A Problem: 21 yr old vs 31 yr old (simple interest solution)

Yearly vs Monthly Contributions

ODE model and its solution

Continuous Model: 21 yr old vs 31 yr old

Continuous Model vs Simple and Continuous Interest

Cyclic Interest Rates: Annual and Monthly Contributions vs Modified ODE Continuous Model

 

		Simple Interest (8%)
	Adam		Bob
Age	Payment	Accumulation End of Year	Payment	Accumulation End of Year
21	$2,400.00	$2,400.00	$0.00	$0.00
22	$2,400.00	$4,992.00	$0.00	$0.00
23	$2,400.00	$7,791.36	$0.00	$0.00
24	$2,400.00	$10,814.67	$0.00	$0.00
25	$2,400.00	$14,079.84	$0.00	$0.00
26	$2,400.00	$17,606.23	$0.00	$0.00
27	$2,400.00	$21,414.73	$0.00	$0.00
28	$2,400.00	$25,527.91	$0.00	$0.00
29	$2,400.00	$29,970.14	$0.00	$0.00
30	$2,400.00	$34,767.75	$0.00	$0.00
31	$0.00	$37,549.17	$2,400.00	$2,400.00
32	$0.00	$40,553.10	$2,400.00	$4,992.00
33	$0.00	$43,797.35	$2,400.00	$7,791.36
34	$0.00	$47,301.14	$2,400.00	$10,814.67
35	$0.00	$51,085.23	$2,400.00	$14,079.84
36	$0.00	$55,172.05	$2,400.00	$17,606.23
37	$0.00	$59,585.81	$2,400.00	$21,414.73
38	$0.00	$64,352.68	$2,400.00	$25,527.91
39	$0.00	$69,500.89	$2,400.00	$29,970.14
40	$0.00	$75,060.96	$2,400.00	$34,767.75
41	$0.00	$81,065.84	$2,400.00	$39,949.17
42	$0.00	$87,551.11	$2,400.00	$45,545.10
43	$0.00	$94,555.20	$2,400.00	$51,588.71
44	$0.00	$102,119.61	$2,400.00	$58,115.81
45	$0.00	$110,289.18	$2,400.00	$65,165.07
46	$0.00	$119,112.32	$2,400.00	$72,778.28
47	$0.00	$128,641.30	$2,400.00	$81,000.54
48	$0.00	$138,932.61	$2,400.00	$89,880.58
49	$0.00	$150,047.22	$2,400.00	$99,471.03
50	$0.00	$162,050.99	$2,400.00	$109,828.71
51	$0.00	$175,015.07	$2,400.00	$121,015.01
52	$0.00	$189,016.28	$2,400.00	$133,096.21
53	$0.00	$204,137.58	$2,400.00	$146,143.91
54	$0.00	$220,468.59	$2,400.00	$160,235.42
55	$0.00	$238,106.07	$2,400.00	$175,454.26
56	$0.00	$257,154.56	$2,400.00	$191,890.60
57	$0.00	$277,726.92	$2,400.00	$209,641.84
58	$0.00	$299,945.08	$2,400.00	$228,813.19
59	$0.00	$323,940.68	$2,400.00	$249,518.25
60	$0.00	$349,855.94	$2,400.00	$271,879.71
	$24,000.00		$72,000.00

Yearly vs Monthly Contributions

Now, let’s take a closer look at Adam’s first 10 years with the account.Let’s compare his annual contributions to monthly, and then continuously (with continuous compounding interest).

Here we have:

·One lump sum of $2400 per year (at the end) for 10 years at 8% interest:

So, let P=2400 and r =.08.

Our calculation is of the form P[(1+r)^9 + (1+r)^8 + … + (1+r) +1] 

So, 2400[(1+.08)^9 + (1+.08)^8 + … + (1+.08) +1]

= 2400[(1-(1+.08)^10)/(1-(1+.08))]

=$34,767.75

But what if he did this:

·$200 every month for 120 months (at the end) at 8% interest:

So, let P= 200 and r = .08

Our calculation is of the form P[(1+r/12)^119 + (1+r/12)^118 + … + (1+r/12) +1] 

So, 200[(1+.08/12)^119 + (1+.08/12)^118 + … + (1+.08/12) +1]

= 200[ (1-(1+.08/12)^120)

=$36,589.21

ODE model and its solution

Now suppose Adam continuously put money into his account:

·$2400 continuously for 10 years at 8% interest:

Here we will need the ODE dS/dt –rS = k

So, let r =.08 and k= 2400

Our differential equation, dS/dt =rS + k, has the solution:

S(t)= (k/r)(e^rt-1) 

To derive this solution,

here, r=p(t)

integrating factor e^ -(antiderivative of p(t))dt

= e^(-rt)

So, e^(-rt) dS/dt – re^(-rt)S = ke^(-rt)

Then, (Se^(-rt))’ = ke^(-rt)

So, Se^(-rt)=ke^(-rt) dt

Thus, S= e^rt{(k/-r)(e^-rt) +C}

S(0)=0iff C=k/r

S=-k/r + (k/r)e^(rt)

That is, S= (k/r)(e^(rt)-1)

So, getting back to our continuous model, we have:

S(t) = (k/r)(e^(rt) – 1)

So, S(10)=(2400/.08){e^(10*.08)-1} 

=$36,766.22

As we can see, the continuous model is greater than the monthly model, which is greater than the annual model.

