APPENDIX 9B: ANNUALIZED RATES OF RETURN AND GROWTH Suppose that Investment A provided a total return of 3.1% during a four-month holding period, and Investment B s total return over a seven-month period was 5.25%. Although B produced the larger total return, it took a significantly longer time to do so. For a proper comparison of the performance of the two investments, we should compare their rates of return for the same time interval. One possibility would be to compare the percent return per month from each investment. However, there is a general preference in business for quoting and comparing percent returns per year (even when the actual holding period is much less than one year). An annualized rate of return is the percent return that would result if a short-term rate of return continues for an entire year. In economics and business, other short-term rates of change are also usually quoted as annualized rates. Two methods are used to annualize short-term returns. They make different assumptions regarding compounding. Simple Annualized Rate This approach simply extends the short-term percent return, on a proportionate basis, to a full year. For example, a return of 2.5% for a three-month holding period would be multiplied by four to give an annualized rate of 10%. This calculation is the reverse of the procedure used in Chapters 6 and 7 to prorate a simple annual interest rate to a partial year. Therefore, the annualized rate of return obtained by this approach is a simple rate of return it assumes that the short-term rate does not compound when extended to a full year. For the general case, the basic proportion is Simple annualized rate Short-term percent return Holding period 1partial year2 Multiplying both sides by short-term percent return, we obtain Simple annualized rate Short-term percent return Holding period Effective Annualized Rate By definition, the duration or term of a short-term investment is less than one year. The proceeds from a short-term investment are available for reinvestment. A rigorous method for annualizing a short-term rate of return should recognize this opportunity for compounding. The usual assumption is that you immediately reinvest the full maturity value in an identical short-term investment. This has the effect of 1. Making the compounding period equal to one term of the investment and 1 year m Term of the investment
2 Appendix 9B: Annualized Rates of Return and Growth 2. Making the periodic rate of return, i, equal to the percent return during one term of the short-term investment. To obtain the effective annualized rate of return, substitute these values for m and i into f (1 i) m 1. Note that m is not likely to be an integer. For example, the proceeds from a 60-day investment can, in principle, be reinvested or compounded 60 6.083 times during a year. The inclusion of compounding makes the effective annualized rate larger than the corresponding simple annualized rate. EXAMPLE 9BA CALCULATING ANNUALIZED RATES OF INFLATION If Statistics Canada reported that the consumer price index (CPI) rose from 106.2 to 106.5 in the previous month, what was a. the simple annualized rate of inflation for the month? b. the effective annualized rate of inflation for the month? The percent increase in the CPI during the month was a. i CPI (final) CPI(initial) CPI(initial) 100% Simple annualized rate Short-term percent change Partial year 12 months 1 month 0.2825% 3.39% b. Now, calculate the annual rate of inflation on the assumption that the inflation rate for the previous month (i 0.2825%) will be repeated and compounded every month for one year (m 12). f 11 i2 m 1 1.002825 12 1 0.0344 3.44% The effective annualized inflation rate is therefore 3.44%. 106.5 106.2 106.2 100% 0.2825% EXAMPLE 9BB ANNUALIZING A SHORT-TERM RATE OF ECONOMIC GROWTH Canada s gross domestic product (GDP) rose from $1050.880 billion in the third quarter to $1058.864 billion in the fourth quarter of the same year. During the fourth quarter, what was a. the simple annualized rate of economic growth (as measured by the GDP)? b. the effective annualized rate of economic growth (as measured by the GDP)? Calculate the rates accurate to the nearest 0.01%. The percent change in the GDP during the fourth quarter was GDP1final2 GDP1initial2 $1058.864 $1050.880 i 100% 100% 0.760% GDP1initial2 $1050.880
Appendix 9B: Annualized Rates of Return and Growth 3 a. Simple annualized rate Short-term percent change Partial year 12 months 3 months 0.760% 3.04% The economy grew at the simple annual rate of 3.04% during the fourth quarter. b. Now, calculate the annual rate of change on the assumption that the rate of GDP growth will be repeated and compounded for four successive quarters. f 11 i2 m 1 11 0.007602 4 1 1.0307 1 3.07% The GDP grew at an effective annualized rate of 3.07% during the fourth quarter. EXAMPLE 9BC CONVERTING A SIMPLE ANNUALIZED RATE TO AN EFFECTIVE ANNUALIZED RATE A $10,000 face value Treasury Bill, with 42 days remaining to maturity, was purchased to yield a 5% rate of simple interest. What effective annualized rate of return was earned on the T-bill? We can calculate the 42-day holding-period return from the given simple annualized rate of return (5%). Then use f (1 i) m 1 to obtain the effective annualized rate. The effective annualized rate is Holding-period return, Number of compounding periods in a year, f 11 i2 m 1 11.00575342 8.6905 1 0.0511 5.11% The effective annualized rate of return was 5.11%. i 42 5% 0.57534% m 42 8.6905 EXAMPLE 9BD CONVERTING A CASH DISCOUNT TO AN ANNUALIZED RATE OF RETURN Cash discounts offered on invoices usually represent an opportunity to earn a large annualized rate of return on investment. Consider the case where Sonya receives an invoice for $100 with terms 2/10, net 30. The two natural alternatives 6 facing Sonya are 1. pay $98 after 10 days, or 2. pay $100, 20 days later (at the end of the 30-day credit period). Investment was placed within quotation marks in the first sentence because the choice can be viewed as an investment decision. To put money to a better use than taking the cash discount, Sonya would need to be able to earn more than $2 on an investment of $98 for 20 days. 6 It is a fundamental tenet of managing accounts payable to use free credit to its limit. If you choose to take the cash discount, you should pay on the last day of the discount period. Otherwise, you should pay on the last day of the credit period.
