Money-weighted and time-weighted rates of return

CFA level I / Quantitative Methods: Basic Concepts / Discounted Cash Flow Applications / Money-weighted and time-weighted rates of return

The money-weighted return is equal to the internal rate of return because it accounts for the timing and amount of all cash flows. So, the money-weighted return can be called as the discount rate that makes the present value of cash outflows equal to the present value of cash inflows.

PV(cash outflows) = PV(cash inflows)

Example 4: Calculating money-weighted return

Jason buys 200 shares of a company at the beginning of the first year for $18 per share. The stock price rises to $24 per share at the end of the first year. Then he further buys 300 more shares of the same company for $24 per share. He sells all the shares at the end of the second year for $21 per share. The stock also pays a dividend of $1 per share at the end of each year. Compute the money-weighted return earned by Jason.

Solution:

Let us first find out the cash flows.

CF₀ = Cash flow at the beginning of first year = -200*18 = -$3,600
CF₁ = Cash flow at the end of first year = 200*1 - 300*24 = -$7,000 (Dividend payment is a cash inflow and buying shares is a cash outflow)
CF₂ = Cash flow at the end of second year = 21*500 + 1*500 = $11,000.

Using the financial calculator:

[CF][2nd][CLR WORK]
3600[+/-][ENTER]
[↓]7000[+/-][ENTER]
[↓][↓]11000[ENTER]
[IRR][CPT]

We will get the IRR as 2.80 percent which is also equal to the time-weighted return earned by Jason.

The time-weighted return measures the compound rate of growth of money invested in the portfolio over a stated measurement period. It is not affected by the cash withdrawals or cash additions to the portfolio. The holding period returns are averaged over time to calculate the time-weighted return. The following steps are followed to calculate the time-weighted return:

(1) Price the portfolio immediately before any addition to or withdrawal of funds from the portfolio. Divide the overall evaluation period into sub-periods based on cash inflows and outflows.
(2) Calculate the holding period return on the portfolio for each sub-period.
(3) Link the holding period returns to obtain an annual rate of return for the year. If the investment is for more than one year, take the geometric mean of the annual returns to obtain the time-weighted rate of return over the measurement period.

Time-weighted return = [(1+ HPR₁)(1+HPR₂)....(1+HPR_N)]^(1/N) - 1
Where
HPR_N =Holding period return of Nth year

Example 5: Calculating time-weighted return

Jason buys 200 shares of a company at the beginning of the first year for $18 per share. The stock price rises to $24 per share at the end of the first year. Then he further buys 300 more shares of the same company for $24 per share. He sells all the shares at the end of the second year for $21 per share. The stock also pays a dividend of $1 per share at the end of each year. Compute the time-weighted return earned by Jason.

Solution:

We need to calculate the geometric average of annual holding period returns to get the time-weighted return.

For the first year,
P₀ = 18*200 = $3,600
P₁ = 24*200 = $4,800
D₁ = 1*200 = 200
HPR₁ = (4,800 - 3,600 + 200)/3,600 = 38.89 percent

For the second year,
P₁ = 24*500 = $12,000 (To calculate the holding period return we will assume that we will buy all the shares at the beginning of the holding period)
P₂ = 21*500 = $10,500
D₂= 1*500 = $500
HPR₂ = (10,500 - 12,000 + 500)/12,000 = -8.33 percent

Time-weighted return = [(1+ HPR₁)(1+HPR₂)]^0.5 - 1 = (1.3889*0.9167)^0.5 - 1 = 12.83 percent.

Jason earned a time-weighted return of 12.83 percent.

Which return is appropriate: The time-weighted return is preferred in the investment industry. The time-weighted return ignores the impact of cash withdrawal and cash addition which is generally not at the manager's discretion. If the cash additions and withdrawals from the portfolio are per the manger's discretion, then money-weighted return should be preferred.

Consider the hypothetical case of two portfolios. Manager A manages Portfolio A, and Manager B manages Portfolio B. Both portfolios were started at the beginning of 1999 when the dot-com bubble was in the formation. The performance of the portfolios is given in the following table.

	Portfolio A			Portfolio B
Year	Amount at the end of year	Amount withdrew at the end of year	Net portfolio value at the end of year	Amount at the end of year	Amount withdrew at the end of year	Net portfolio value at the end of year
1998	100,000	0	100,000	100,000	0	100,000
1999	300,000	0	300,000	180,000	150,000	30,000
2000	350,000	0	350,000	31,000	25,000	6,000
2001	170,000	0	170,000	2,000	0	2,000
2002	95,000	0	95,000	1,000	0	1,000
2003	110,000	110,000	0	900	900	0
	MWR = 1.92 percent			MWR = 65.25 percent
	TWR = 13.34 percent			TWR = -22.53 percent

	Annual holding period return (in percent)
Year	Portfolio A	Portfolio B
1st	200.00	80.00
2nd	16.67	3.33
3rd	-51.43	-66.67
4th	-5.00	-50.00
5th	15.79	-10.00

In the above table, the money is regularly withdrawn from Portfolio B because the portfolio performed very poorly as compared to the peers while no amount was withdrawn from Portfolio A as it performed better than the peers. If we look carefully, we will notice that the Portfolio A has performed better than Portfolio B in every single year by a good margin. But still, its money-weighted return (MWR) is less than that of Portfolio A. That is because of the high cash outflows from Portfolio B before the extremely poor performance of the portfolio due to dot com bubble bust. Any sane investor would choose Manager A over Manager B, but if we go by money-weighted return, the Manager B has performed better. That's why the time-weighted return is preferred when the cash flows are not at the discretion of the managers.

When the cash flow is added (withdrawn) before the superior return period, then the money-weighted return is greater (less) than time-weighted return.
When the cash flow is added (withdrawn) before the poor return period, then the time-weighted return is greater (less) than money-weighted return.

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