In the problems 1 to 3, find the present value of the given (ordinary) annuity.
1. $3500000 every 6 months for 5 years at the rate of 16% compounded semi-annually.
2. $7000000 per month for 10 months at the rate of18% compounded monthly.
3. $1700000 per year for 5 years at the rate of 14% compounded annually.
I will do the first one, the other two are done the same way
a)
PV = 3500000( 1 - 1.08^-10)/.08
= $23 485 284.09
BTW, what's with the huge numbers?
To find the present value of an annuity, we need to calculate the discounted value of the future cash flows. The formula to calculate the present value of an annuity is:
PV = PMT x (1 - (1 + r/n)^(-nt)) / (r/n),
Where:
PV = Present Value,
PMT = Payment amount per period,
r = Interest rate per period,
n = Number of compounding periods per year,
t = Number of years.
Let's solve each problem step by step:
1. $3500000 every 6 months for 5 years at the rate of 16% compounded semi-annually.
- PMT = $3,500,000 (payment every 6 months)
- r = 16% per year = 0.16
- n = 2 (since it is compounded semi-annually)
- t = 5 (5 years)
Using the formula, we have:
PV = $3,500,000 x (1 - (1 + 0.16/2)^(-2*5)) / (0.16/2)
Calculating this expression, we obtain the present value of the annuity.
2. $7000000 per month for 10 months at the rate of 18% compounded monthly.
- PMT = $7,000,000 (payment per month)
- r = 18% per year = 0.18
- n = 12 (since it is compounded monthly)
- t = 10 months / 12 (convert months to years)
Using the formula, we have:
PV = $7,000,000 x (1 - (1 + 0.18/12)^(-12*(10/12))) / (0.18/12)
Calculating this expression will give us the present value of the annuity.
3. $1700000 per year for 5 years at the rate of 14% compounded annually.
- PMT = $1,700,000 (payment per year)
- r = 14% per year = 0.14
- n = 1 (since it is compounded annually)
- t = 5 years
Using the formula, we have:
PV = $1,700,000 x (1 - (1 + 0.14/1)^(-1*5)) / (0.14/1)
Calculating this expression will provide us with the present value of the annuity.