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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.

3

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.

14

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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19

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.

24

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.

26

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.

27

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.

30

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.