Calculating Discrete Holding Period Returns

What are some of the basic properties of calculating returns?

• Expressed as a percentage gain or loss on an investment
• Can be calculated for the holding period, or, on an annualised basis
• Can be expressed as a discrete return, or, on a continuously compounded basis

How do we calculate holding period returns?

Holding Period Return = Sell price / Buy price - 1

Or

Holding Period Return = (Sell price - Buy price) / Buy price

For a price series

Holding Period Return = End value / Start value - 1

1. The holding period return is simply the selling price divided by the buying price minus 1.
2. Or, looking at the middle formula, it’s the profit as a percentage of the cost.
3. If you currently hold an investment and you want to calculate the periodic returns since you made the investment, simply take a price (end value), divided by the previous price (start value) and subtract 1.

Using the information below, how do we calculate the holding period for both investors?

• Investor A - has made $100,000 in one year
• Investor B - has made $50,000 in six months

1. The holding period for Investor A is 10%, 100,000/1,000,000
2. The holding period for Investor B is 50%, 50,000/100,000

Investor B has a greater return in a shorter time, and thus has clearly done better.

How do we calculate annualised returns?

If the holding period is less than a year, we use the top formulae. And if the holding period is greater than a year, we use the bottom one.

Annual Return = Holding period return x 365 / holding period (days)

Or

Annual Return = (Sell price / Buy price)^{(1/years held) - 1}

(Multi-period TVM formula re-arranged to solve for r)

For the following examples, what is the holding period return and annual return?

• Buy Price = 250
• Sell Price = 265
• Holding Period = 30 days

1. Holding period return: (265/250) - 1 = 6%
2. Annual return: (6 x 365)/30 = 73%

• You bought shares 10 years ago for 150, and you sold them for 650.

1. Holding period return: (650/150) - 1 = 333.3%
2. Annualised return - As the holding period is greater than one year, we do the following calculation: (650/150)^{(1/10) - 1} = 15.79%