calculate the time- weighted return.
In this example, we might use the internal rate of return to compare the portfolio against the index.
This is because the problem of different cashflows is less acute than it is when comparing different funds because we have already notionally mirrored the actual cashflows within the calculation of the notional index tracking fund. In other words, the timing and amounts of the cashflows are the same for both funds.
However, there could still be some distortion. For example, if the relative performance of the actual fund is greatest when the fund is large, then the internal rate of return will magnify the extent of the out-performance. (Even here, it may be debatable whether this “distortion” is really undesirable. If you were responsible for a fund, you might be very keen for out-performance to occur when the fund was at its largest!)

22

Assessment based on amounts 


Let’s now look at the $47 million invested in the same way as the index, together with the $4 million new money. We need some more information on the level of the index throughout the year and also on the amount of investment income that it would have produced. This data is often given in the form of the capital index value and the dividend yield at the time: 

Although we have carried out a reasonable assessment of the portfolio against the index based on an internal rate of return, the calculation is not ideal because the calculations are not robust enough to be used for other portfolios over the period. This is because the internal rate of return for the index will vary when we look at portfolios that have different cashflows. The solution is to use the time-weighted rate of return, or the linked internal rate of return as an approximation

23

Comparison of portfolio performance with a benchmark portfolio

Assessing the performance of a portfolio against a predetermined benchmark portfolio (often called a “notional fund”) is similar to assessing against a published index. The only difference is that rather than develop a notional fund based on a particular market index, the notional fund is defined in some other, predetermined manner (eg a mix of more than one index, such as 50% in a fixed interest index and 50% in an equity index).
This is best illustrated with a worked example.

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Worked example

The trustees of a pension scheme decide (after advice from the scheme actuary on the nature of the liabilities) that the assets should be invested 60% in domestic equities, 15% in overseas equities and 25% in fixed interest securities.
The trustees appoint a fund manager who is given some freedom to move away from the benchmark within some parameters (eg within a maximum tracking error).
Although the fund manager has some freedom, the regular performance assessments will be based on the 60/15/25 benchmark portfolio. For each sector, the most appropriate index will be used.
At the start of the year, the pension scheme’s assets will be notionally invested in the indices in the 60/15/25 proportions and subsequent net cashflows will be invested in the same proportions. The final total of the notional fund can then be compared with the final total of the actual fund.
The fund manager will hope that the actual portfolio exceeds the notional fund.

25

Worked example

The trustees of a pension scheme decide (after advice from the scheme actuary on the nature of the liabilities) that the assets should be invested 60% in domestic equities, 15% in overseas equities and 25% in fixed interest securities.
The trustees appoint a fund manager who is given some freedom to move away from the benchmark within some parameters (eg within a maximum tracking error).
Although the fund manager has some freedom, the regular performance assessments will be based on the 60/15/25 benchmark portfolio. For each sector, the most appropriate index will be used.
At the start of the year, the pension scheme’s assets will be notionally invested in the indices in the 60/15/25 proportions and subsequent net cashflows will be invested in the same proportions. The final total of the notional fund can then be compared with the final total of the actual fund.
The fund manager will hope that the actual portfolio exceeds the notional fund.

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Complications

are the four standard issues that we have mentioned throughout this chapter. They are all relevant here.
Cashflows, investment income, tax and expenses inevitably, various issues which may make the assessment more complex.

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For the comparison of the actual fund with the notional fund to be valid, we must make appropriate allowance within the notional fund for:

  contributions in and repayments out, as actually experienced by the portfolio 

  investment income and capital gains, as would have been experienced in the 
notional fund 

  taxes on income and capital gains, as would have been experienced in the notional fund 

  expenses, as would have been experienced in the notional fund. 