Continuous model: 21 yr old vs 31 yr old

Now, applying this model to Adam and Bob, we have:

Adam (at age 21):

Contributes $2400 continuously/year for 10 years

Amount in account at age 60 at 8% interest:

S(t)= (k/r)*{e^(rt)-1}

S(10)=(2400/.08){e^(10*.08)-1}

=$36,766.22

So, amount at age 60 is, 

S(10)*{e^(.08*30)} 

=$405,280.61

Bob (at age 31):

Contributes $2400 continuously/year for 30 years

Amount in account at age 60 at 8% interest:

S(t)= (k/r)*{e^(rt)-1}

S(30)= (2400/.08){e^(30*.08)-1}

=$300,695.29

Continuous Model vs Simple and Continuous Interest

Comparing this to their simple interest model, we see that both have a better account with the continuous interest and contributions, and Adam still has more than Bob. If we compare the continuous interest and contributions model to the one below which just has continuous interest, we see that, although continuous interest is better than simple interest, continuous contributions with continuous interest is even better.

 

		Continuous Interest
	Example A		Example B
Age	Payment	Accumulation End of Year	Payment	Accumulation End of Year
21	$2,400.00	$2,400.00	$0.00	$0.00
22	$2,400.00	$4,999.89	$0.00	$0.00
23	$2,400.00	$7,816.32	$0.00	$0.00
24	$2,400.00	$10,867.31	$0.00	$0.00
25	$2,400.00	$14,172.42	$0.00	$0.00
26	$2,400.00	$17,752.80	$0.00	$0.00
27	$2,400.00	$21,631.38	$0.00	$0.00
28	$2,400.00	$25,832.99	$0.00	$0.00
29	$2,400.00	$30,384.55	$0.00	$0.00
30	$2,400.00	$35,315.19	$0.00	$0.00
31	$0.00	$38,256.48	$2,400.00	$2,400.00
32	$0.00	$41,442.75	$2,400.00	$4,999.89
33	$0.00	$44,894.40	$2,400.00	$7,816.32
34	$0.00	$48,633.52	$2,400.00	$10,867.31
35	$0.00	$52,684.07	$2,400.00	$14,172.42
36	$0.00	$57,071.97	$2,400.00	$17,752.80
37	$0.00	$61,825.32	$2,400.00	$21,631.38
38	$0.00	$66,974.57	$2,400.00	$25,832.99
39	$0.00	$72,552.69	$2,400.00	$30,384.55
40	$0.00	$78,595.39	$2,400.00	$35,315.19
41	$0.00	$85,141.37	$2,400.00	$40,656.48
42	$0.00	$92,232.54	$2,400.00	$46,442.64
43	$0.00	$99,914.32	$2,400.00	$52,710.71
44	$0.00	$108,235.89	$2,400.00	$59,500.84
45	$0.00	$117,250.54	$2,400.00	$66,856.49
46	$0.00	$127,016.00	$2,400.00	$74,824.77
47	$0.00	$137,594.79	$2,400.00	$83,456.70
48	$0.00	$149,054.65	$2,400.00	$92,807.57
49	$0.00	$161,468.98	$2,400.00	$102,937.24
50	$0.00	$174,917.26	$2,400.00	$113,910.58
51	$0.00	$189,485.60	$2,400.00	$125,797.85
52	$0.00	$205,267.30	$2,400.00	$138,675.19
53	$0.00	$222,363.42	$2,400.00	$152,625.04
54	$0.00	$240,883.41	$2,400.00	$167,736.73
55	$0.00	$260,945.89	$2,400.00	$184,107.03
56	$0.00	$282,679.30	$2,400.00	$201,840.76
57	$0.00	$306,222.83	$2,400.00	$221,051.49
58	$0.00	$331,727.24	$2,400.00	$241,862.22
59	$0.00	$359,355.82	$2,400.00	$264,406.22
60	$0.00	$389,285.52	$2,400.00	$288,827.83
	$24,000.00		$72,000.00

 

Cyclic Interest Rates: Annual and Monthly Contributions vs Modified ODE Continuous Model Now, these models are unrealistic since the interest rate won't stay constant. So, let's vary the interest rate. I chose the cycle below by picking rates between 2 and 8. So, let's go back and compare Adam's three models using the cycle below, where the interest rate moves through the cycle once per year. Below that, we will see that monthly contributions is still better than yearly. Now, for the continuous interest and contributions model, we must modify the ODE since r is no longer constant, but a function of time. So, using sine regression I found the sine function which best fit the interest rate cycle. This is shown below the yearly and monthly comparison chart. Here we let the interest rate be cyclic as follows:
	{8, 4, 2.1, 3.4, 4.9, 6.1, 7.4, 8,…}
YEAR	Year End	Month End
1	$2,400.00	$2,489.99
2	$4,992.00	$5,035.92
3	$7,488.00	$7,565.94
4	$9,938.40	$10,264.98
5	$12,420.00	$13,234.06
6	$14,937.60	$16,532.54
7	$17,484.00	$20,281.39
8	$20,061.60	$24,454.72
9	$22,653.60	$27,895.54
10	$25,149.60	$30,910.25
	dS/dt=r(t)S+k
	r(t)=a*sin(bx+c)+d
	a=0.02874081409827
	b=0.89546468358745
	c=1.4916468315558
	d=0.0512853542211
	Continuous
	$2,463.90
	$5,061.72
	$7,803.08
	$10,698.34
	$13,758.64
	$16,995.95
	$20,423.15
	$24,054.07
	$27,903.62
	$31,987.84