Dollar Weighted Returns
The simplest form of performance measurement is the dollar weighted re- turn (DWR), which measures the amount of money gained or lost from the beginning of a review period to the end of a review period, as a percentage of the original dollars invested.
Formula 4.1 (Dollar Weighted Return)
where EMV = Ending market value.
BMV = Beginning market value.
Note: All market values should include corporate actions that have an impact on the value of the portfolio, such as dividends and interest.
Example Let’s assume that on December 31, we invested $1,000,000 in CAM’s small-cap value product and that the value of that investment in- creased during the month, finishing at $1,100,000 on January 31. Let’s assume that we did not make any contributions or distributions during that period (cash flows are important and will be discussed later in the chapter). Using formula 4.1, we calculate the following return for the month of January:
The answer, 0.10, is in decimal format. To convert it to percentage for- mat, simply multiply it by 100.
DWR= −
=
=
$ , , $ , ,
$ , ,
$ ,
$ , , .
1 100 000 1 000 000 1 000 000
100 000
1 000 000 0 10
DWR EMV BMV
= BMV−
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Formula 4.2 (Conversion of Decimal Return to Percentage Format) (Decimal Return ×100) = Percentage Return
Using the information from our example, we use formula 4.2 to calcu- late the percentage return: (0.10 ×100) = 10.0%.
So, our initial $1,000,000 investment grew by $100,000 in absolute dollar terms or by 10% in percentage terms.
Chain-Linking Returns
In the previous example, we calculated the dollar weighted return for our
$1 million investment in CAM for the month of January. Let’s extend the example to include performance for the following month. To calculate the February return, we use the market value at the end of January as the be- ginning market value ($1,100,000). The value of the portfolio at the end of February was $1,200,000. Using the formula for dollar weighted return, we calculate the following:
February Return
In percentage terms, the February return was: 0.091 ×100 = 9.10%.
Total Period Return (January and February)
In percentage terms, the total period return was: 0.2 ×100 = 20.0%.
Let’s summarize:
Initial market value (12/31): $1,000,000
Ending market value (2/28): $1,200,000
% Return Gain/Loss
January return: 10.00% $100,000
February return: 9.10% $100,000
Total period return: 20.00% $200,000
$ , , $ , ,
$ , ,
$ ,
$ , , .
1 200 000 1 000 000 1 000 000
200 000
1 000 000 0 20
−
=
=
$ , , $ , ,
$ , ,
$ ,
$ , , .
1 200 000 1 100 000 1 100 000
100 000
1 100 000 0 091
−
=
= (in decimal format)
42 EQUITY AND FIXED INCOME MANAGER ANALYSIS
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What we immediately notice is that the returns from January and Feb- ruary do not equal the total period return when added together (19.10%
versus 20.00%). This is due to compounding—while the dollar gain was the same in each month ($100,000), the percentage required to achieve that return in February was lower than it was in January because we started the month with a higher beginning market value. The same is true when the portfolio decreases in value.
To properly combine the returns for January and February to match the total period return, we must chain-link the returns using the follow- ing formula:
Formula 4.3 (Chain-Linked Return)
CLR = [(1 + PDR1)×(1 + PDR2)×. . . ×(1 + PDRn)] –1 where PDR1= Period decimal return for period 1.
PDR2= Period decimal return for period 2.
PDRn= Period decimal return for final period.
Note: The periods in question can be monthly, quarterly, or annually, or can cover any period you specify. In addition, the periods can measure different time intervals. For example, PDR1can measure a single month, while PDR2can measure a six-month period. In the aforementioned exam- ple, the total chain-linked return would cover a seven-month period.
Example Using the summary data from the previous example for January and February, we calculate the chain-linked return as follows:
CLR = [(1 + January Decimal Return) ×(1 + February Decimal Return)] – 1 CLR = [(1 + 0.10) ×(1 + 0.091)] – 1
CLR = [(1.10) ×(1.091)] = 1.20
CLR = 1.20 – 1 = 0.20 (in decimal format)
CLR = 0.20 ×100 = 20.0% (in percentage format)
So we can arrive at the correct total period return by using the dollar weighted average for the entire period or by chain-linking the individual period (in this case monthly) returns. Using the chain-linking formula, we can calculate what the total period return would have been if the return for both January and February was 10%.
