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 Portfolio Return 

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A portfolio's Total Return is the best measure of investment progress. Total Return [TR] is the sum of capital gains or losses, and investment income (interest and dividends). It might seem simple at first glance to compute Total Return but, in a real portfolio with cash flows (capital deposits and withdrawals, interest and dividends) coming in at different times, the computation becomes more difficult.

You can't just combine the Total Return numbers published in a mutual fund listing or a stock guide because those numbers assume you made no capital additions to or withdrawals from your position, and that you reinvested all dividends as they were paid. We'll develop the correct procedure in Four Steps and give you several numerical examples so you can follow the process.

Step 1: Portfolio with Capital Gains/Losses [No Cash Flows; No Flow Timing]

In this overly simplified case, presume that the only value changes in the portfolio comes from capital gains or losses. That is, there are NO cash flows...

Let     BV = the beginning dollar value of the portfolio, and

          EV = the ending dollar value of the portfolio.

Then the Total Return [TR], expressed as a percentage, is

TR = 100 (EV - BV) / BV

Suppose a portfolio has a Beginning Value [BV] of $100,000 and an Ending Value [EV] of $125,000 in some period of time, then its Total Return in that time period would be...

TR = 100 ($125,000 - $100,000) / $100,000 = 25%

Step 2: Portfolio with Cash Flows [No Flow Timing]

Now let's suppose there are cash flows [deposits, withdrawals, interest, dividends] during the calculation period.  An approximate way to compute their effect is to subtract the total deposits received during the period and add back the total withdrawals, both against the Ending Value. This effectively isolates the amount of capital that was working during the period.

Let                                     Cash Flow [CF] = Total Deposits - Total Withdrawals

Then the Total Return formula becomes...

TR = 100 [EV - (BV + CF)] / BV

To the numerical parameters in the Step 1 example, let there also be a $12,000 cash deposit,  a $5,000 cash withdrawal, and total Interest and Dividends of $3,000 received.

Then                                          CF = $12,000 + $3,000 - $5,000 = $10,000  

and the Total Return is

TR = 100 [$125,000 - ($100,000 + $10,000)] / $100,000 =  15%

The Total Return is lower is this case because we effectively had $10,000 more capital at work in the period.

Step 3: Portfolio with Time-Weighted Cash Flows

The trouble with the computation in Step 2 is that it doesn't take account of the timing of the cash flows. It should make a difference to the Total Return if a cash deposit comes in on the first day of a calculation period [so the money has the whole period to work] or on the last day. To take account of this timing we introduce Time-Weighted Cash Flow [TWCF].

Suppose a Deposit of "D" dollars [this might be a capital deposit or a dividend payment] comes into the portfolio on the  n-th  day of a calculation period that has N days in total. Then these dollars have  N+1-n  days to work during the N-day period and their Time-Weighted Cash Flow contribution is

Contribution of Deposit "D" = D [(N+1-n) / N]

This satisfies our intuition because if the deposit is made on the first day of the period, n=1, and the full contribution will work for the entire period since the factor D(N+1-1)/N = DN/N = D. On the other hand, a deposit made on the last day of the period, the N-th day, will make a D(N+1-N)/N = D/N contribution to the cash flow.

Similarly, a Withdrawal of "W" dollars coming in the the m-th day of an N-day period makes a contribution...

Contribution of Withdrawal "W" = W [(N+1-m) / N]

If we sum all these Time-Weighted Cash Flow contributions, we have...

Time-Weighted Cash Flow [TWCF]= (Sum of all time-weighted Deposits) - (Sum of all time-weighted Withdrawals)

Now the Total Return formula to use is

TR = 100 [EV - (BV + CF)] / [BV + TWCF]

Using the same numerical examples as in Steps 1 & 2, let's additionally assume the $12,000 cash deposit is made on the 5th day of a 30-day month,  the $5,000 cash withdrawal is made on the 15th of the month, and total Interest and Dividends of $3,000 are received on the 22nd day. Then the Time-Weighted Cash Flow is...

TWCF = $12,000 [(30+1- 5)/30] + $3,000 [(30+1- 22)/30] - $5,000 [(30+1- 15)/30] or

TWCF = $12,000 [0.867] + $3,000 [0.3] - $5,000 [0.533] = $8,639.

Putting this in the Total Return Formula and using the previous values for EV, BV and CF gives...

TR = 100 [$125,000 - ($100,000 + $10,000)] / [$100,000 + $8,639] = 13.8%

This Total Return is lower than we computed in Step 2 because the time-weighted deposits were greater than the time-weighted withdrawals. Had the $5,000 withdrawal been made at the very beginning of the period of the month and the $12,000 deposit at the very end, the TWCF would have been negative and the Total Return would have been higher than in Step 2.

Step 4: "Geometric Linking" of Multi-Period Total Returns

The last issue we must consider is how to "link" Total Returns from several time periods to get a composite Total Return for the total time span of those periods. This arises, for instance, if you want to compute an Annual Total Return from twelve monthly Total Returns, or from four quarterly Total Returns. It also applies to linking Annual Total Returns over a number of years.

It's NOT appropriate to simply algebraically sum Total Returns from several time periods. Take the example of a $100 investment that loses 50% in a first time period, then gains 50% in a second time period. A simple algebraic sum of those two Total Returns says there'd be a zero change [-50%+50% = 0]... in other words, the investor would predict he'd have his original $100 at the end of the second period. In fact, the first period loss of 50% would leave the investor with just $50, while the second period gain of 50% [on a base of $50!] would just boost his final stake to $75. Hence, this investor's true two-period Total Return was -25%, that is, a 25% loss!

The procedure that correctly combines multi-period total returns is called "geometric linking." It works like this. Let the Total Return in a first period [of arbitrary time length] be designated TR1, the Total Return in a second period be TR2, etc. Additionally TR1, TR2,...  must be expressed as DECIMALs, not percentages here! [That is, if a 23% Total Return was achieved in the first period, then TR1 = 0.23; if a -12.4% Total Return was achieved in a second period, then TR2 = -0.124] The n-period Total Return, in percent, is then appropriately given by

n-Period Total Return = 100 [(1+TR1) (1+TR2) (1+TR3) ... (1+TRn) - 1]

That is, we add "1" to the decimal Total Return for each period, multiply those factors together, subtract "1" from the product of terms for all periods, and multiply that by 100 to get an answer in percent.

Consider the simple two-period example we discussed above. In the first period, the Total Return was -50% and in the second period, the Total Return was +50%. Using the n-Period formula

2-period Total Return = 100 [(1- 0.5) (1+ 0.5) - 1] = 100 [(0.5)(1.5) -1] = -25%

That's the answer logic said we should get. "Geometric linking" effectively takes account of the fact that each period's return is applied to the ending value of the prior period.

How many time periods should one use? Ideally, starting a new time period on every transaction date gives the best accuracy. That's not practical for individual investors since one must have a complete evaluation of all asset values for the beginning and end of every time period. Mutual Funds do calculate their Total Return every business day so they use about 250 periods to calculate an Annual Total Return. Investment Managers typically do monthly Total Return computations and geometrically link twelve of those to get Annual Total Return. Individual investors with relatively low transaction frequency should get a good estimate if they calculate Total Return quarterly and geometrically link four quarters to derive an Annual Total Return.

 

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