Question

1. Ms. Early Saver has decided to invest $1,000 at the end of each year for the next 10 years, then she will just let the amount compound for 25 additional years. Her brother, Late Saver, has a different investment program: He will invest nothing for the next 10 years but will invest $1,000 per year (at the end of each year) for the following 25 years. If we assume a 3% percent rate of return, compounded annually, which investment program will be worth more 35 years from now? If instead, the assumed interest rate is 8% percent rate which investment program will be worth more in 35 years?

   Are the results with different interest rates different? Using your results elaborate on the power of compounding and how it impacts savings in the long run.

Answer 1

You need to calculate the Future value of noth perons Investment plans

1. FV of Ms. Saver, $1000 for 10 years and then accumulated amount for 25 years. Rate =3%

First calculate the Fv after 10 years.

Fv = Annuity + Annuity x cumulative annuity factor @3% for 9 years

[ For annuity factors refer you annuity factor tables or can calculate at calculator. It is sum of (1.03) +(1.03)² +........+ (1.03)⁹ ]

= 1000 + 1000 x 10.4638793

= 11463.8793

Now calculate the Fv of 11463.8793 after 25 years i.e. at Time 35.

FV = 11463.8793(1.03)²⁵

FV = 11463.8793 x 2.09377793

FV = 24002.82

2. Fv of Mr. Saver, $1000 for 25 years from time 10 to time 35

Fv = Annuity + Annuity x cumulative annuity factor @3% for 24 years

[ For annuity factors refer you annuity factor tables or can calculate at calculator. It is sum of (1.03) +(1.03)² +........+ (1.03)²⁴ ]

= 1000 + 1000 x 35.45926413

FV = 36459.26

3. Now calculate the both steps in 1 and 2 with 8 % rate

For Ms. Saver

First calculate the Fv after 10 years.

Fv = Annuity + Annuity x cumulative annuity factor @8% for 9 years

[ For annuity factors refer you annuity factor tables or can calculate at calculator. It is sum of (1.08) +(1.08)² +........+ (1.08)⁹ ]

= 1000 + 1000 x 13.48656245

= 14486.5625

Now calculate the Fv of 14486.5625 after 25 years i.e. at Time 35.

FV = 14486.5625 (1.08)²⁵

FV = 14486.5625 x 6.848475

FV = 99210.86

For Mr saver now:

Fv = Annuity + Annuity x cumulative annuity factor @8% for 24 years

[ For annuity factors refer you annuity factor tables or can calculate at calculator. It is sum of (1.08) +(1.08)² +........+ (1.08)²⁴ ]

= 1000 + 1000 x 72.10593968

FV = 73105.94

Final Interpetation:

Investment Values at 3% rate

Ms Saver = 24002.82 Mr Saver = 36459.26

And at 8%

Ms Saver = 99210.86 Mr Saver = 73105.94

Looking at the above it is clearly visible that results are different at both rates. At 3% rate Mr. Saver has more money and at 8% Ms. Saver has more money at Time 65.

This is because the effect of compunding. In 1st Scenario rate of compunding is low i.e. 3% so Ms. saver only save 10 installments of 1000 and then lumpsum invested for 25 years but Mr. saver invested 25 intallemnts so his resulted figure is more because he invested more money than Ms. saver. Ms saver investment is only 1000 x 10 = 10000 and Mr. Saver investment 1000 x 25 = 25000. So having more investment amount and low rate Mr. Saver get more money.

But the things got changed when rate increased to 8%. Now you can see the power of compunding Ms. Saver who invested $10000 in 10 years and then lumpsum amount for 25 years and get the investment value higher than Mr. Saver. Mr. Saver who invested $25000 still having lower money. Ms. Saver money invested from starting and getting 8% interest and every year 8% more on invested money and Interest also. Interest on Interest effect make a bigger amount when invested for long term. So always look to start your investment at early age with higher rate of interests and for longer durations.