(1.10)×(1.10) = 1.21 1.21 – 1 = 0.21 0.21×100 = 21.0%
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Due to compounding the total period return would have been 21.0%
and the absolute dollar gain would have been $210,000. Using these for- mulas, we can start with the returns and work our way back to the dollar amounts or start with the dollar amounts and work our way to the returns.
Cumulative Returns
Now that we can calculate dollar weighted returns and have discussed the formula for chain-linking the individual period returns, we can put to- gether a cumulative return table and graph. This is a basic, often-used method of displaying an investment manager’s performance over a period of time. There are two ways of showing this information: (1) using actual dollar growth or (2) using the returns to illustrate the theoretical growth of a dollar (in other words, how an initial investment of $1 has grown over the review period).
The actual calculation for cumulative return is a continuous applica- tion of the chain-linking formula already outlined, where the initial return is the first period return and each subsequent cumulative return chain-links the previous cumulative return figure to the current return period. The for- mula is broken out to better illustrate its application:
Formula 4.4 (Cumulative Return)
Period Formula Notes
Period 1 PDR1 Measures period 1
Period 2 (1 + PDR1)×(1 + PDR2) = PDR1 . . . 2 Measures periods 1–2 Period 3 (1 + PDR1 . . . 2)×(1 + PDR3) = PDR1 . . . 3 Measures periods 1–3 Period 4 (1 + PDR1 . . . 3)×(1 + PDR4) = PDR1 . . . 4 Measures periods 1–4 Final period (1 + PDR1 . . . n–1)×(1 + PDRn) Measures periods 1–n where PDR1 . . . n–1 = Next-to-last period.
PDRn = Last period.
Each progressive cumulative return is calculated by simply chain-linking the previous total period return to the next period return. This calculation can continue over the entire performance period of for some subset of the period. However, it is critical that the performance be linked in calendar or- der (from first period to last) to get an accurate picture of the actual growth.
We have thus far calculated the dollar weighted return for CAM’s small-cap value portfolio for January and February. We will now extend this example to include returns for an entire calendar year. The data listed in Exhibit 4.2 is dollar weighted and assumes that no cash flows into or out of the portfolio took place during the year.
44 EQUITY AND FIXED INCOME MANAGER ANALYSIS
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EXHIBIT 4.2Returns for an Entire Calendar Year ReturnReturnCumulative PeriodMonth(Decimal)(%)FormulaReturn PDR1January0.10010.0%PDF110.0% PDR2February0.0919.1%(1 + PDF1)×(1 + PDF2) = PDF1...220.0% PDR3March0.0343.4%(1 + PDR1...2)×(1 + PDR3) = PDR1...324.1% PDR4April0.0121.2%(1 + PDR1...3)×(1 + PDR4) = PDR1...425.6% PDR5May0.0373.7%(1 + PDR1...4)×(1 + PDR5) = PDR1...530.2% PDR6June0.0232.3%(1 + PDR1...5)×(1 + PDR6) = PDR1...633.2% PDR7July0.0040.4%(1 + PDR1...6)×(1 + PDR7) = PDR1...733.8% PDR8August0.0060.6%(1 + PDR1...7)×(1 + PDR8) = PDR1...834.6% PDR9September0.0424.2%(1 + PDR1...8)×(1 + PDR9) = PDR1...940.2% PDR10October0.0818.1%(1 + PDR1...9)×(1 + PDR10) = PDR1...1051.6% PDR11November0.0292.9%(1 + PDR1...10)×(1 + PDR11) = PDR1...1156.0% PDR12December0.0020.2%(1 + PDR1...11)×(1 + PDR12) = PDR1...1256.3%
45
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The formula column in Exhibit 4.2 clearly shows how the cumulative return calculation works by displaying the formula’s incremental adjust- ments throughout the calendar-year period examined in the example.
Exhibit 4.3 takes the resulting cumulative returns and displays them in graphical format. The cumulative graph could also be displayed by apply- ing the period returns to a theoretical dollar value. For example, if we set the beginning value at $1, we would have ended the 12-month period with
$1.56 (because the cumulative return for the period was 56.3%). The growth of a dollar is depicted in Exhibit 4.4.
