Money-weighted and time-weighted rates of return are two methods for measuring investment performance. The money-weighted rate considers cash flows, so it depends on an individual investor's activities. The time-weighted rate ignores cash flows and measures only market performance over a period. It is preferred for evaluating managers because it is not affected by deposits or withdrawals outside their control. The document provides examples of calculating each type of return for portfolios with cash flows over multiple periods.

1. TWRR & LIRR
Sourav Sir’s Classes

2. Money Vs. Time-Weighted Return
Money-weighted and time-weighted rates of return are two methods of measuring per-
formance, or the rate of return on an investment portfolio. Each of these two approaches
has particular instances where it is the preferred method. Given the priority in today's en-
vironment on performance returns (particularly when comparing and evaluating money
managers), the CFA exam will be certain to test whether a candidate understands each
methodology.
Money-Weighted Rate of Return
A money-weighted rate of return is identical in concept to an internal rate of return: it is
the discount rate on which the NPV = 0 or the present value of inflows = present value of
outflows. Recall that for the IRR method, we start by identifying all cash inflows and out-
flows. When applied to an investment portfolio:
Outflows
1. The cost of any investment purchased
2. Reinvested dividends or interest
3. Withdrawals
Inflows
1.The proceeds from any investment sold
2.Dividends or interest received
3.Contributions
Example:
Each inflow or outflow must be discounted back to the present using a rate (r) that will
make PV (inflows) = PV (outflows). For example, take a case where we buy one share of
a stock for $50 that pays an annual $2 dividend, and sell it after two years for $65. Our
money-weighted rate of return will be a rate that satisfies the following equation:
PV Outflows = PV Inflows = $2/(1 + r) + $2/(1 + r)2
+ $65/(1 + r)2
= $50
Solving for r using a spreadsheet or financial calculator, we have a money-weighted rate
of return = 17.78%.
Exam Tips and Tricks
Note that the exam will test knowledge of the concept of money-
weighted return, but any computations should not require use of a fi-
nancial calculator
It's important to understand the main limitation of the money-weighted return as a tool
for evaluating managers. As defined earlier, the money-weighted rate of return factors all
cash flows, including contributions and withdrawals. Assuming a money-weighted return
is calculated over many periods, the formula will tend to place a greater weight on the

3. performance in periods when the account size is highest (hence the label money-
weighted).
In practice, if a manager's best years occur when an account is small, and then (after the
client deposits more funds) market conditions become more unfavorable, the money-
weighted measure doesn't treat the manager fairly. Here it is put another way: say the ac-
count has annual withdrawals to provide a retiree with income, and the manager does rel-
atively poorly in the early years (when the account is larger), but improves in later peri-
ods after distributions have reduced the account's size. Should the manager be penalized
for something beyond his or her control? Deposits and withdrawals are usually outside of
a manager's control; thus, a better performance measurement tool is needed to judge a
manager more fairly and allow for comparisons with peers - a measurement tool that will
isolate the investment actions, and not penalize for deposit/withdrawal activity.
Time-Weighted Rate of Return
The time-weighted rate of return is the preferred industry standard as it is not sensitive to
contributions or withdrawals. It is defined as the compounded growth rate of $1 over the
period being measured. The time-weighted formula is essentially a geometric mean of a
number of holding-period returns that are linked together or compounded over time (thus,
time-weighted). The holding-period return, or HPR, (rate of return for one period) is
computed using this formula:
Formula 2.8
HPR = ((MV1 - MV0 + D1 - CF1)/MV0)
Where: MV0 = beginning market value, MV1 = end-
ing market value,
D1 = dividend/interest inflows, CF1 = cash flow re-
ceived at period end (deposits subtracted, withdraw-
als added back)
For time-weighted performance measurement, the total period to be measured is broken
into many sub-periods, with a sub-period ending (and portfolio priced) on any day with
significant contribution or withdrawal activity, or at the end of the month or quarter. Sub-
periods can cover any length of time chosen by the manager and need not be uniform. A
holding-period return is computed using the above formula for all sub-periods. Linking
(or compounding) HPRs is done by
(a) adding 1 to each sub-period HPR, then
(b) multiplying all 1 + HPR terms together, then
(c) subtracting 1 from the product:
Compounded time-weighted rate of return, for N holding periods

