Question

2. An investment of $425 is made at a nominal rate of 4% per year, with interest compounded monthly. How long will it take that investment to double?

a) Write the equation that you need to solve in order to answer this problem.

b) Show how to solve your equation TWO ways: graphically and symbolically. To solve graphically, clearly sketch the window you saw, give the dimensions of the window, label your curves, and give the coordinates of any point(s) of importance.

When solving symbolically, I want you to first find the EXACT answer, and then give an appropriate decimal approximation.

You may use natural logs and/or common logs, but logs of no other bases. Be sure the answer to the question asked is stated in a short sentence at the end of your work.

Answer 1

. In order to compute the timeframe in which the investment doubles , we need to compute the APR effective rate and use the TVM or time value of money considerations to find the time in which the investment doubles

In order to calculate the effective annual rate we have

Eff rate ( 1+nom rate /m )^m -1 where m is the number compoundings

We have a nominal rate of 4% pa

M = monthly compoundings 12

So Eff rate = 4.074% pa.

So as per TVM considerations

PV = FV (1+R)^N

Or

425 = 2x425 (1+0.04074)^n

Taking Log ( Natural Logarithm )

We have

Log ( 0.5) = n x log (1.04074)

Or n = 17.358189 years

Approximated to 2 decimals we have   17.36 years

If we plot the equation graphically

YEAR	PV	INTEREST	FV	Times
1	425	17.3145	442.3145	1.04074
2	442.3145	18.01989	460.3344	1.08314
3	460.3344	18.75402	479.0884	1.127267
4	479.0884	19.51806	498.6065	1.173192
5	498.6065	20.31323	518.9197	1.220988
6	518.9197	21.14079	540.0605	1.270731
7	540.0605	22.00206	562.0626	1.3225
8	562.0626	22.89843	584.961	1.376379
9	584.961	23.83131	608.7923	1.432452
10	608.7923	24.8022	633.5945	1.490811
11	633.5945	25.81264	659.4071	1.551546
12	659.4071	26.86425	686.2714	1.614756
13	686.2714	27.9587	714.2301	1.680541
14	714.2301	29.09773	743.3278	1.749007
15	743.3278	30.28318	773.611	1.820261
16	773.611	31.51691	805.1279	1.894419
17	805.1279	32.80091	837.9288	1.971597
18	837.9288	34.13722	872.066	2.05192
19	872.066	35.52797	907.594	2.135515
20	907.594	36.97538	944.5694	2.222516

The graph obtained as follows

The point in FV at the doubling point can be seen from the graph

Thanks