Exhibit 4.5 reviews CAM’s historical cumulative performance since its inception (as a growth of a dollar chart) and compares it to the broad
46 EQUITY AND FIXED INCOME MANAGER ANALYSIS
EXHIBIT 4.3 Cumulative Performance Chart Month
Return
0%
10%
20%
30%
40%
50%
60%
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
EXHIBIT 4.4 Cumulative Growth of $1 Month
Dollar Growth
$1.0
$1.1
$1.2
$1.3
$1.4
$1.5
$1.6
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
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small-cap and small-cap value indexes. The results should not be surprising given the performance figures stated in Exhibit 4.1.
Annualized Returns
What if I told you that over the past 10 years a portfolio had gained 100%
(doubling in value over the period). While this information is useful in de- termining an absolute dollar gain, it does not tell us anything about the typical portfolio’s annual returns, which is generally a better-understood measure. To solve this we could simply divide the total return (100%) by the number of years (10) to come up with an average return of 10% per year; but, due to compounding, this would be inaccurate. The proper way to determine a given product’s annual return (taking the effects of com- pounding in consideration) is to apply the following formula:
Formula 4.5 (Annualized Return)
[(1 + DWR) ^ (1 ÷YRS)] – 1 where DWR = Dollar weighted return (in decimal format).
YRS = Total number of years of period under review.
^ = Raised to the ( ) power. For example, 3 ^ (2) is translated as 3 raised to the 2nd power, the result of which is equal to 3 times 3, or 9.
Performance Analysis 47
EXHIBIT 4.5 Cumulative Performance Comparison Date
Growth of $1
CAM
S&P Small-cap Value Index S&P Small-cap Index
Jan-98 Jun-98 Nov-98 Apr-99 Sep-99 Feb-00 Jul-00 Dec-00 May-01 Oct-01 Mar-02 Aug-02 Jan-03 Jun-03
$0.70
$0.90
$1.10
$1.30
$1.50
$1.70
$1.90
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Note: The exponent can be stated to reflect differences in the fre- quency of the underlying return periods. For example: If the underlying re- turn periods are stated in a monthly format, we can apply the following:
(12÷# of months); if stated in quarterly format, we can apply the follow- ing: (4 ÷# of quarters).
Applying formula 4.5 to our example, we calculate the following an- nualized return (100% in decimal format is 1.00):
Annualized Return = [(1 + 1.00) ^ (1 ÷10)] – 1
= [(2.00) ^ (0.10)] – 1
= 1.07177 – 1 = 0.07177 In percent format:
Annualized Return = 0.07177 ×100 = 7.17%
The impact of compounding is clearly evident when you compare the average return (10%) to the annualized return (7.17%). An invest- ment firm can, therefore, state its product’s historical returns in cumula- tive and/or annualized terms. When analyzing performance figures, make sure you are aware of the performance methodology. By simply annualizing returns for products with different time periods, we can standardize each product’s return and make any comparisons more meaningful.
Adjusting for Cash Flows
So far, we have calculated returns based on the assumption that there were no contributions or distributions of assets during the measurement periods. However, in the real world, cash flows are a part of everyday business. Cash flows into or out of an investment manager’s portfolio can be the result of many different investment decisions. For example, an investor might contribute assets to or redeem assets from a specific investment product based on asset allocation changes, funding require- ments, or concerns regarding the underlying investment professionals.
Whatever the reason, cash flows can have a big impact on return calcu- lations.
To illustrate how cash flows can impact portfolio returns let’s review the following example:
BMV (Jan. 1): $1,000,000 EMV (Jan. 31): $2,000,000
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Using the formula for dollar weighted return (4.1), we calculate the following return:
A one-month return of 100%—that is amazing . . . or is it?
What I left out from this example was the fact that there was a
$900,000 contribution to this portfolio at the end of business on January 30. Given this information, it is clearly unfair to use the dollar weighted re- turn formula to determine the portfolio’s return for January.
At this point, you might be inclined to simply adjust the dollar weighted return by the amount of the cash flow. This can be done in the following way:
Formula 4.6 (Cash Flow Adjusted Dollar Weighted Return—CFADWR)
where Net cash flows = Inflows – Outflows
After all, the portfolio gained $1,000,000 in value over the month, but
$900,000 was due to an additional contribution—not the investment man- ager’s investment acumen. Investment managers should be credited only for the value that they add to the portfolio—not what is given to them.
