Modified Dietz method: Difference between revisions

Revision as of 05:34, 5 February 2019

Revision as of 06:34, 5 March 2019

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|isbn=978-0-470-47371-9

|pages=828–

|quote=Peter O. Dietz published his seminal work, Pension Funds: Measuring Investment Performance, in 1966. The Bank Administration Institute (BAI), a U.S.-based organization serving the financial services industry, subsequently formulated rate-of-return calculation guidelines based on Dietz's work.}}</ref> is a measure of the ''ex post'' (i.e. historical) performance of an [[investment portfolio]] in the presence of external flows. (External flows are movements of value such as transfers of cash, securities or other instruments in or out of the portfolio, with no equal simultaneous movement of value in the opposite direction, and which are not income from the investments in the portfolio, such as interest, coupons or dividends.) To calculate the modified Dietz return, divide the gain or loss in value, net of external flows, by the average capital over the period of measurement. The result of the calculation is expressed as a percentage [[rate of return|return]] over the holding period. The average capital weights individual cash flows by the amount of time from when those cash flows occur until the end of the period.

To calculate the modified Dietz return, divide the gain or loss in value, net of external flows, by the average capital over the period of measurement. The average capital weights individual cash flows by the length of time between those cash flows until the end of the period. Flows which occur towards the beginning of the period have a higher weight than flows occurring towards the end. The result of the calculation is expressed as a percentage [[rate of return|return]] over the holding period.

This method has the practical advantage over [[internal rate of return]] (IRR) method that it does not require repeated trial and error to get a result.<ref name="Feibel2003">{{cite book

|author=Bruce J. Feibel

|title=Investment Performance Measurement

|url=https://books.google.com/books?id=fzAVZvAGP7cC&pg=PA41

|date=21 April 2003

|publisher=John Wiley & Sons

|isbn=978-0-471-44563-0

|pages=41–

|quote=One of these return calculation methods, the Modified Dietz method, is still the most common way of calculating periodic investment returns.}}</ref>

The cash flows used in the formula are weighted based on the time they occurred in the period. For example, if they occurred in the beginning of the month they would have a higher weight than if they occurred at the end of the month. This is different from the [[simple Dietz method]], in which the cash flows are weighted equally regardless of when they occurred during the measurement period, which works on an assumption that the flows are distributed evenly throughout the period.

With the advance of technology, most systems can calculate a true time-weighted return by calculating a daily return and geometrically linking in order to get a monthly, quarterly, annual or any other period return. However, the modified Dietz method remains useful for performance attribution, because it still has the advantage of allowing modified Dietz returns on assets to be combined with weights in a portfolio, calculated according to average invested capital, and the weighted average gives the modified Dietz return on the portfolio. Time weighted returns do not allow this.

==GIPS==

This method for return calculation is used in modern portfolio management. It is one of the methodologies of calculating returns recommended by the Investment Performance Council (IPC) as part of their Global Investment Performance Standards (GIPS). The GIPS are intended to provide consistency to the way portfolio returns are calculated internationally.<ref>{{cite web|title=Global Investment Performance Standards (GIPS®) Guidance Statement on Calculation Methodology|url=http://www.gipsstandards.org/standards/guidance/documents/develop/calcmethod.pdf|publisher=IPC|accessdate=13 January 2015}}</ref>

==Origin==

The method is named after Peter O. Dietz.<ref>{{cite book

|title=The C.F.A. Digest

Line 72 ⟶ 62:

This assumes that the flow happens at the end of the day. If the flow happens at the beginning of the day, the flow is in the portfolio for an additional day, so use the following formula for calculating the weight:

:<math> W_i = \frac{C -D_i + 1}{C}</math>

==Comparison with time-weighted return and internal rate of return==

The modified Dietz method has the practical advantage over the [[true time-weighted rate of return]] method, in that the calculation of a modified Dietz return does not require portfolio valuations at each point in time whenever an external flow occurs. The [[internal rate of return]] method shares this practical advantage with the modified Dietz method.

With the advance of technology, most systems can calculate a true time-weighted return by calculating a daily return and geometrically linking in order to get a monthly, quarterly, annual or any other period return. However, the modified Dietz method remains useful for performance attribution, because it still has the advantage of allowing modified Dietz returns on assets to be combined with weights in a portfolio, calculated according to average invested capital, and the weighted average gives the modified Dietz return on the portfolio. Time weighted returns do not allow this.

The modified Dietz method also has the practical advantage over [[internal rate of return]] (IRR) method that it does not require repeated trial and error to get a result.<ref name="Feibel2003">{{cite book

|author=Bruce J. Feibel

|title=Investment Performance Measurement

|url=https://books.google.com/books?id=fzAVZvAGP7cC&pg=PA41

|date=21 April 2003

|publisher=John Wiley & Sons

|isbn=978-0-471-44563-0

|pages=41–

|quote=One of these return calculation methods, the Modified Dietz method, is still the most common way of calculating periodic investment returns.}}</ref>

The modified Dietz method is based upon a simple rate of interest principle. It approximates the [[internal rate of return]] method, which applies a compounding principle, but if the flows and rates of return are large enough, the results of the Modified Dietz method will significantly diverge from the internal rate of return.