4. = [(1 + HPR1)*(1 + HPR2)*(1 + HPR3) ... *(1 + HPRN)] - 1.
The annualized rate of return takes the compounded time-weighted rate and standardizes
it by computing a geometric average of the linked holding-period returns.
Formula 2.9
Annualized rate of return = (1 + compounded
rate)1/Y
- 1
Where: Y = total time in years
Example: Time-Weighted Portfolio Return
Consider the following example: A portfolio was priced at the following values for the
quarter-end dates indicated:
Date Market Value
Dec. 31, 2003 $200,000
March 31, 2004 $196,500
June 30, 2004 $200,000
Sept. 30, 2004 $243,000
Dec. 31, 2004 $250,000
On Dec. 31, 2004, the annual fee of $2,000 was deducted from the account. On July 30,
2004, the annual contribution of $20,000 was received, which boosted the account value
to $222,000 on July 30. How would we calculate a time-weighted rate of return for 2004?
Answer:
For this example, the year is broken into four holding-period returns to be calculated for
each quarter. Also, since a significant contribution of $20,000 was received intra-period,
we will need to calculate two holding-period returns for the third quarter, June 30, 2004,
to July 30, 2004, and July 30, 2004, to Sept 30, 2004. In total, there are five HPRs that
must be computed using the formula HPR = (MV1 - MV0 + D1 - CF1)/MV0. Note that
since D1, or dividend payments, are already factored into the ending-period value, this
term will not be needed for the computation. On a test problem, if dividends or interest is
shown separately, simply add it to ending-period
value. The ccalculations are done below (dollar amounts in thousands):
Period 1 (Dec 31, 2003, to Mar 31, 2004):
HPR = (($196.5 - $200)/$200) = (-3.5)/200 = -1.75%.

5. Period 2 (Mar 31, 2004, to June 30, 2004):
HPR = (($200 - $196.5)/$196.5) = 3.5/196.5 = +1.78%.
Period 3 (June 30, 2004, to July 30, 2004):
HPR = (($222 - $20) - $200)/$200) = 2/200 = +1.00%.
Period 4 (July 30, 2004, to Sept 30, 2004):
HPR = ($243 - $222)/$222 = 21/222 = +9.46%.
Period 5 (Sept 30, 2004, to Dec 31, 2004):
HPR = (($250 - $2) - $243)/$243 = 5/243 = +2.06%
Now we link the five periods together, by adding 1 to each HPR, multiplying all terms,
and subtracting 1 from the product, to find the compounded time- weighted rate of return:
2004 return = ((1 + (-.0175))*(1 + 0.0178)*(1 + 0.01)*(1 + 0.0946)*(1 + 0.0206)) - 1 =
((0.9825)*(1.0178)*(1.01)*(1.0946)*(1.0206)) - 1 = (1.128288) - 1 = 0.128288, or
12.83% (rounding to the nearest 1/100 of a percent).
Annualizing: Because our compounded calculation was for one year, the annualized fig-
ure is the same +12.83%. If the same portfolio had a 2003 return of 20%, the two-year
compounded number would be ((1 + 0.20)*(1 + 0.1283)) - 1, or 35.40%. Annualize by
adding 1, and then taking to the 1/Y power, and then subtracting 1: (1 + 0.3540)1/2
- 1 =
16.36%.
Note: The annualized number is the same as a geometric average, a concept covered in
the statistics section.
Example: Money Weighted Returns
Calculating money-weighted returns will usually require use of a financial calculator if
there are cash flows more than one period in the future. Earlier we presented a case where
a money-weighted return for two periods was equal to the IRR, where NPV = 0.
Answer:
For money-weighted returns covering a single period, we know PV (inflows) - PV (out-
flows) = 0. If we pay $100 for a stock today, and sell it in one year later for $105, and
collect a $2 dividend, we have a money-weighted return or IRR = ($105)/(1 + r) + ($2)/(1
+ r) - $100 = $0. r = ($105 + $2)/$100 - 1, or 7%.
Money-weighted return = time-weighted return for a single period where the cash flow is

6. received at the end. If the period is any time frame other than one year, take (1 + the re-
sult), multiply by 1/Y and subtract 1 to find the annualized return.
Assumptions of Time-weighted returns
Time-weighted returns assume that all cash distributions (i.e. dividends, interest, etc) are
reinvested back into the portfolio. It also eliminates the effect of cash flows in and out of
the portfolio, in essence treating the portfolio as if there were a single investment at the
beginning of the measurement period.
How are Time-weighted returns calculated?
A quick example would help illustrate the point. Assume your portfolio's value was
$1,000 at the beginning of the month, and $1900 at the end of the month. On the 10th day
of the month, you deposit $250, and on the 20th day of the month you deposit another
$250. The overall value of your portfolio (after the deposits are made) on the 10th day is
$1,300, and $1,700 on the 20th day. Therefore, there are three "sub-periods"- the first in-
cludes days 1-10, the second days 11-20, and finally the third is for days 21-30.
In order to calculate the time weighted return, we first need to calculate the return of each
subperiod.
 The return of subperiod one is: [ ($1,300-$250)-$1,000 ] / $1,000 = 5%.
 The return of subperiod two is: [ ($1,700-$250)-$1,300 ] / $1,300 = 11.5%
 The return of subperiod three is: [ ($1,900-$1,700) ] / $1,700 = 11.8%
Finally, we compound the returns together to calculate the overall time-weighted rate of
return:
 Time-weighted rate of return: [(1+0.05)*(1+0.115)*(1+0.118)]^0.33 -1 = 9.39%
r

