Quantitative Methods - Money Vs. Time-Weighted Return

Money-weighted and time-weighted rates of return are two methods of measuring performance, 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 environment 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 outflows. When applied to an investment portfolio:

1. The cost of any investment purchased
2. Reinvested dividends or interest
3. Withdrawals

1.The proceeds from any investment sold
2.Dividends or interest received

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 financial 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 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 account has annual withdrawals to provide a retiree with income, and the manager does relatively poorly in the early years (when the account is larger), but improves in later periods 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 = ending market value,
D1 = dividend/interest inflows, CF1 = cash flow received at period end (deposits subtracted, withdrawals 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

= [(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?

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%.

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 figure 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.

For money-weighted returns covering a single period, we know PV (inflows) - PV (outflows) = 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 received at the end. If the period is any time frame other than one year, take (1 + the result), multiply by 1/Y and subtract 1 to find the annualized return.

Calculating Yield
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