Question

Are the following statements equivalent to each other?

1.a continuous rate of payment of $100 per year

2. a continuous rate of payment of $50 semi-annually

Answer 1

No, These statement are not equivalent to each other as this a concept of time value and continuous compounding .

In continuous compounding , it is assumed that interest is compounded and added back to Initial value over infinite number of Times.

The concept also includes time value which States that value of money changes over a period of time due to factors like inflation.

In this case let's assume that interest rate is 10 Percent and we are investing for one year

(1)If we are depositing the amount of 100 for one year , At the end of year the overall amount will be( 100+ .10*100) $= 110.

(2) If we deposit 50 for 6 months at the interest of 10 percent annually. It would be( 50+.05*50) so it would be 52.50 at the end of 6 months and if we invest $50 again after end of 6 months then, it would be( 52.50+50)+ (102.50*.05). So that would result into (102.50+5.125)=$ 107.625

So we can say that due to difference in time value of money concept, the return on both the amount when invested annually and semiannually is different .

This is because the money invested all at once will have more effects of compounding and time value while the money which is invested in parts will be relatively less compounded.