Which is the better deal, $10,000 invested at 5%, compounded yearly, for 20 years, or $5,000 invested at 10%, compounded continuously, for 20 years?

Question

Which is the better deal, $10,000 invested at 5%, compounded yearly, for 20 years, or $5,000 invested at 10%, compounded continuously, for 20 years?

Answer 1

Use the formula A=P(1+r)^t for the first deal and A=Pe^rt for the second deal. 

 

The first deal: P=10,000, r=0.05, and t=20.

 

The second deal: P=5,000, r=0.1, and t=20.

Answer 2

At 5% interest compounded annually the investment increases by a factor of 1.05 every year.  After the first year the investment will be $10,000*1.05, after the second year it will be $10,000*1.05*1.05=$10,000*1.05² and so on.   The pattern is that after n years the investment will be $10,000*1.05ⁿ.  After 20 years the investment will have grown to $10,000*1.05²⁰=$26,532.98.

 

The formula for continuous interest is A=Pe^rt, where A is the final amount, P is the principle, $5,000 in this case, r is the rate of interest, 0.1 in this case, and t is the time in years, 20 in this case.  So A=$5,000*e^0.1*20=$36,945.28.

 

So it better to invest $5,000 at a 10% compounded continuously.

 

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