The modified Dietz method 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 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 return over the holding period.
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.
Origin The method is named after Peter O. Dietz. The original idea behind the work of Peter Dietz was to find a quicker, less computer-intensive way of calculating an IRR as the iterative approach using the then-quite-slow computers that were available was taking a significant amount of time; the research was produced for BAI, Bank Administration institute. The modified Dietz method is a linear IRR.
Formula
The formula for the modified Dietz method is as follows:
where
is the starting market value
is the ending market value
is the net external inflow for the period (so contributions to a portfolio are treated as positive flows while withdrawals are negative flows)
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. Assuming that the flow happens at the end of the day, 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, unless you adopt the convention that the start date is the same as the end date of the previous period) 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, the flow is in the portfolio for an additional day, so use the following formula for calculating the weight:
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. Conversely, if there exists a portfolio valuation at any point in time, the implied modified Dietz valuation of cashflows at that point in time is quite unlikely to agree with the actual valuation.
With the advance of technology, most systems can calculate a 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.
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 start value
is the end value
is the total length of time period
and
is the time between the start of the period and flow
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 to the equation:
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,
To calculate the average capital over the two-year period,
so the modified Dietz return is:
The (unannualized) internal rate of return in this example is 125%:
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, together with the fact that the returns are large. If there are no flows, there is no difference between the modified Dietz return, the unannualized IRR, or any other method of calculating the holding period return. If the flows are small, or if the returns themselves are small, then the difference between the modified Dietz return and the unannualized internal rate of return is small.
The IRR is 50% since:
but the unannualized holding period return, using the IRR method, is 125%. Compounding an annual rate of 50% over two periods gives a holding period return of 125%:
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. Note that in the example above, the flow occurs midway through the overall period, which matches the assumption underlying the simple Dietz method. This means the simple Dietz return and modified Dietz return are the same in this particular example.
Adjustments If either the start or the end value is zero, or both, the start and/or end dates need to be adjusted to cover the period over which the portfolio has content.
Example Suppose we are calculating the 2016 calendar year return, and that the portfolio is empty until a transfer in of EUR 1m cash in a non-interest bearing account on Friday 30 December. By the end of the day on Saturday 31 December 2016, the exchange rate between euros and Hong Kong dollars has changed from 8.1 HKD per EUR to 8.181, which is a 1 percent increase in value, measured in Hong Kong dollar terms, so the right answer to the question of what is the return in Hong Kong dollars is intuitively 1 percent.
However, blindly applying the modified Dietz formula, using an end-of-day transaction timing assumption, the day-weighting on the inflow of 8.1m HKD on 30 December, one day before the end of the year, is 1/366, and the average capital is calculated as:
+ inflow × = 0 + 8.1m × 1/366 = 22,131.15
and the gain is:
- - = 8,181,000 - 0 - 8,100,000 = 81,000
so the modified Dietz return is calculated as:
= = 366 %
So which is the correct return, 1 percent or 366 percent?
Adjusted time interval The only sensible answer to the example above is that the holding period return is unambiguously 1 percent. This means the start date should be adjusted to the date of the initial external flow. Likewise, if the portfolio is empty at the end of the period, the end date should be adjusted to the final external flow. The end value is effectively the final external flow, not zero.
The return annualized using a simple method of multiplying-up 1 percent per day by the number of days in the year will give the answer 366 percent, but the holding period return is still 1 percent.
Example corrected The example above is corrected if the start date is adjusted to the end of the day on 30 December, and the start value is now 8.1m HKD. There are no external flows thereafter.
The corrected gain or loss is the same as before:
- = 8,181,000 - 8,100,000 = 81,000
but the corrected average capital is now:
+ = 8.1m
so the corrected modified Dietz return is now:
= = 1 %
Second example Suppose that a bond is bought for HKD 1,128,728 including accrued interest and commission on trade date 14 November, and sold again three days later on trade date 17 November for HKD 1,125,990 (again, net of accrued interest and commission). Assuming transactions take place at the start of the day, what is the modified Dietz holding-period return in HKD for this bond holding over the year to-date until the end-of-day on 17 November?
Answer The answer is that firstly, the reference to the holding period year to-date until the end-of-day on 17 November includes both the purchase and the sale. This means the effective adjusted holding period is actually from the purchase at the start of the day on 14 November until it is sold three days later on 17 November. The adjusted start value is the net amount of the purchase, the end value is the net amount of the sale, and there are no other external flows.
= 1,128,728
= 1,125,990
There are no flows, so the gain or loss is:
− = 1,125,990 − 1,128,728 = −2,738
and the average capital equals the start value, so the modified Dietz return is:
= = −0.24 % 2 d.p.
Contributions - when not to adjust the holding period This method of restricting the calculation to the actual holding period by applying an adjusted start or end date applies when the return is calculated on an investment in isolation. When the investment belongs inside a portfolio, and the weight of the investment in the portfolio, and the contribution of that return to that of the portfolio as a whole is required, it is necessary to compare like with like, in terms of a common holding period.
Example Suppose that at the beginning of the year, a portfolio contains cash, of value $10,000, in an account which bears interest without any charges. At the beginning of the fourth quarter, $8,000 of that cash is invested in some US dollar shares (in company X). The investor applies a buy-and-hold strategy, and there are no further transactions for the remainder of the year. At the end of the year, the shares have increased in value by 10% to $8,800, and $100 interest is capitalized into the cash account.
What is the return on the portfolio over the year? What are the contributions from the cash account and the shares? Furthermore, what is the return on the cash account?
Answer The end value of the portfolio is $2,100 in cash, plus shares worth $8,800, which is in total $10,900. There has been a 9 percent increase in value since the beginning of the year. There are no external flows in or out of the portfolio over the year.
= 0
so
= = $10,000
so the return is:
= = 9 %