However, the cash flow adjustment does not take into account the timing of any cash flows. In our example, if the portfolio was given the $900,000 on the first business day of the measurement period, then the 5.26% return would be fair because the portfolio manager or team would have had the money to invest over the entire measurement period as opposed to one day (January 31) in our example. But what if the money was added to the port- folio toward the end of the measurement period? In this instance, it would be unfair (and inaccurate) to calculate the return based on the assumption that the manager had the assets for the entire period.
So we need to apply a formula that takes contributions and distribu- tions into account and adjusts them based on their timing. In this chapter, we discuss two variations of the return calculation that take cash flows and
CFADWR EMV BMV Net Cash Flows BMV Net Cash Flows CFADWR
CFADWR CFADWR
= − −
+
−
= − −
−
=
−
= − = =
1 2 000 000 1 000 000 900 000
1 000 000 1
100 000 1 900 000 1
1 0526 1 0 0526 5 26
$ , , $ , , $ ,
$ , ,
$ ,
$ , ,
. . . %
$ , , $ , ,
$ , ,
$ , ,
$ , , . %
2 000 000 1 000 000 1 000 000
1 000 000
1 000 000 1 00 100 100
−
=
= × =
Performance Analysis 49
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their timing within a period into account: (1) the modified Dietz formula and (2) the time weighted formula. The former represents a very close ap- proximation of cash flow adjusted returns, while the latter provides for a more exact result.
Modified Dietz
The modified Dietz formula is named for its creator and is best described as a quick and relatively accurate approximation of the return for a portfo- lio over a discrete period of time. It adjusts for cash flows coming into and going out of a portfolio by adjusting the cash flow amounts based on the actual amount of time that the assets were invested in the portfolio. In ad- dition, it can incorporate as many cash flows as needed.
Formula 4.7 (Modified Dietz Return)
where EMV = Ending market value.
BMV = Beginning market value.
Cont = Sum of all contributions.
Dist = Sum of all distribution.
NTWCF = Net time weighted cash flows.
where CF1= Cash flow #1.
CFn= Cash flow #n.
Note: Contributions are positive numbers; distributions are negative numbers.
The numerator of this formula simply calculates the dollar gain in the portfolio (EMV – BMV) and then adjusts that figure for the net cash flows (by subtracting all contributions and adding all distributions). The denom- inator is where the actual time adjustments take place (NTWCF).
The time weighted cash flow (TWCF) adjusts each cash flow to reflect only the amount of time the assets were actually invested in the portfolio (or available for investment in the portfolio). The net TWCF (NTWCF) would be the sum of the results of each individual TWCF. Using the previ-
NTWCF of Days Invested
Total # of Days in Period Cash Flow
CF CF1
=
×
∑
n # MDR EMV BMV Cont Dist
BMV NTWCF
= − − +
+
∑
( )50 EQUITY AND FIXED INCOME MANAGER ANALYSIS
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ous example, we would apply formula 4.6 to calculate the return using the modified Dietz methodology:
BMV: $1,000,000
EMV: $2,000,000
Contribution: $900,000 Contribution date: January 30
Distribution: None
Distribution date: N/a
Let’s calculate the modified Dietz return using the following steps:
Step 1: Calculate the TWCF for each cash flow that took place during the measurement period.
The number of days the contributed assets were available for invest- ment was 1 because the contribution took place on January 30—leaving only the 31st for any portfolio investments. The total number of days in the period (January) was 31, so the fraction of days available for invest- ment relative to the total number of days in the period was 1/31. We sim- ply multiply this fraction by the actual dollar amount of the cash flow ($900,000) to calculate the TWCF. The $29,032.26 represents the por- tion of the $900,000 contribution that we are actually going to apply to- ward the portfolio’s return for the month. Now, let’s calculate the modified Dietz return.
MDR=
= × =
$ ,
$ , 100 000, . . . %
1 029 032 26 0 0972 100 9 72
MDR= − − +
+
$ , , $ , , $ , $
$ , , $ , .
