On his 30th birthday, Bob wanted to start preparing for retirement on his 60th birthday by investing $10,000 a year at a rate of 10% per annum. On second thought, he realized that since he has so much time before retirement, he could delay starting the investment so that he now starts on his 40th birthday. In the first case, Bob would have invested for 30 years while in the second case he will only invest for 20 years. Thus, we would expect that the amount of money he will have after the 30-year investment will be more than the amount he will have after the 20-year investment. Which of the following best describes the relationship between the two investments?
A. The 20-year investment will yield a return that is about the same as what the 30-year investment will yield B. The 20-year investment will yield a return that is about two-thirds of what the 30-year investment will yield C. The 20-year investment will yield a return that is about one-half of what the 30-year investment will yield D. The 20-year investment will yield a return that is about one-third of what the 30-year investment will yield E. None of the above
Which of the following options will make more money?
Option 1: A one million dollar payment today
Option 2: A payment scheme where you start with 1 cent but the amount of money you are paid doubles every day for one month–that is, you are paid 1 cent today, 2 cents tomorrow, 4 cents the day after, and then 8 cents, and so on for the next one month (thirty days), which one would you choose?
A. One million dollars today B. 1 cent today, 2 cents tomorrow, 4 cents the day after, and then 8 cents, and so on for the next one month C. They will make the same amount of money D. There isn’t enough information to tell
Bob invests $200 today while Sue invests $2,000. However, Bob adds $200 to his investment after every year while Sue just leaves her initial $2,000 investment alone. If the money they invest grows at 10% per annum, which of the investments will have made more money after 20 years?
A. Bob’s investment B. Sue’s investment C. Both investments will make the same amount of money D. There isn’t enough information to tell
One dollar growing at a rate of 10% annually will become two dollars after ________. A. about 2 years B. about 5 years C. about 7 years D. about 10 years E. none of the above
Solution
Answer: C.
Let’s look at how the one dollar is going to grow over time. At the end of year 1, it will be $1.00 × 1.1 = $1.10 At the end of year 2, it will be $1.10 × 1.1 = $1.21 At the end of year 3, it will be $1.21 × 1.1 = $1.331 At the end of year 4, it will be $1.331 × 1.1 = $1.4641 At the end of year 5, it will be $1.4641 × 1.1 = $1.61051 At the end of year 6, it will be $1.61051 × 1.1 = $1.771561 At the end of year 7, it will be $1.771561 × 1.1 = $1.9487171 At the end of year 8, it will be $1.9487171 × 1.1 = $2.14358881
From the calculations above, we can see that at the end of year 7, the one dollar will have grown to $1.9487171, which is almost two dollars while at the end of year 8 it will have grown to $2.14358881. This means that somewhere between year 7 and year 8 it will be exactly $2.00. To find the exact time where it will be two dollars, we would have to solve for x in the exponential equation 1.1x = 2. Solving for x gives x = (log 2) / (log 1.1) = 7.27254089734.
The rule of 72
This problem illustrates the rule of 72 which gives the approximate number of years it would take for an amount of money growing at a given rate to double. By this rule, an amount of money growing at a rate of x% per annum will double in 72/x years. Thus, by that rule, it would take 72/10 = 7.2 years for money growing at 10% per annum to double.
Compound interest has been called the most powerful force in the universe. It has also been called the eighth wonder of the world. Interestingly, both of these quotes are attributed to Albert Einstein. Though we may not be able to verify that he actually said these things, we must wonder what would cause such profound statements to be made about the phenomenon of compounding. Let’s take the quiz below to test our understanding of the concept and to stimulate a discussion on it. The solutions have been provided in the articles linked at the end.
Q1. How quickly can you double your money?
One dollar growing at a rate of 10% annually will become two dollars after ________.
A. about 2 years B. about 5 years C. about 7 years D. about 10 years E. none of the above
Q2. Earlier investments vs. later investments
Bob invests $200 today while Sue invests $2,000. However, Bob adds $200 to his investment after every year while Sue just leaves her initial $2,000 investment alone. If the money they invest grows at 10% per annum, which of the investments will have made more money after 20 years?
A. Bob’s investment B. Sue’s investment C. Both investments will make the same amount of money D. There isn’t enough information to tell
Q3. One huge payment now vs. growing payments over time
Which of the following options will make more money?
Option 1: A one million dollar payment today
Option 2: A payment scheme where you start with 1 cent but the amount of money you are paid doubles every day for one month–that is, you are paid 1 cent today, 2 cents tomorrow, 4 cents the day after, and then 8 cents, and so on for the next one month (thirty days), which one would you choose?
A. One million dollars today B. 1 cent today, 2 cents tomorrow, 4 cents the day after, and then 8 cents, and so on for the next one month C. They will make the same amount of money D. There isn’t enough information to tell
Q4. Starting investing at 30 vs. starting at 40
On his 30th birthday, Bob wanted to start preparing for retirement on his 60th birthday by investing $10,000 a year at a rate of 10% per annum. On second thought, he realized that since he has so much time before retirement, he could delay starting the investment so that he now starts on his 40th birthday. In the first case, Bob would have invested for 30 years while in the second case he will only invest for 20 years. Thus, we would expect that the amount of money he will have after the 30-year investment will be more than the amount he will have after the 20-year investment. Which of the following best describes the relationship between the two investments?
A. The 20-year investment will yield a return that is about the same as what the 30-year investment will yield B. The 20-year investment will yield a return that is about two-thirds of what the 30-year investment will yield C. The 20-year investment will yield a return that is about one-half of what the 30-year investment will yield D. The 20-year investment will yield a return that is about one-third of what the 30-year investment will yield E. None of the above
Q5. One huge payment now vs. smaller fixed payments over time
On your birthday your rich uncle offers you two choices. In the first option, he offers to give you GHC 10,000 on that day. In the second option, he offers to give you GHC 1,000 that day and every subsequent birthday for the next 20 years. Which option would you choose and why?