The modified Dietz return is the solution <math>R</math> to the equation:

:<math>B = A \times (1+R)+ \sum_{i=1}^n F_i \times (1+R \times \frac{T - t_i}{T})</math>

where

:<math>A</math> is the start value

:<math>B</math> is the end value

:<math>T</math> is the total length of time period

and

:<math>t_i</math> is the time between the start of the period and flow <math>i</math>

Compare this with the (unannualized) [[internal rate of return]] (IRR). The IRR (or more strictly speaking, an un-annualized holding period return version of the IRR) is a solution <math>R</math> to the equation:

:<math>B = A \times (1+R)+ \sum_{i=1}^n F_i \times (1+R)^ \frac{T - t_i}{T}</math>

For example, suppose the value of a portfolio is $100 at the beginning of the first year, and $300 at the end of the second year, and there is an inflow of $50 at the end of the first year/beginning of the second year. (Suppose further that neither year is a leap year, so the two years are of equal length.)

To calculate the gain or loss over the two-year period,

:<math>\text {gain or loss} = B - A - F = 300 - 100 - 50 = $150\text{.}</math>

To calculate the average capital over the two-year period,

:<math>\text {average capital} = A + \sum \text {weight} \times \text {flow} = 100 + 0.5 \times 50 = $125\text{,}</math>

so the modified Dietz return is:

:<math>\frac {\text {gain or loss}}{\text {average capital}} = \frac {150}{125} = 120\%</math>

The (unannualized) internal rate of return in this example is 125%:

:<math>300 = 100 \times (1 + 125\%)+ 50 \times (1+125\%)^ \frac{2 - 1}{2} = 225 + 50 \times 150\% = 225 + 75</math>

so in this case, the modified Dietz return is noticeably less than the unannualized IRR. This divergence between the modified Dietz return and the unannualized internal rate of return is due to a significant flow within the period, and the fact that the returns are large.

==The simple Dietz method==

The modified Dietz method is different from the [[simple Dietz method]], in which the cash flows are weighted equally regardless of when they occurred during the measurement period. The [[simple Dietz method]] is a special case of the Modified Dietz method, in which external flows are assumed to occur at the midpoint of the period, or equivalently, spread evenly throughout the period, whereas no such assumption is made when using the Modified Dietz method, and the timing of any external flows is taken into account.

==Adjustments==

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==Fees==

To measure returns net of fees, allow the value of the portfolio to be reduced by the amount of the fees. To calculate returns gross of fees, compensate for them by treating them as an external flow, and exclude accrued fees from valuations.

==Comparison with time-weighted return and internal rate of return==

The modified Dietz method has the practical advantage over the [[true time-weighted rate of return]] method, in that the calculation of a modified Dietz return does not require portfolio valuations at each point in time whenever an external flow occurs. The [[internal rate of return]] method shares this practical advantage with the modified Dietz method.

The modified Dietz method has the practical advantage over the [[internal rate of return]] method, in that there is a formula for the modified Dietz return, whereas iterative numerical methods are usually required to estimate the internal rate of return.

The modified Dietz method is based upon a simple rate of interest principle. It approximates the [[internal rate of return]] method, which applies a compounding principle, but if the flows and rates of return are large enough, the results of the Modified Dietz method will significantly diverge from the internal rate of return.

The modified Dietz return is the solution <math>R</math> to the equation:

:<math>B = A \times (1+R)+ \sum_{i=1}^n F_i \times (1+R \times \frac{T - t_i}{T})</math>

where

:<math>A</math> is the start value

:<math>B</math> is the end value

:<math>T</math> is the total length of time period

and

:<math>t_i</math> is the time between the start of the period and flow <math>i</math>

Compare this with the (unannualized) [[internal rate of return]] (IRR). The IRR (or more strictly speaking, an un-annualized holding period return version of the IRR) is a solution <math>R</math> to the equation:

:<math>B = A \times (1+R)+ \sum_{i=1}^n F_i \times (1+R)^ \frac{T - t_i}{T}</math>

For example, suppose the value of a portfolio is $100 at the beginning of the first year, and $300 at the end of the second year, and there is an inflow of $50 at the end of the first year/beginning of the second year. (Suppose further that neither year is a leap year, so the two years are of equal length.)

To calculate the gain or loss over the two-year period,

:<math>\text {gain or loss} = B - A - F = 300 - 100 - 50 = $150\text{.}</math>

To calculate the average capital over the two-year period,

:<math>\text {average capital} = A + \sum \text {weight} \times \text {flow} = 100 + 0.5 \times 50 = $125\text{,}</math>

so the modified Dietz return is:

:<math>\frac {\text {gain or loss}}{\text {average capital}} = \frac {150}{125} = 120\%</math>

The (unannualized) internal rate of return in this example is 125%:

:<math>300 = 100 \times (1 + 125\%)+ 50 \times (1+125\%)^ \frac{2 - 1}{2} = 225 + 50 \times 150\% = 225 + 75</math>

so in this case, the modified Dietz return is noticeably less than the unannualized IRR. This divergence between the modified Dietz return and the unannualized internal rate of return is due to a significant flow within the period, and the fact that the returns are large.

==Annual rate of return==

Note that the Modified Dietz return is a holding-period return, not an annual rate of return, unless the period happens to be one year. Annualisation, which is conversion of the holding-period return to an annual rate of return, is a separate process.

==The simple Dietz method==

Note also that the [[simple Dietz method]] is a special case of the Modified Dietz method, in which external flows are assumed to occur at the midpoint of the period, or equivalently, spread evenly throughout the period, whereas no such assumption is made when using the Modified Dietz method, and the timing of any external flows is taken into account.

==Money-weighted return==