# Rate of Return on Portfolio

Actual or Realized or Historic or Ex-post facto ROR on a portfolio for a one year investment is : Realized Rp = (Wealth at the end - Wealth at the beginning) / Wealth at the beginning. Symbol Rp will be used to denote rate of return on a portfolio. Wealth at the end includes both the wealth in the form of shares at ending price and the wealth in the form of cash received during the year as dividends. This can be written as: Realized Rp = (W1 – W0) / W0

The above formulation for realized Rp is correct if during the year there were no inflows or outflows of cash in this portfolio. In real life only closed – end mutual funds have this quality; otherwise open end mutual funds as well as the portfolios managed by individuals may , and do, experience inflows and outflows within the year. In such cases realized Rp calculations are not tha straight forward. Two alternative methods of calculating actual / realized Rp are propose: time weighted Rp and Rupee weighted Rp. Time weighted Rp is correct method. Let us see why? Time Weighted Returns vs Rupee weighted returns Of Portfolio Time weighted Rp

It is common practice the investors and also for the open ended mutual funds to receive additional funds and invest those in the portfolio already constructed; these portfolios may also experience outflows during a period. For example:

Now MV of shares in a portfolio is 100 million rupees, by the end of the first month it declines to 96 million; but at that time another 5 million is injected in the portfolio and additional shares are bought using these new funds; then at the end of 2nd month MV of shares in the portfolio is 103 million. Question: what is the monthly ROR and what is the time weighted Rp for the holding period of 2-month. ROR for first month = (ending wealth - Beginning wealth) / beginning wealth

= (96 - 100 )/ 100

= - 4%

ROR for the 2nd month= (103 - 101) / 101 = 1.98%

Please note Beginning of 2nd month due to cash in flows wealth in portfolio = 96 + 5 = 101. Time weighted 2 –month holding period Rp =

[(1 + Rp of first month)*(1 + Rp of 2nd month)] – 1

[(1+ -0.04)*(1 + 0.0198)] -1

0.977 -1

-0.021 = -2.1%

Note it is geometric mean returns.

Rupee weighted Rp

Relevant cash flows are

MV of portfolio now 100; further investment by you at the end of first month 5; MV of portfolio shares at the end of 2nd month 103 million. First a monthly ROR is calculated; then it would be converted into a 2-month holding period ROR. Monthly ROR would be an IRR; to calculate that go to CASH mode of FC -100; in data editor enter -100 exe (as if you build the portfolio by investing in shares at time zero) -5 exe ( as if you invested further after one month)

103 exe ( MV of shares in the portfolio at the end of 2nd month) Esc

Solve IRR

You get -0.980297% as monthly IRR. Please note in decimal form it is 0.980296 /100 = 0.00980297 To convert it into a 2-month holding period Rp = (1 + monthly ROR)2 -1

= (1 + -0.00980297)2 -1

= (0.990197)2 -1

= 0.980498-1

= -0.0195

= -1.95%

This procedure gives 2-month holding period returns with 2- monthly compounding. There is another possibility of calculating 2-monthly returns with monthly compounding, if done that way the answer is slightly different: To convert it into a 2-month holding period Rp = monthly ROR *2

= -0.980297% * 2

= -1.96%

The difference between time weighted 2-month holding period returns (-2.1%) and rupee weighted 2-month holding period returns (-1.95%) may appear negligible; but it is not. In fact time weighted Rp is correct. Another example

A portfolio has MV of 50 million now, then by the end of first month it declined to 25 million, but at that time you as investor injected another 25 million; and at the end of 2nd month MV of shares in the portfolio was 100 million. Time weighted Rp :

ROR of first month = (25 - 50)...

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