When measuring returns over a single period the holding period return (HPR) is used. HPRs can be calculated using the following formula:
If an investor started with a $1,000 portfolio, that is now worth $1,100, we could calculate the HPR as follows:
HPRs can also be adjusted to take into account dividends or interest received during the period. Assume in the example above the portfolio generated $25 in income:
In this instance, the HPR increases from 10% to 12.5%.
In real life, it is often common for portfolio managers to make additional investments throughout the year. Let’s assume that a PM created a portfolio with an initial investment of $1,500,000. Throughout the course of the year the PM made three additional deposits of $60,000 each.
When calculating HPRs given the assumptions above, the beginning and ending values must be taken into account before and after the deposit was made. We can see how the $60,000 deposits affected the beginning and ending market values on the day the deposit was received in the table below:
| Sub-Period | Cf | Beg MV | End MV |
| Jan 1, 2020 to Jan 31, 2020 | |||
| Jan 1, 2020 to Jan 15, 2020 | $ – | $ 1,500,000.00 | $ 1,550,000.00 |
| Jan 16, 2020 to Jan 31, 2020 | $ 60,000.00 | $ 1,610,000.00 | $ 1,615,000.00 |
| Feb 1, 2020 to Feb 29, 2020 | $ – | $ 1,615,000.00 | $ 1,650,000.00 |
| Mar 1, 2020 to Mar 31, 2020 | $ – | $ 1,650,000.00 | $ 1,625,000.00 |
| Apr 1, 2020 to Apr 30, 2020 | |||
| Apr 1, 2020 to Apr 7, 2020 | $ – | $ 1,625,000.00 | $ 1,630,000.00 |
| Apr 8, 2020 to Apr 30, 2022 | $ 60,000.00 | $ 1,690,000.00 | $ 1,685,000.00 |
| May 1, 2020 to May 31, 2020 | $ – | $ 1,685,000.00 | $ 1,700,000.00 |
| Jun 1, 2020 to Jun 30, 2020 | $ – | $ 1,700,000.00 | $ 1,710,000.00 |
| Jul 1, 2020 to Jul 31, 2020 | |||
| Jul 1, 2020 to Jul 5, 2020 | $ – | $ 1,710,000.00 | $ 1,712,000.00 |
| Jul 6, 2020 to Jul 31, 2020 | $ 60,000.00 | $ 1,772,000.00 | $ 1,760,000.00 |
| Aug 1, 2020 to Aug 28, 2020 | $ – | $ 1,760,000.00 | $ 1,750,000.00 |
Let’s examine the effect the deposits had in the month of January. On January 15th, the portfolio had an ending market value of $1,550,000, the deposit of $60,000 received on January 16th is credited to the ending market value on the 15th, for a total beginning market value of $1,610,000 on the 16th.
If the ending market value on January 31st was $1,615,000, in order to calculate the HPR for the month, we must calculate the HPR for each of these two sub-periods and geometrically link them together as follows:
| Sub-Period | Cf | Beg MV | End MV | Monthly Ret | HPR |
| Jan 1, 2020 to Jan 31, 2020 | 3.65% | ||||
| Jan 1, 2020 to Jan 15, 2020 | $ – | $ 1,500,000.00 | $ 1,550,000.00 | 3.33% | |
| Jan 16, 2020 to Jan 31, 2020 | $ 60,000.00 | $ 1,610,000.00 | $ 1,615,000.00 | 0.31% |
Based on the table above, the HPR return after geometrically linking the sub-periods of 1/1/20 – 1/15/20 and 1/6/20 to 1/31/20 is 3.65%. Geometrically linking returns can be done as follows:
Let’s use the same methodology of geometrically linking returns to calculate the HPRs of this portfolio over the eight month period:
| Sub-Period | Cf | Beg MV | End MV | Monthly Ret | HPR |
| Jan 1, 2020 to Jan 31, 2020 | 3.65% | ||||
| Jan 1, 2020 to Jan 15, 2020 | $ – | $ 1,500,000.00 | $ 1,550,000.00 | 3.33% | |
| Jan 16, 2020 to Jan 31, 2020 | $ 60,000.00 | $ 1,610,000.00 | $ 1,615,000.00 | 0.31% | |
| Feb 1, 2020 to Feb 29, 2020 | $ – | $ 1,615,000.00 | $ 1,650,000.00 | 2.17% | 2.17% |
| Mar 1, 2020 to Mar 31, 2020 | $ – | $ 1,650,000.00 | $ 1,625,000.00 | -1.52% | -1.52% |
| Apr 1, 2020 to Apr 30, 2020 | 0.01% | ||||
| Apr 1, 2020 to Apr 7, 2020 | $ – | $ 1,625,000.00 | $ 1,630,000.00 | 0.31% | |
| Apr 8, 2020 to Apr 30, 2022 | $ 60,000.00 | $ 1,690,000.00 | $ 1,685,000.00 | -0.30% | |
| May 1, 2020 to May 31, 2020 | $ – | $ 1,685,000.00 | $ 1,700,000.00 | 0.89% | 0.89% |
| Jun 1, 2020 to Jun 30, 2020 | $ – | $ 1,700,000.00 | $ 1,710,000.00 | 0.59% | 0.59% |
| Jul 1, 2020 to Jul 31, 2020 | -0.56% | ||||
| Jul 1, 2020 to Jul 5, 2020 | $ – | $ 1,710,000.00 | $ 1,712,000.00 | 0.12% | |
| Jul 6, 2020 to Jul 31, 2020 | $ 60,000.00 | $ 1,772,000.00 | $ 1,760,000.00 | -0.68% | |
| Aug 1, 2020 to Aug 28, 2020 | $ – | $ 1,760,000.00 | $ 1,750,000.00 | -0.57% | -0.57% |
| Jan 1, 2020 to Aug 28, 2020 HPR | 4.66% |
Now, we can geometrically link each of the months together for a total HPR of 4.66% for the period of January 1st to August 28th.
The Excel model used to calculate multiple HPRs can be found here.
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