How to find the actual doubling time with the rule of 72.

Question

I have a programming assignment in C# from my professor that involves the Rule of 72. He clearly says that in order to find the amount of time in years it will take for an amount to double, you have to divide the rate into 72, so it would look like 72/r. Now my issue is, he does not say how to find the actual doubling time. This part is driving me crazy. How can you find the ACTUAL doubling time when you only have the rate? For those that want to see what the assignment is ill show you.

Assignment 5

Rule of 72: This rule is used to approximate the time required for prices to double due to inflation.

If the inflation rate is r % then the Rule of 72 estimates that prices will double in 72/r years. For instance at tan inflation rate of 6% prices double in about 72/6 or 12 years. Write a program to test the accuracy of this rule. For each interest rate from 1% to 20% the program should display the rounded value of 72/r and the rounded value of the actual number of years required for prices to double at an r% inflation rate. Assume prices increase at the end of each year.

Output Window:

Interest rate: 1.00%

Rule of 72 doubling time: 72 years to double.

Actual Doubling time: 70 years to double.

Interest rate: 2.00%

Rule of 72 doubling time: 36 years to double.

Actual Doubling time: 36 years to double.

Interest rate: 3.00%

Rule of 72 doubling time: 24 years to double.

Actual Doubling time: 24 years to double.

I tried to group the output together but for some reason I cant. I hope this doesnt upset anyone willing to help me. SO right now, all i have is 72/r to give me the doubling years according to the rule of 72. How do i find the actual?

Answer 1

The process that is modeled by the rule of 72 is Compound Interest, which works like this: during each compounding period, the money in your account increases by some percentage; this increase is pulled back into your account, where next time around, it too will be increased by some percentage.

So if you have $\yen1,000,000$ in your account and it grows at $5\%$/year, then after 1 year you'll have $\yen1,000,000\times1.05=\yen1,050,000$; after two years you'll have $\yen1,050,000\times1.05=\yen1,102,500$; and so on and so forth. Doing this, we discover that after 15 multiplications -- 15 years -- we're at $\yen2,078,928$, and our money has doubled. This matches up pretty well with the rule of 72, which says that it should take $14.4$ years.

Your likely goal in this problem is to apply this multiplication over and over until you find that your money has doubled, and say how many repetitions it took. Note that it doesn't matter how much money you start with, so your best bet is to pretend you start with 1 money.

Of course, it's also possible to come up with an exact doubling time without loops: $t=\ln(2)/\ln(1+r)$. The rule of 72 comes from the fact that, for small $r$, $\ln(1+r)\approx r$. It also includes an adjustment so it works with percentages, and another adjustment so it's nicer to work with in your head.

Answer 2

Start with initial amount $A$.

At the end of one year the value will be $A \left (1+\frac r {100} \right)$.

At the end of two years the value will be $A \left (1+\frac r {100} \right)^2$.

At the end of n years the value will be $A \left (1+\frac r {100} \right)^n$.

You want the amount to double, so it will be $2A$.

We therefore need to solve $2A=A \left (1+\frac r {100} \right)^n$.

First we divide by $A$ to get $2= \left (1+\frac r {100} \right)^n$.

Then take logarithms to get $\log (2) = n \log\left (1+\frac r {100} \right)$.

Then take logarithms to get $n = \frac{\log (2)}{\log\left (1+\frac r {100} \right)}$.

r   n   n rounded   ratio 
0.1 693.4936964 694 69.4
0.2 346.9200485 347 69.4
0.3 231.3954607 232 69.6
0.4 173.6331381 174 69.6
0.5 138.9757216 139 69.5
0.6 115.8707581 116 69.6
0.7 99.36719646 100 70.0
0.8 86.9895109  87  69.6
0.9 77.36240945 78  70.2
1.0 69.66071689 70  70.0
1.1 63.35932173 64  70.4
1.2 58.10814962 59  70.8
1.3 53.66484141 54  70.2
1.4 49.85628343 50  70.0
1.5 46.55552563 47  70.5
1.6 43.6673555  44  70.4
1.7 41.11896345 42  71.4
1.8 38.85371982 39  70.2
1.9 36.82691697 37  70.3
2.0 35.00278878 36  72.0
2.1 33.35238175 34  71.4
2.2 31.85200663 32  70.4
2.3 30.48209405 31  71.3
2.4 29.22633621 30  72.0
2.5 28.07103453 29  72.5
2.6 27.00459792 28  72.8
2.7 26.01715251 27  72.9
2.8 25.10023494 26  72.8
2.9 24.24654925 25  72.5
3.0 23.44977225 24  72.0
3.1 22.70439665 23  71.3
3.2 22.00560358 23  73.6
3.3 21.34915826 22  72.6
3.4 20.73132414 21  71.4
3.5 20.14879168 21  73.5
3.6 19.5986191  20  72.0
3.7 19.0781826  20  74.0
3.8 18.58513463 19  72.2
3.9 18.11736836 19  74.1
4.0 17.67298769 18  72.0
4.1 17.25028146 18  73.8
4.2 16.8477015  17  71.4
4.3 16.46384368 17  73.1
4.4 16.09743147 17  74.8
4.5 15.74730184 16  72.0
4.6 15.41239288 16  73.6
4.7 15.09173308 16  75.2
4.8 14.78443185 15  72.0
4.9 14.48967133 15  73.5
5.0 14.20669908 15  75.0
5.1 13.93482168 14  71.4
5.2 13.67339904 14  72.8