We were so far concerned with single-period returns. Now, we will talk a bit about multi-period returns. Let's say on November 1, 2011 you bought some shares for a price of $50.00 and after one month on December 1, 2011 they were worth $52.00. So, if you had sold your shares at that time your return would be ($52.00 - $50.00) / $50 = 4.00%. But, you did not. Instead, you held them one more month and sold them on January 3, 2012 for a price of $53.30. This implies a monthly return of ($53.30 - $52.00) / $52.00 = 2.50% in December.
The question is, what is your holding period return (HPR) over the 2 months of November and December? There are 2 ways of finding the answer. First, the simple way. Use your purchasing price of $50 in November and selling price of $53.30 in January to calculate your HPR as ($53.30 - $50.00) / $50 = 6.60%. But, you need to be careful. You can't use this way if there were dividend payments and/or stock splits during your holding period. You would get a wrong answer.
The difficult way involves: (1) calculating returns month by month, (2) compounding monthly gross returns, and (3) subtracting 1 at the end. You should prefer this way, even though it is harder, because if your holding period is long (1 year or more), it is quite likely that there were dividend payments or stock splits during the time you owned the stock, in which case the simple way discussed above would not work.
Ok, let's do it the harder way. We have already calculated the monthly net returns in the first paragraph: 4.00% in November and 2.50% in December. Then, compounding monthly gross returns and subtracting 1 yields:
which is equal to the figure we obtained using the simple way. In general, the holding period return over T number of periods is calculated as:
Π is the multiplication symbol. It tells us to compound the returns from the first period t=1 to the last period t=T and Rt is the total return in period t.
Next tutorial: Dividend reinvestment
Previous tutorial: Nominal vs real rate of return
The question is, what is your holding period return (HPR) over the 2 months of November and December? There are 2 ways of finding the answer. First, the simple way. Use your purchasing price of $50 in November and selling price of $53.30 in January to calculate your HPR as ($53.30 - $50.00) / $50 = 6.60%. But, you need to be careful. You can't use this way if there were dividend payments and/or stock splits during your holding period. You would get a wrong answer.
The difficult way involves: (1) calculating returns month by month, (2) compounding monthly gross returns, and (3) subtracting 1 at the end. You should prefer this way, even though it is harder, because if your holding period is long (1 year or more), it is quite likely that there were dividend payments or stock splits during the time you owned the stock, in which case the simple way discussed above would not work.
Ok, let's do it the harder way. We have already calculated the monthly net returns in the first paragraph: 4.00% in November and 2.50% in December. Then, compounding monthly gross returns and subtracting 1 yields:
(1 + 4.00%) * (1 + 2.50%) - 1 = 6.60%
HPRT = Π ( 1 + Rt ) - 1 where t ∈ [1, T]
Next tutorial: Dividend reinvestment
Previous tutorial: Nominal vs real rate of return
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