Modified Dietz Method - Discussion/derivation

Discussion/derivation

1) 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.

2) 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.

3) 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:

EMV = ending market value

BMV = beginning market value

T = total length of time period

t_i = time between the start of the period and flow

Compare this with the internal rate of return, which is the solution to the equation:

4) 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.

5) 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.

6) Modified Dietz is an example of a money (or dollar) 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:

7) 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.

8) 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% (4 d.p.) and the portfolio return is 20%. The asset weights w_i (as opposed to the time weights W_i) 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 + (1-w)*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 (rCash.Cells.Count - 1) TempSum = TempSum + (((iPeriod - Days(i)) / iPeriod) * Cash(i)) Next i MDIETZ = (dEndValue - dStartValue - SumCash) / (dStartValue + TempSum) 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 (double emv, double bmv, double cashFlow, int numCD, int numD) { /* 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; if (numCD <= 0) { throw new ArithmeticException ("numCD <= 0"); } for (int i=0; iModified Dietz Method