the effects of compounding\nthe compounding of gains from investments causes them to increase dramatically…

the effects of compounding\nthe compounding of gains from investments causes them to increase dramatically over time. this can be calculated using the rule of 72, which is an easy way to see how long it would take for an investment to double at different rates of return.\nthe rule of 72 simply says that by dividing 72 by an annual rate of return, you find how long it will take an investment to double in value.\nfor example: if you have a bond paying 10%, it will take 7.2 years to double (72/10 = 7.2). the rule of 72 is a handy rule of thumb to use especially when comparing investments or investing over the long term, as this exercise illustrates.\ninvestment a: a stock mutual fund that earns 15% per year, on average. how long will it take for an investment at this rate to double in price?\nhow many times will an investment in it have doubled at this rate in\n5 years: \n10 years: \n15 years: \n20 years: \ntake the number of times an investment has doubled at this rate in 25 years and apply that to $1,000. what will that $1,000 be worth? \ninvestment b: a bond earns 9% per year, on average. how long will it take for an investment at this rate to double in price?\nhow many times will an investment in it have doubled at this rate in\n5 years: \n10 years: \n15 years: \n20 years: \ntake the number of times an investment has doubled at this rate in 25 years and apply that to $1,000. what will that $1,000 be worth? \ninvestment c: a bank account earns 5% per year, on average. how long will it take for an investment at this rate to double in price?\nhow many times will an investment in it have doubled at this rate in\n5 years: \n10 years: \n15 years: \n20 years: \ntake the number of times an investment has doubled at this rate in 25 years and apply that to $1,000. what will that $1,000 be worth?

the effects of compounding\nthe compounding of gains from investments causes them to increase dramatically over time. this can be calculated using the rule of 72, which is an easy way to see how long it would take for an investment to double at different rates of return.\nthe rule of 72 simply says that by dividing 72 by an annual rate of return, you find how long it will take an investment to double in value.\nfor example: if you have a bond paying 10%, it will take 7.2 years to double (72/10 = 7.2). the rule of 72 is a handy rule of thumb to use especially when comparing investments or investing over the long term, as this exercise illustrates.\ninvestment a: a stock mutual fund that earns 15% per year, on average. how long will it take for an investment at this rate to double in price?\nhow many times will an investment in it have doubled at this rate in\n5 years: \n10 years: \n15 years: \n20 years: \ntake the number of times an investment has doubled at this rate in 25 years and apply that to $1,000. what will that $1,000 be worth? \ninvestment b: a bond earns 9% per year, on average. how long will it take for an investment at this rate to double in price?\nhow many times will an investment in it have doubled at this rate in\n5 years: \n10 years: \n15 years: \n20 years: \ntake the number of times an investment has doubled at this rate in 25 years and apply that to $1,000. what will that $1,000 be worth? \ninvestment c: a bank account earns 5% per year, on average. how long will it take for an investment at this rate to double in price?\nhow many times will an investment in it have doubled at this rate in\n5 years: \n10 years: \n15 years: \n20 years: \ntake the number of times an investment has doubled at this rate in 25 years and apply that to $1,000. what will that $1,000 be worth?

Answer

Explanation:

Step1: Calculate time to double for each investment

Use the Rule of 72 formula $t=\frac{72}{r}$, where $t$ is the time to double and $r$ is the annual - rate of return.

Step2: Calculate number of doublings in given time periods

For a given time period $T$, the number of doublings $n=\frac{T}{t}$.

Step3: Calculate final value of $1000$ investment

If an investment doubles $n$ times, the final value $V = 1000\times2^{n}$.

Investment A (15% annual return)

  • Time to double: $t_A=\frac{72}{15}=4.8$ years
  • Number of doublings in 5 years: $n_{A5}=\frac{5}{4.8}\approx1.04$
  • Number of doublings in 10 years: $n_{A10}=\frac{10}{4.8}\approx2.08$
  • Number of doublings in 15 years: $n_{A15}=\frac{15}{4.8}\approx3.125$
  • Number of doublings in 20 years: $n_{A20}=\frac{20}{4.8}\approx4.17$
  • Number of doublings in 25 years: $n_{A25}=\frac{25}{4.8}\approx5.21$
  • Final value of $1000$ in 25 years: $V_A = 1000\times2^{5.21}\approx36379.3$

Investment B (9% annual return)

  • Time to double: $t_B=\frac{72}{9}=8$ years
  • Number of doublings in 5 years: $n_{B5}=\frac{5}{8}=0.625$
  • Number of doublings in 10 years: $n_{B10}=\frac{10}{8}=1.25$
  • Number of doublings in 15 years: $n_{B15}=\frac{15}{8}=1.875$
  • Number of doublings in 20 years: $n_{B20}=\frac{20}{8}=2.5$
  • Number of doublings in 25 years: $n_{B25}=\frac{25}{8}=3.125$
  • Final value of $1000$ in 25 years: $V_B = 1000\times2^{3.125}\approx9050.9$

Investment C (5% annual return)

  • Time to double: $t_C=\frac{72}{5}=14.4$ years
  • Number of doublings in 5 years: $n_{C5}=\frac{5}{14.4}\approx0.35$
  • Number of doublings in 10 years: $n_{C10}=\frac{10}{14.4}\approx0.69$
  • Number of doublings in 15 years: $n_{C15}=\frac{15}{14.4}\approx1.04$
  • Number of doublings in 20 years: $n_{C20}=\frac{20}{14.4}\approx1.39$
  • Number of doublings in 25 years: $n_{C25}=\frac{25}{14.4}\approx1.74$
  • Final value of $1000$ in 25 years: $V_C = 1000\times2^{1.74}\approx3090.9$

Answer:

Investment A:

  • Time to double: 4.8 years
  • Number of doublings in 5 years: 1.04
  • Number of doublings in 10 years: 2.08
  • Number of doublings in 15 years: 3.125
  • Number of doublings in 20 years: 4.17
  • Number of doublings in 25 years: 5.21
  • Final value of $1000$ in 25 years: $$36379.3$

Investment B:

  • Time to double: 8 years
  • Number of doublings in 5 years: 0.625
  • Number of doublings in 10 years: 1.25
  • Number of doublings in 15 years: 1.875
  • Number of doublings in 20 years: 2.5
  • Number of doublings in 25 years: 3.125
  • Final value of $1000$ in 25 years: $$9050.9$

Investment C:

  • Time to double: 14.4 years
  • Number of doublings in 5 years: 0.35
  • Number of doublings in 10 years: 0.69
  • Number of doublings in 15 years: 1.04
  • Number of doublings in 20 years: 1.39
  • Number of doublings in 25 years: 1.74
  • Final value of $1000$ in 25 years: $$3090.9$