4 Appendix 9B: Annualized Rates of Return and Growth At what simple and effective annualized rates of return on investment would Sonya be indifferent between taking the cash discount after 10 days versus investing $98 for 20 days and then paying the full price on the last day of the credit period? Sonya will be indifferent if $98 can earn exactly $2 in 20 days. The percent return for the 20 days would have to be i $2 100% 2.041% $98 The corresponding simple annualized rate of return would be 2.041% 37.25% 20 and the effective annualized rate would be f (1 i) m 1 (1.02041) /20 1 0.4459 44.59% The cash discount represents a simple annualized return on investment of 37.25% and an effective annualized return on investment of 44.59%. 7 EXAMPLE 9BE ANNUALIZED RETURNS ON MONEY MARKET MUTUAL FUNDS A money market mutual fund invests in Treasury Bills and other short-term debt investments. The rate of return on the portfolio changes from day to day as old investments (earning old rates) mature and are replaced by new investments (earning current rates). Each day, major newspapers report the current yield and the effective yield for each of approximately 100 money market mutual funds. The current yield is the simple annualized rate of return; the effective yield is the effective annualized rate of return. In both cases, the fund s percent return for the previous seven days is annualized. Suppose that ABC Money Market fund had a return of 0.089% for the most recent seven days. What current yield and effective yield would be reported for the fund in the financial pages? Current yield Simple annualized rate Short-term percent return Holding period days 0.089% 7 days 4.64% Effective yield Effective annualized return (1 i) m 1 (1.00089) /7 1 1.00089 52.143 1 0.0475 4.75% The financial pages would report a current yield of 4.64% and an effective yield of 4.75%. 7 Here is another interpretation of the numbers. Suppose the purchaser does not have the cash on hand to take advantage of the cash discount. An option is to borrow $98 for 20 days to pay the bill. It would be to Sonya s advantage to borrow at any effective interest rate below 44.59% rather than forgo the discount and pay $100 on the last day of the 30-day credit period.
Appendix 9B: Annualized Rates of Return and Growth 5 EXERCISE 9B Answers to the odd-numbered problems are at the end of the book. Calculate percentages accurate to the nearest 0.01%. 1. If the consumer price index rose by 0.5% over a two-month period, what were the simple and effective annualized rates of inflation during the two-month period? 2. The consumer price index rose from 131.2 to 132.1 during the second quarter of a year. What was the effective annualized rate of inflation during the quarter? 3. If the money supply increased from $331.12 billion to $333.81 billion in a single month, what were the simple and effective annualized rates of increase in the money supply during the month? 4. If the S&P/TSX Composite Index declined from 14,614 to 14,238 over a 50-day period, what were the simple and effective annualized rates of decline in the index during the period? 5. The Calgary Real Estate Board reports that house prices increased by 5% during the first seven months of the year. If prices continue to rise at the same rate for the subsequent five months, what will be the (compounded) increase for the entire year? 6. A bank pays a simple interest rate of 4.1% on 30-day to 179-day GICs of at least $100,000. What is the effective annualized rate of return a. on a 40-day GIC? b. on a 160-day GIC? 7. A T-bill with 125 days remaining to maturity is discounted to yield 4.6% simple interest. What is the effective annualized rate of return on the T-bill? 8. Neil s common stock portfolio increased in value over a two-month period from $78,900 to $84,300. What were the simple and effective annualized rates of total return over the period? 9. Danielle s shares in the Industrial Growth Fund, an equity mutual fund, dropped in price from $12.86 to $12.56 over a three-month period. What were the simple and effective annualized rates of return during the period? 10. If the holding-period return on a money market mutual fund for the most recent seven days is 0.081%, what current (simple) yield and effective annualized yield will be quoted for the fund in the financial press? 11. If the holding-period return on a money market mutual fund for the most recent seven days is 0.097%, what current (simple) and effective annualized yields will be quoted for the fund in the financial pages? 12. The current (simple annualized) yield on a money market mutual fund, based on the return for the most recent seven days, is 4.12%. What effective (annualized) yield will be reported for the fund? 13. The current (simple annualized) yield, based on the holding-period return for the most recent seven days, is reported for a money market mutual fund as 4.54%. What is the fund s corresponding effective (annualized) yield? 14. The terms of payment on an invoice are 3/10, n/90. What are the simple and effective annualized rates of return earned by taking the cash discount on the last day of the discount period instead of paying the full price on the last day of the credit period? 15. An income tax preparation service discounts income tax refunds at the statutory maximum amount of 15% on the first $300. For example, a taxpayer eligible for a $200 tax refund can sell the refund to the discounter for immediate payment of $170. What is the discounter s effective annualized rate of return if the time taken by the Canada Revenue Agency (CRA) to process the taxpayer s return and refund the tax is a. 25 days? b. 50 days? (Note: With the CRA s EFILE electronic tax return filing system, the turnaround time should be less than three weeks.)