TimeWeighted Return on TWR
To calculate the timeweighted return, proceed as follows: the entire estimated period (for example, four months, as in the case of the Omega Foundation) is divided into intervals that begin and end with the next flow of money (inflow or outflow), and Within the interval, the flow of money flows does not occur. Then, for each interval, the value (1 + r ) is calculated, while the final sum of money (at the end of the interval, before the occurrence of a new cash flow) is taken as the numerator, and the initial (at the beginning of the interval , at the time of receipt of the cash flow). After that, the results are converted into monthly or annual returns.
Consider the example of the Omega Foundation, while we will conditionally assume that on April 30, at the end of the first month, the net asset value of the fund attributable to the investor is 94.8 million rubles. Conditional data on the value of net assets in other dates are given in Table. 5.3.
As follows from Table. 5.3, the entire life of the portfolio is divided into a number of intervals.
Table 5.3
Calculating weighted profitability time
Date 
The value of the portfolio P t, thousand rubles. 
Money flows CF t, thousand rubles 

1 + r 
March 31 
90000 
0 

April 1 
+5000 
94800 
0.99879 

April 30th 
94,800 
90000 + 5000 

May 1 
+5000 
99600 
0.99800 

May 31 
99600 
94800 + 5000 

June 1 
+5000 
106300 
1,01625 

June 30 
106300 
99600 + 5000 

July 1 
+5000 
115000 
1,03324 

July 31 
115000 
106300 + 5000 
The first interval begins April 1, when the original amount of 90 million rubles. a first installment of 5000 thousand rubles is added. The first interval is soldered on April 30, on the eve of the next cash inflow of money, when the value of the investor's portfolio is 94,800 thousand rubles. Note that during this first interval, no movement of funds occurs in the investor's portfolio.
The second interval begins May 1, when the investor makes another installment of 5000 thousand rubles, and ends on May 31. And again during this interval there is no movement of money.
For the same approaches, the third and fourth intervals are formed.
The data in column 2 is chosen conditionally to show the essence of the method. As follows from Table. 5.3, the returns are calculated for each interval of receipt of cash flows. The yield for April and May is less than one, which indicates a negative profitability (losses) in these months (on April 1 the fund had 95 thousand rubles, and on April 30  94.8 thousand rubles.)
As in the example of the foundation "Omega" the duration of the intervals is the same (one month), then according to the calculated values (1 + r ), it is easy to find the average monthly return.
Average arithmetic monthly return:
Average geometric monthly return:
An important feature of a timeweighted profitability is the simplicity of its estimates for the computed values of the values (1 + r t) for N intervals, even for different interval lengths, if as a basis select geometric the average yield. In this case, based on the successive multiplication of N values (1 + r t), the geometric average yield per day is calculated, which is then transformed into an annual average geometric yield:
where K  is the total number of days in the N intervals.
Geometric profitability for the year: .
Example 5.4. Assume that the previously discussed Vega March 7 to 9.00 had on the account 121.56 million rubles. (the figure is chosen conditionally). Then in the first six days of March, the return of the Vega fund " will be
For the remaining twentyfive days of March, the return will be
If we multiply these quantities one by one and extract a root of the 31st power, we get the value (1 + r) per day. By raising the value to the 365th degree, we get the average geometric profitability for the year:
The calculation of the average arithmetic profitability, if the estimated intervals have different duration, requires the use of special techniques. Suppose, for example, that the portfolio is formed for 35 days, and for the first eight days the value (1 + r ) for the estimated portfolio was 1.014, and for the next 27 days  1.004.
Then the average daily yield can be calculated, for example, like this:
If we then multiply this by 365, we get the average arithmetic yield for the year: 0.0005116 • 365 = = 0.1867/18, or 18.6748%.
Monetary Weighted Return MWR
As it was already mentioned, MWR is used if the evaluation of the portfolio return is conducted in a relatively short period of time, and the manager is able to independently influence the cash flows of the portfolio. In such cases, it is possible not to take into account the possibility of reinvesting the revenues received from the portfolio, abstracting from the need to assign each cash flow to a specific date, and conditionally assume that the yield does not change over the estimated time interval. Such assumptions make it easier to calculate MWR : the monetary weighted return is defined as the ratio of the income received from the portfolio over the estimated period of time to the average value of invested capital AIC:
The income provided by the portfolio over the estimated time interval is calculated as follows:
where MV is the final market value of the portfolio; МV start  the initial market value of the portfolio; NCF  net cash flow for the estimated interval.
Net cash flow NCF is the sum of all inflows to the portfolio, minus the sum of all outflows from the portfolio:
where C t  all inflows to the portfolio, which include additional deposits of cash, as well as the purchase of securities and the acquisition of other assets; W t  all outflows from the portfolio, including withdrawals of money, sale of securities and other assets, payment of dividends and coupon amounts.
In order to adequately reflect the dynamics of the portfolio contents, it is advisable to separately maintain a securities account and a cash account. The principal difference between the additional contribution to the portfolio from the purchase of securities is that the contribution leads to changes only in the cash account, whereas the purchase of securities results in an outflow of money (a reduction in the cash account) and in the inflow of securities papers).
The value of the average invested capital AIC can be calculated in various ways, assuming that
For example, the Dietz Approximation is built on the assumption that the net cash flow NCF is observed in the middle of the estimated period, so the weighted cash flow is 1/2 NCF:
Accordingly,
The moneyweighted yield MWR provides correct estimates if the cash flows for the portfolio exceed 10% of the market value of the portfolio. If there are no cash flows within the estimated time period, then i MWR actually turns into a timeweighted return TWR. The internal rate of return (IRR) can be considered as averaged value MWR over a long period of time, when reinvestment of the revenues obtained plays a significant role.
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