Time-Weighted Returns

Time-Weighted Return (TWR) measures the performance of the portfolio manager. The amount of funds invested is 'neutralized' in the calculation of TWR because contributions and withdrawals by the client are not under the control of the fund manager. The time-weighted return over a certain period depends only on the length of this period an not on the amount invested - the return is 'time-weighted'.

'True' Time-Weighted Returns

The 'True Time-Weighted Method'  for the period 0 to T can be calculated by valuing the portfolio whenever contributions occur:

r(T) = prod[MV(t+1)/{MV(t)+C(t)}] - 1

This requires re-valuing the portfolio each time a net contribution occurs (which does not necessary mean 'daily valuation').

If the portfolio does not include the cash position, then every buy and sell decision creates a cash flow in the portfolio. Thus, the portfolio has to be re-valued every time a transaction takes place.

The term 'true' TWR is misleading insofar as TTWR are only a necessary, but not a sufficient conditions to calculate "true" returns: TTWR are free of distortions caused by approximations, but can still be "wrong" due to other reasons (for example, valuation issues, inconsistent price sources/prices, liquidity issues etc.).

Modified-Dietz Method

The modified Dietz method overcomes the need to know the valuation of the portfolio on the date of each cash flow by assuming a constant rate of return during the period. Each cash flow is weighted by the amount of time it is held in the portfolio. The formula is given by...

r(T) = {MV(T)-MV(0)-sum[C(t)]}/{MV(0)+sum[w(t)*C(t)]}

r(T)... Modified Dietz Return
MV(T)... Ending market value
MV(0)... Beginning market value
C(t)... Net contribution occurring on day t
w(t)... weight of the net contribution on day t...

w(i) = {T - t} / T

T... Total number of days
t...  day the net contribution occurs

The Modified Dietz method assumes that net contributions are invested at the end of the respective day they occur.

Dietz Method

The original Dietz method (also known as 'Midpoint Dietz Method') is obtained by setting w(i)=w= 0.5 for all i, implying that Dietz returns assume that all net contributions take place in the middle of the period...

r(T) = {MV(T)-MV(0)-C}/{MV(0)+0.5*C]}

or

r(T) = {MV(T)-0.5*C}/{MV(0)+0.5*C]} - 1

ICAA Method

ICAA: Investment Counsel Association of America

"Standards of Measurement and Use for Investment Performance Data", 1971

r(T) = {MV(T)-MV(0)-C-I)}/{MV(0)+0.5*C]}

I... income (all coupons, dividends, ...)

Same as Dietz method, but different treatment of investment income.

Relationship between TWR & MWR

There is a great deal of confusion out there about using IRR/MWR approximations to calculate TWR and in general the relationship between time- and money-weighted returns. Here are the facts...

• IRR is always a MWR, never a TWR. Being able to express MWR/TWR as mathematical approximations merely relates to a numerical relationship, and does not affect the fundamental difference between MWR and TWR of whether return is repoted inclusing or excluding the effect of the timing of net contributions.
• When calculating the IRR for small sub periods, accounting for net contributions and finally chain-linking them, the resulting return is a TWR approximation. The IRR calculated with the same data set over the whole period will give a MWR and most likely a very different result.
• The BAI Method is a methodology for approximating TWR with the eopIRR. The portfolio is valued every time a net contribution occurs and eopIRR is calculated. The time-weighted period returns are derived by chain-linking the eopIRR.
• In the Modified-BAI Method, time periods are set independently of the occurrence of net contributions. Usually, monthly or quarterly time periods are chosen. The IRR is then calculated for each month or quarter and chain-linked. Large cash flows lower the quality of the approximation. It is therefore recommended to value the portfolio during the time period if the net contribution exceed about 10% of the portfolio.
• The Midpoint-BAI Method also uses fixed time periods, but assumes that net contributions occur in the middle of the period. The IRR expression then simplifies to...
  
  MV(T) = MV(0)*{1+r(T)} + C*{1+r(T)}^0.5

The fact that IRR can be used to calculate TWR as well as MWR is practical and was very important when market valuation and computing power were expensive. These days have gone. Today, whenever possible, the 'true' time-weighted returnshould be calculated if the return independent of the timing of net controbutions is of interest.