28

Maintaining the notional 60/15/25 split 


Maintaining the notional 60/15/25 split 
The initial split of the notional fund is simple: just follow the split as defined by the trustees. However, it is not so straightforward once the market values of the different sectors start to move in different directions. For example, a fund that starts at 60/15/25 might soon move to 65/13/22 if domestic equities increase rapidly in value. Should the notional fund be rebalanced or left in the new proportions? 
In specifying the benchmark portfolio it is necessary to set out how new money and investment income are to be invested and how often the benchmark is to be rebalanced. Care will need to be taken that the calculations allow correctly for these factors. 
A pragmatic basis to follow is as follows: 

  the notional fund starts with the 60/15/25 split 

  all investment income from a particular sector is reinvested within that sector 

  all contributions in and payments out are split in the 60/15/25 split 

  no attempt is made to rebalance the notional fund during the period of assessment (eg one year). 
This basis is effectively the same as the trustees saying to the fund manager: “Divide the fund into three distinct sub-funds. Keep the three funds separate during the year. Split all cashflows by the original proportions of 60/15/25. ”

29

Performance attribution

Fundamentally, there are two ways that a fund manager can out-perform a benchmark portfolio:
1. by choosing the right investment sectors (eg equities, property, fixed interest) 

2. having chosen the sector, choosing the right stocks (eg IBM, Microsoft). 

Some investment managers may be very good at sector selection, while others may be very good at stock selection. If we analyse the performance of a portfolio into the components of stock and sector selection, then we will be better placed to understand the relative strengths and weaknesses of each investment manager. This process of attributing performance to stock and sector selection is called performance attribution or attribution analysis.

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How to find the sector and stock components

Suppose that we want to carry out the analysis for a particular portfolio over a given year. We start by considering the size of the actual fund (in monetary units) at the end of the year. This is the amount that has been generated from:
  the actual sector split (eg 68/11/21) 

  the actual stocks chosen by the fund manager. 
Call this amount Fundvalueactual / actual or FAA . 
Now consider the end of year amount that would have been generated by the notional benchmark portfolio that we discussed in Section 3. This is the amount that would have been generated from: 

  the notional sector split (eg 60/15/25) 

  the notional stocks (eg the market indices for each sector). 
Call this amount Fundvaluenotional / notional or FNN .
Then FAA - FNN is the overall amount of profit (or loss if negative) generated by the 
fund manager’s choice of sectors and stocks.

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overall investment performance of a fund can be divided into:

  sector selection, ie the extent to which the fund’s proportions in the various sectors will have affected performance. (Sector selection performance is sometimes known as asset allocation performance. ) 

  stock selection, ie within any one sector, have the selected stocks performed better or worse than the sector as a whole? 
As an example, consider a pension fund that is only invested in two sectors, UK equities and gilts. Suppose the trustees’ guidelines to the investment manager are that the proportions of the fund invested in equities and gilts should be 85% and 15% respectively. The investment manager is permitted to depart to some extent from these guidelines and the actual proportions adopted were 90% and 10%.

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alternative method of splitting the stock and sector selection components of relative performance is to

substitute for N2 the value of the fund that would have
been attained if the actual stocks selected were held but the sector proportions were the same as the benchmark. The two methods will give slightly different answers because the two methods are effectively assuming different weights on the two components that make up the total relative return.
In practice, the method most commonly used by investment performance analysts involves the following approach:
  the first method outlined above, ie splitting overall performance into stock and sector performance using an intermediate fund based on actual sectors and notional stocks 

  calculating percentage returns rather than monetary amounts 

  attributing the sector and stock selection profits to each of the asset classes, rather than to aggregate categories – we will consider this in detail in the next section. 
So, this is the approach the examiners are likely to want you to follow! 
A good example of what the examiners expect can be found in the Examiners’ Report to the examination paper for ST5 April 2006, Question 5. 
Section 4.5 of this chapter includes a simple example to illustrate this approach. 
The technique can be adapted to investigate the influence of other factors thought to have a bearing on overall performance. However, there must be a benchmark against which the particular element of “style” which is of interest can be judged. 
For example, where a fund includes investment in both domestic and overseas equities, the attribution analysis could breakdown any under-performance or over-performance as compared to the appropriate benchmark into three elements:
1. stock selection profit 

2. sector selection profit 

3. currency selection profit. 


