The modified Dietz method is a measure of the historical performance of an
investment portfolio in the presence of external flows. 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 for the time 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.
This method has the practical advantage over Internal Rate of Return that it
does not require repeated trial and error to get a result.
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 in the
past 15 years, 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.
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 as part of their Global Investment Performance
Standards. The GIPS standard is intended to standardize the way portfolio returns
are calculated internationally. The method is named after Peter O.
Dietz. Formula
The formula for the modified Dietz method is as follows:
where is the ending market value
is the beginning market value is the net external inflow for the
period and
the sum of each flow multiplied by its weight
The weight is the proportion of the time period between the point in time
when the flow occurs and the end of the period. can be calculated as
where is the number of calendar days during
the return period being calculated, which equals end date minus start date
plus 1 is the number of days from the start of
the return period until the day on which the flow occurred. This assumes that
the flow happens at the end of the day. If the flow happens at the beginning of
the day, use the following formula for calculating weight:
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 to the equation:
where is the ending market value
is the beginning market value is the total length of time period
and is the time between the start of the
period and flow Compare this with the internal rate of
return. The IRR is a solution to the equation:
For example, suppose the value of a portfolio is 100 USD at the beginning of
the first year, and 300 USD at the end of the second year, and there is an
inflow of 50 USD at the end of the first year/beginning of the second year.
To calculate the gain or loss over the two-year period,
To calculate the average capital over the two-year period,
so the Modified Dietz return is: The internal rate of return in this
example is 125%: so in this case, the Modified Dietz
return is noticeably less than the IRR. This divergence between the Modified
Dietz return and the 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
not an annual rate of return, unless the period happens to be one year.
Annualisation, which is conversion of the 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 The Modified Dietz method is an example
of a money weighted methodology. In particular, if the Modified Dietz return
on two portfolios are and , measured over a common matching time interval,
then the Modified Dietz return on the two portfolios put together over the
same time interval is the weighted average of the two returns:
where the weights of the portfolios depend on the Average Capital over the
time interval: Linked Return versus True Time-Weighted
Return An alternative to the Modified Dietz
method is to link geometrically the Modified Dietz returns for shorter
periods. This method is classed as a time-weighted method, but does not
produce the same results as the true time weighted method, which requires
valuations at the time of each cash flow.
Issues There are sometimes difficulties when
calculating or decomposing portfolio returns, if all transactions are treated
as occurring at a single point during the day. Whatever method is applied to
calculate returns, an assumption that all transactions take place
simultaneously at a single point in time each day can lead to errors.
For example, consider a scenario where a portfolio is empty at the start of a
day, so that BMV = 0. There is then an external inflow during a day of F =
$100. By the close of the day, market prices have moved, and EMV = $99.
If all transactions are treated as occurring at the end of the day, then
there is zero start value BMV, and zero value for Average Capital, so no
Modified Dietz return can be calculated. Some such problems are resolved if the
Modified Dietz method is further adjusted so as to put purchases at the
open and sales at the close, but more sophisticated exception-handling
produces better results. There are sometimes other difficulties
when decomposing portfolio returns, if all transactions are treated as
occurring at a single point during the day.
For example, consider a fund opening with just $100 of a single stock that is
sold for $110 during the day. During the same day, another stock is purchased for
$110, closing with a value of $120. The returns on each stock are 10% and
120/110 - 1 = 9.0909% and the portfolio return is 20%. The asset weights wi
required to get the returns for these two assets to roll up to the portfolio
return are 1200% for the first stock and a negative 1100% for the second:
w*10/100 +*10/110 = 20/100 → w = 12. Such weights are absurd, because the
second stock is not held short. Excel VBA function for modified Dietz
return Public Function MDIETZ(dStartValue As
Double, dEndValue As Double, iPeriod As Integer, rCash As Range, rDays As Range)
As Double 'Jelle-Jeroen Lamkamp 10 Jan 2008
Dim i As Integer: Dim Cash() As Double: Dim Days() As Integer
Dim Cell As Range: Dim SumCash As Double: Dim TempSum As Double
'Some error trapping If rCash.Cells.Count rDays.Cells.Count
Then MDIETZ = CVErr(xlErrValue): Exit Function
If Application.WorksheetFunction.Max(rDays)
> iPeriod Then MDIETZ = CVErr(xlErrValue): Exit Function
ReDim Cash(rCash.Cells.Count - 1) ReDim Days(rDays.Cells.Count - 1)
i = 0 For Each Cell In rCash
Cash(i) = Cell.Value: i = i + 1 Next Cell
i = 0 For Each Cell In rDays
Days(i) = Cell.Value: i = i + 1 Next Cell
SumCash = Application.WorksheetFunction.Sum(rCash)
TempSum = 0 For i = 0 To
TempSum = TempSum +) / iPeriod) * Cash(i))
Next i MDIETZ = /
End Function The above VBA program is designed to use
with Excel. Here is a Java program written for general purposes.
Java Method for Modified Dietz Return private static double modifiedDietz {
/* emv: Ending Market Value * bmv: Beginning Market Value
* cashFlow[]: Cash Flow * numCD: actual number of days in
the period * numD[]: number of days between
beginning of the period and date of cashFlow[]
*/ double md = -99999; initialize modified
dietz with a debugging number try {
double[] weight = new double[cashFlow.length];
if (numCD