7. Example: Time-weighted rate of return for Investor 1
Investor 2 initially invested $250,000 on December 31, 2013 in the exact same portfolio
as Investor 1. On September 15, 2014, their portfolio was worth $290,621. They then
withdrew $25,000 from the portfolio, bringing the portfolio value down to $265,621. By
the end of 2014, the portfolio had decreased to $250,860.
Using the same process, Investor 2 ends up with the exact same time-weighted rate of
return for the year.
Example: Time-weighted rate of return for Investor 2

8. INTERNAL RATE OF RETURN
IRR is essentially a money-weighted return since cash contributions to the portfolio de-
termine the return of the portfolio. Total return, on the other hand, is a time-weighted re-
turn, in that the timing of cash contributions to the portfolio is irrelevant since the portfo-
lio is re-evaluated whenever there are cash inflows or outflows. It is time-weighted be-
cause only the time period over which the return is calculated matters. Think of time-
weighted return as the return on the prices of the securities in the portfolio and money-
weighted return as the return you receive on your money, based on when you invested it
during the time period.
IRR
Realized return (internal rate of return) is calculated consistently for both monthly and
daily data.
Suppose:
= the initial market value of a portfolio
= the ending market value of a portfolio
= a series of interim cash flows
then the Internal Rate of Return is the rate that equates the sum of net present value of all
cash flows to zero:
where are times when there are interim cash contributions

9. and are entered with a negative sign because they represent cash
outflows for the portfolio.
With an iterative algorithm, we find the that solves the equation and present it as the
realized return/IRR for the portfolio.
Total Return
Total Return is calculated differently for monthly and daily data
For monthly data, total return is calculated by geometrically linking the IRR for each in-
terim month. The approximation is used to avoid portfolio re-evaluation whenever there
are cash inflow or outflows. Generally speaking, the shorter the sub-sample period, the
more accurate the approximation is.
For daily data, we keep track of the portfolio value for each trading day. Obviously, the
portfolio is always re-evaluated when there are cash inflow or outflows. That’s why the
total return calculation for daily data is very accurate.
The difference between IRR and Total Return
These two returns are meant to be different. IRR is a money-weighted return, in that the
interim cash contributions to a portfolio will change the IRR or the portfolio. Because of
this, it is account-specific. On the other hand, total return is time-weighted return: the
timing of cash contributions is irrelevant and total return captures solely the market per-
formance during a specific time period, thus is market-specific.
An example will help to illustrate this. Suppose the investment horizon is three years and
the portfolio is composed of one security. Suppose the price of the security sits flat
($100) for the first two years, and doubles (to $200) on the third year. There are $100
cash contributions at the end of both the first and second years. The total return for the
three-year time period is 100% despite the interim cash contributions. The annualized
return for the portfolio is 26%.
On the other hand, the interim cash contribution matters for IRR calculation. Let’s con-
sider two scenarios, in one, the investor puts in an additional $100 at the end of the first
year, and in the second scenario, the investor puts in the additional $100 at the end of the
second year.
Suppose:

10. = IRR for the first scenario
= IRR for the second scenario
then the following equations must hold in calculating IRR:
For the first scenario:
For the second scenario:
In the second scenario, since the additional contribution spends less time idle earning no
return, the overall portfolio return is higher than the return for the first portfolio. This
example illustrates that cash contributions with different timing cause different IRR for
the portfolio even though the underlying securities earn the same total return during that
specified time period.

11. Regardless of the amounts both investors contributed or withdrew from the portfolio,
they ended up with the exact same return. This is precisely the result that should be ex-
pected. The time-weighted rate of return is not affected by contributions and withdrawals
into and out of the portfolio, making it the ideal choice for benchmarking portfolio man-
agers or strategies. If we compare their return to the returns of the MSCI Canada IMI
Index over the same period (which their portfolio manager was attempting to track), we
also get the same result of 9.79%.