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Attributing stock selection and sector selection profits Introduction

can sometimes go further in our performance attribution than Example 4.1 suggests. In particular, having split out the overall performance between stock selection and sector selection performance, we can then split each of these in turn into the contribution from each of the asset classes involved. We will illustrate this by means of a simple example involving the returns over a single year on a fund invested only in equities and bonds.

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Split of stock selection profit

As suggested above, we can split the total stock selection profit of 3.8% (from the solution to Question 15.15) between the individual contributions of equities and bonds. Recall that the figure of 3.8% is derived by considering the difference in performance between:
  the actual fund (actual sectors and actual stocks), and 

  a notional fund based on the actual sectors and notional stocks. 


Thus, to determine the contribution of each asset class to the overall stock selection profit we calculate:
actual sector weight  (actual sector return – notional sector return)
So, in this instance and given that the actual equity weighting was wEA = 80% , the
contribution of equities to this total stock selection profit is given by: wEA ¥ (rEA - rEN ) = 0.8 ¥ (15.0 -10.0) = +4%
Likewise, the contribution of bonds is given by: wBA ¥ (rBA - rBN ) = 0.2 ¥ (4.0 - 5.0) = -0.2%
So, the two individual contributions sum to the total stock selection profit of 3.8%. In this instance, the actual equities chosen out-performed the equity index by 5%, which given the actual equity weighting of 80%, produced the equity contribution of 4%. Similarly, the actual bonds under-performed the bond index, so producing a negative contribution to the total stock selection profit.

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Split of sector selection profit

In addition, recall that the total sector selection profit of 1.5% is derived (in the solution to Question 15.15) by considering the difference in performance between:
  a notional fund based on the actual sectors and notional stocks, and 

  the benchmark fund (based on notional sectors and notional stocks). 
Thus, to determine the contribution of each asset class to the overall sector selection profit we calculate: 
(actual sector weight – notional sector weight)
 (notional sector return – overall benchmark return) 
So, here the contribution of equities to the total sector selection profit is: (wEA -wEN)¥(rEN -rNN)=(0.8-0.5)¥(10.0-7.5)=+0.75% 
This figure reflects the fact that the equity index out-performed the overall benchmark fund by 2.5%. Given that the fund was 30% overweight in equities, this sector choice contributed +0.75% to performance. Here we have used the notional returns in order to strip out any effect due to stock selection.

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Risk-adjusted performance measures

The measurement of the investment performance of portfolios usually concentrates on the return, often relative to some benchmark. However, in order to get a true picture of performance we also need to pay attention to the risk dimension.
This is because the investment return achieved will typically depend on the risk incurred by the investor.
MPT and the CAPM give us a framework for doing this.
The risk-adjusted performance measures are not very commonly used in the UK but their use is more widespread in the USA. One problem with the measures described below is that they only allow for risk defined in terms of variance of return and do not allow for actuarial risk or downside risk.
The appropriate measure of risk for an investor to use, within the MPT framework, depends on whether the portfolio being considered represents all his assets or just a part of them. Where the portfolio represents the whole of the investor’s wealth the appropriate measure is the standard deviation. If it is a subset of his assets, the appropriate measure is the portfolio beta.
The reason for this is that the beta of a portfolio is a measure of its risk relative to a well-diversified portfolio and adjusting the return using beta tells us how good the manager is at picking out-performing securities, given the level of systematic risk assumed. Using standard deviation to adjust the return allows us to measure how well-diversified the whole portfolio is as well as how good the manager is at individual stocks that produce an excess return relative to their betas.
The measures that interpret risk in terms of the standard deviation are based on the capital market line equation, which applies only to efficient portfolios. Recall that the CAPM suggests that a rational investor should hold a well-diversified portfolio of risky assets (the market portfolio in fact) that is efficient.
In contrast the measures that interpret risk in terms of beta are based on the security market line equation, which applies to all portfolios, efficient or otherwise. In particular, it will apply to subsets of any efficient portfolio, which need not by themselves be efficient.

