Question

A series of 16 equal semi-annual payments of $ 2,000 each are made for 8 years. What would be
the future value of this series of payments, immediately after the last semi-annual payment of $
2,000 ? The annual interest rate is 8 % compounded continuously.

Answer 1

In the above question, fixed payments have been made for specified number of years which means that it is an annuity payment. Further, the annual interest rate is compunded continuously. The formula of future value of series of payments (i.e. annuity) in case of continuous compounding is as follows:

FV= CF * (e^rt -1$^{_{}}$) / (e^r -1)

wherein, FV= Future Value

CF= Cash Flow

r= rate

t= time

In our question, 16 semi annual payments of $2000 each have been made for a period of 8 years at annual interest rate of 8% continuous compounding. Since semi anuual payments have been made, the annual rate of 8% will be divided by 2, hence the rate (r) shall be 4%(0.04). Further, over a period of 8 years, payments have been made in 16 semi annual installments, hence the time (t) shall be 16. Cash flow(CF) is $2000.

To remember: the value of (e) that has been mentioned in the formula is always 2.7183.

Therefore,

FV= CF * (e^rt -1$^{_{}}$) / (e^r -1)

FV= 2000 * (2.7183^0.04*16 -1$^{_{}}$) / (2.7183^0.04 -1)

= 2000 * (1.89654 - 1) / (1.0408159-1)

= 2000 * (0.89654) / (0.0408159)

= 2000 * 21.9655

= $43,931

If you have a scientific calculator, you can calculate e^rt and e^rusing it easily. Otherwise, I have shown the calculations below:

2.7183^0.04*16= 2.7183^0.64

^{Now, $\sqrt{}$} (press root) of 2.7183 12 times.

Then, subtract 1 from the above achieved value

Then, multiply with power i.e. 0.64 as shown above

Now, Add 1

Then, press (X) (=) 12 times i.e press (multiply =) 12 times