2 000 000 1 000 000 900 000 0 1 000 000 29 032 26
MDR MVE MVB Cont Dist
MVB NTWCF
= − − +
+
∑
( )TWCF of Days Invested
Total # of Days in Period Cash Flow TWCF
CF CF1
=
×
=
×
=
∑
#$ , $ , .
n
1
31 900 000 29 032 26
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We can now compare these results to the results that we achieved us- ing the cash flow adjusted dollar weighted return (formula 4.6), which we calculated to be 5.26%. This is in stark contrast to the return we cal- culated using the modified Dietz formula, which was 9.72%. The differ- ence in returns clearly highlights the impact that cash flow timing has on performance.
Time Weighted Returns
Time weighted returns utilize many of the formulas already discussed in this chapter to create a very accurate way of calculating returns that ac- count for cash flows during the review period. The only caveat is that it is necessary to value the portfolio every time there is a cash flow into or out of the portfolio, and this is not always possible. Single-manager portfolios can be valued relatively easily (depending on the underlying assets), but multiple-manager portfolios can be a bit of a challenge. For a pension plan, it would be nearly impossible to value every manager across every asset class each time monies are added, subtracted, or reallocated. Liquidity also plays a part, as some portfolio holdings might not trade on a daily basis and, as a result, may not have daily pricing available. Lastly, there is a cost factor involved in valuing a portfolio (or series of portfolios) each time a cash flow takes place. As a result, most pension plans calculate perfor- mance only at regular intervals (most often on a monthly basis).
In this case, the modified Dietz formula is a better choice because it represents a close approximation based on the amount and timing of the cash flows. The modified Dietz formula does not require that the portfolio be revalued on the date of each cash flow.
Formula 4.8 (Time Weighted Return)
TWR = [(1 + PDR1)×(1 + PDR2)×. . . ×(1 + PDRn)] – 1
where PDR1= Period decimal return from start date to end of business on the date previous to the initial cash flow.
PDR2= Period decimal return from the end of business on the date previous to the initial cash flow to end of business on the date previous to the second cash flow.
PDRn= Period decimal return from the end of business on the date previous to the last cash flow to the end of the
measurement period.
This formula is the same as the one used to calculate chain-linked re- turns (formula 4.3). The only difference is that we are adjusting each pe-
52 EQUITY AND FIXED INCOME MANAGER ANALYSIS
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riod return based on cash flows into and out of the portfolio. Whenever a cash flow occurs, we must value the portfolio as of the instant prior to the cash flow. To keep things simple, we will value the portfolio as of the end of the previous business day. So if a cash flow takes place on the 15th of a given month, we will value the portfolio as of the end of the previous business day. If the 15th falls on a Wednesday, then we will value the port- folio as of the end of business on Tuesday; if the 15th falls on a Monday, we will use the previous Friday as the valuation date. The same holds true for holidays.
From a total performance perspective, valuing the portfolio on the prior business day makes sense; however, it might be unfair to value the underlying investment manager’s portfolio under any of these three conditions:
1. Very large cash flow.The time weighted return formula as stated previ- ously assumes that the assets contributed to a portfolio are invested in- stantly on the date of the cash flow. In the real world, this is often not practical. Liquidity and market conditions may force an investment manager to invest newly contributed cash over a period of days, weeks, and sometimes months.
2. Liquidity of both the market and the underlying investments slated for purchase.An example of problems due to market liquidity would in- clude periods of time when investors are holding back from the market or when investors are panic selling. In either case, it might be difficult or unwise to rush and purchase securities. An example of problems due to individual security liquidity would include the purchase of small-cap or micro-cap stocks that typically trade less frequently than large-cap stocks. In this case, a rush to purchase less liquid small-cap or micro-cap stocks might cause an artificial run-up in their prices, re- sulting in a negative impact on the underlying portfolio.
3. Contribution dates.Some investment managers specify certain dates in which they will accept new assets or make portfolio distributions. We see this more often in traditional commingled funds, hedge funds, and private equity funds. The cash flow dates can be stated as monthly, quarterly, semiannually, or at any interval the investment manager and underlying clients contractually agree to.
To better understand the time weighted return calculation, we calcu- late the return for the following portfolio based on the market values and cash flows indicated.
Example We made a $1,000,000 investment in the CAM portfolio on De- cember 31. The portfolio’s value at the end of January was $2,000,000.
Performance Analysis 53
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