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Four risk-adjusted return measures can be specified according to whether

beta or standard deviation is the appropriate measure of risk and according to whether the required level of risk is pre-specified or not.

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Uses of performance measurement

To improve future performance
First, data collected during performance monitoring can form the inputs for planning future strategy, ie by finding out what has been successful in the past, investors should be better able to determine what might perform best in the future.
Secondly, if fund managers know that their performance is being measured, it might give them an extra incentive to maximise the returns of the funds they manage.
Comparison of the rate achieved against a target rate
Many funds will have one or more “target” rates of return. For example, the trustees of a pension fund will want to know the rate of return achieved on the investments compared with the rate of return assumed in the actuarial valuation.
Similarly, the actuaries and managers of a life insurance company will need to know what rate of return has been achieved on the fund compared with the rate assumed in premium rate, bonus distribution and reserve calculations.
Comparison against the performance of other portfolios, an index and/or a benchmark portfolio
Those responsible for the funds will want to know how the performance of the portfolio compares with other portfolios. On the basis of this information, they are able to make decisions regarding the future investment of the assets, eg should a new fund manager be hired?
Also, by analysing the performance against a notional portfolio, it may be possible to identify some relative strengths and/or weaknesses of individual fund managers (eg in sector or stock selection).
There may also be other factors that depend on the performance of the fund. For example, the fees paid to the fund manager may be linked to the performance of the fund.
These reasons have been given as if from the viewpoint of those responsible for the management of a fund. Similar reasons can be devised from other perspectives. For example, fund managers (eg unit trust managers) will want to advertise the good rates of return they have achieved.
To appraise and remunerate investment managers
Following from the above, performance measures can be used in the appraisal and remuneration of managers.

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Limitations of performance measurement

Projection of past results
The fact that a particular result was attained in the past does not mean that it will occur in the future. There is a random element in investment returns and it may be difficult to determine how much a fund manager’s results are due to method and how much to luck. Furthermore a technique that proved successful in a particular set of circumstances may not work so well in changed circumstances in the future.
So, past performance may be a poor guide to the future and it may not be easy to distinguish good luck from skill.
Risk
In the long term we would expect a riskier strategy to produce higher average returns. The measurement of relative performance should therefore take account of the degree of risk taken on by a fund manager.
When a fund manager invests largely in high-risk investments, there are likely to be implications for the relative performance of the fund manager in performance measurement tables:
  In the very long term, the manager should achieve a higher rate of return. 

  In short-term periods, the results will probably be more volatile. There may be 
some periods of excellent results and other periods of very poor results. 
So, a fund manager who takes a high-risk strategy could quite easily appear at the top of both the short-term and long-term performance tables. But, because high levels of risk may be undesirable for some funds, users of the performance tables will want to be able to identify those fund managers who have generally invested in high-risk investments. 
Beta values can be used as one indication of the level of risk (in the traditional sense of variability of return) of a portfolio. They provide a useful insight into how different portfolios may have performed and may be used to construct risk-adjusted performance measures, as outlined in Section 5 above. 


