Calculating Returns

Worksheet: Calculating Returns The Basics Say that you started with $2,000 and a year later you have $2,500. You can calculate the return in two ways: 1) 2,500/2,000 = 1.25 This is the return relative. Subtract 1 to get the return. = 0.25 or 25% 2) (2,500-2,000)/2000 = 500/2000 = 0.25 or 25% Negative returns are handled the same way. You started with $2,000 and a year later you have $1,500: 1) 1,500/2,000 = .75 (return relative). Subtract 1 to get the return of -0.25 or -25% 2)(1,500-2,000)/2,000 = -500/2000 = -0.25 or -25% Remember, when calculating returns, you use the initial value as the base. Income vs. Capital Gains. For some purposes (such as taxes) it is important to distinguish where the returns came from, either from income or capital gains. For example, you bought a stock for $32, one year later you received a dividend of $2 and then you sold the stock for $36. Payments to holders of the asset (such as dividends) are income, while increases in the value of the asset are capital gains. In this case, you earned $2 of income and $4 of capital gains. Your return from income was (2/32) = 0 .0625 or 6.25% and your return from capital gains was (4/32) = 0.125 or 12.5% Your total return was the sum of the two, 0.1875 or 18.75% Investments of Less Than One Year When measuring investment returns, it’s typical to use annual rates to make it easier to compare returns across different investments. When investing for less than one year, we will need to convert the return to an annual rate. Say that we invested $1,000 for 3 months and earned 2%. The return over the time of the investment, the 2%, is called the holding period return. To put it on an annual basis (the annualized return), we ask how much we would have if we earn the same rate for the entire year. To do this, we use our compounding formula. If we are investing at an annual rate x for n years, our return for entire period, r, is given by: (1+x)n = 1+r Since there are 4 three-month periods in a year, n equal 4. (1+0.02)4 – 1 = r = 0.0824 or 8.24% Earning 2% for three months implies earning 8.24% for an entire year. However; it’s important to remember that this assumes that you can earn the same return for the remainder of the year, which may not be true. Dealing with Multiple Years If you are making an investment with a payoff several years in the future, you also need to know how to convert your return to an annual rate. Say that you started with $2,000 and ended with $3,000 two years later. That’s $1,000 over two years, or $500 per year, or approximately a 25% annual return. However, it’s only an approximation because we’ve ignored the effect compounding. To do this correctly, we follow a two step procedure. Step 1: Determine the holding period return. In this case, you earned $1,000 on $2,000 for a 50% holding period return. Step 2. Convert this to an annual number. The annualization formula is just a rearrangement of our compounding formula. Rearranging to find x, x = (1+r)(1/n) – 1 In our problem, (1+0.50)(1/2) – 1 = 0.2247 or 22.47% Multiple Cash Flows Say that you’ve made payments into a fund for a number of years and you want to know the return over the entire investment. When calculating your return, you want to include all of payments you made, not just the initial investment. In this situation, you can use the internal rate of return, or IRR, to measure the return on your investment. The IRR is the interest rate that sets the present value of all the payments equal to 0. If you invested $300 at the start of the year each year for three years and have $1,500 at the end of the third year, your IRR is the solution to this equation, −300 − 300 (1+𝑟) − 300 (1+𝑟)2 + 1500 (1+𝑟)3 =0 In practice, you would want to use the IRR function of Excel to solve this (the answer is 27.82%). Problems 1. You started with $11,000 and one year later you have $12,500. What was your rate of return? 2. You started with $8,000 and one year later you have $7,200. What was your rate of return? 3. You bought a stock for $20 and a year later sold it for $18. You were paid $3 in dividends. What was your capital gain rate of return, you income rate of return and your total rate of return? 4. You started with $130,000 and four years later you have $150,300. What was your annual rate of return? 5. You started with $20,000 and two years later you have $18,000. What was your annual rate of return? 6. You started with $190,000 and six months later you have $210,000. What was your annual rate of return? Answers 1. 13.64% 2. -10% 3. -10%, 15%, 5% 4. 3.69% 5. -5.13 6. 22.16%