Timescale
Determining the frequency of performance measurement calculations requires a delicate balance between assessing performance frequently enough so that problems can be spotted and corrected and avoiding spurious conclusions based on too short a measurement period.
It might take five years to obtain data that gives a reliable verdict on a particular fund manager. However, the trustees of a pension fund should not have to wait for five years before they realise that the assets are being poorly managed.
In practice, many pension fund investment valuations are carried out each quarter, with analyses over a variety of periods (eg 3 months, 1 year, 3 years, 5 years, 10 years).
The users of these analyses therefore have:
  the regular data they need to stay informed 

  the longer-term data to help make judgements. 
They should resist the temptation of making bold conclusions from the very short-term data. 
Differing fund objectives 
Different funds may have different objectives and constraints. Comparisons between such funds may not be valid. 
The main difference will often be that the liabilities underlying one fund may differ from those underlying another. There may also be other reasons why different funds cannot be directly compared: 

  different constraints imposed by the directors or trustees 

  different taxation positions (eg this may apply for insurance companies where 
the tax positions may vary from one office to another). 
Comparisons between different unit-linked funds and collective investment vehicles is clearly sensible only if they have similar investment objectives. 
You may also come across “different cashflow” and “different size of fund” as reasons why different funds cannot be validly compared. In practice, these two factors should not really invalidate comparison between the investment returns achieved by different portfolios. 

Impact on fund manager behaviour
Knowledge of how and how often he will be assessed is likely to influence the investment strategy of a manager. This may not be in the fund’s best interests. For example, frequent monitoring can encourage a short-term approach to investment.
Some people will argue that this may mean that the long-term performance of the fund could be sacrificed. Others would argue that it is not a problem because the long run is simply a series of short runs.
More generally, the investment management decisions should be driven entirely a desire to meet the investor’s objectives. It is therefore important to ensure that the mandate given to the investment manager is consistent with the investor’s objectives (as was discussed in Chapter 9).
Cost
Users of performance measurement services must balance the value of the service against the cost. Also, for a number of assets (eg property), valuation is difficult, time-consuming and very subjective. Detailed, frequent calculations based on subjective valuations are inappropriate.

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Relative merits of different ways of assessing portfolios

8.1  Performance relative to published indices 
The main advantage of assessing the performance relative to published indices is that it is relatively easy to do. By definition, the data for published indices is readily available - and it should be reliably accurate. 
However, this advantage can be easily outweighed by a major potential disadvantage: the published index might not be appropriate. There may be no single index which is consistent with the objectives of the investor. 
For example, comparing the returns achieved by a life insurance company’s with-profit fund with the returns achieved from an index holding of domestic equities will not be appropriate because the objectives and constraints imposed on the insurance company would probably make a 100% holding in domestic equities inappropriate. 

. 8.2  Performance relative to other portfolios 
This type of comparison is appropriate if the funds being compared have the same objectives and the same factors influencing investment strategy. It also gives an indication of the cost or benefit of following a particular strategy, relative to that adopted by other funds. 
For example, considering pension funds, it would be quite sensible to compare the returns achieved by different pension funds where the:
  funds have the same liability profile (by type and term) 

  funds have similar levels of solvency 

  funds are of similar size 

  fund managers have been given similar levels of freedom by the trustees (or at least, similar instructions) 

  fund managers have adopted the same level of risk 

  funds have similar levels of cashflow (although use of time-weighted rate of 
return should mean that differences will not invalidate comparison). 
Taken to these extremes, you may find that, for each pension scheme, there are very few other pension schemes that can be used for a valid comparison. In practice, the comparisons tend to be driven more by what data is available. 


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..

However, it may be totally inappropriate to compare the performance of funds that have very different investment objectives, eg insurance company with-profit funds because:
  there is generally more variation in the type and term of liabilities 

  the investors may have quite different tax positions 

  the data is generally not widely available. 
Overall, performance measurement relative to other portfolios is not ideal. 

..

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Performance relative to a benchmark portfolio

Benchmark portfolios can be constructed to reflect the objectives of the fund. They should also be constructed in such a way that the data necessary for comparisons is easily obtained.
They can therefore overcome the problems we identified with the other two forms of comparison.
By having a benchmark portfolio that reflects the liabilities of the fund, the danger of giving the fund manager conflicting objectives is also avoided. This would occur where the basis for assessment encourages the fund manager to adopt a strategy that is not necessarily consistent with the objectives of the fund.