Can someone please help me with the following question. I am not sure even where to start.
A 5-year annuity of 10 $9,000 seminannual payments will begin 9 years from now, with the first payment coming 9.5 years from now. If the discount rate is 11 percent compounded semiannually, what is the value of this annuity five years from now? What is the value three years from now? What is the current value of the annuity?
Treat it like a "present value" problem with the 9 year as you focal date.
PV = 9000(1 - (1.055)^-10)/0.055
= $67,836.63 at the 9 year mark
for its value now
Vaule = 67,836.63(1.055)^-9
= $41,899.12
the value 3 years from now
= 67,836.63(1.055)^-6
= 49,199.69
To solve this problem, we can break it down into three parts: finding the value of the annuity five years from now, finding the value three years from now, and finding the current value of the annuity.
1. Value of the annuity five years from now:
The first step is to determine the present value of the annuity five years from now. To do this, you can use the present value of an annuity formula:
PV = PMT * [1 - (1 + r)^(-n)] / r
Where:
PV = Present Value
PMT = Periodic Payments
r = Interest Rate per Period
n = Number of Periods
In this case, the periodic payments (PMT) are $9,000, the interest rate (r) is 11% compounded semiannually (which means the effective rate per period is 11% / 2 = 5.5%), and the number of periods (n) is 10 semesters.
Now plug these values into the formula:
PV = $9,000 * [1 - (1 + 5.5%)^(-10)] / 5.5%
After simplifying the equation, you will get the present value of the annuity five years from now.
2. Value of the annuity three years from now:
To find the value of the annuity three years from now, we will make use of the concept of time value of money. We need to discount the value of the annuity five years from now to the present value, and then compound it for another two years to calculate its value three years from now. For this, you'll need to use the future value of an annuity formula:
FV = PV * (1 + r)^n
Where:
FV = Future Value
PV = Present Value
r = Interest Rate per Period
n = Number of Periods
In this case, the present value (PV) will be the value of the annuity five years from now that you calculated in step 1. The interest rate (r) is still 5.5% per semester, and the number of periods (n) is 6 semesters (since three years is six semesters).
Now plug in these values into the formula:
FV = PV * (1 + 5.5%)^6
After calculating this equation, you will get the value of the annuity three years from now.
3. Current value of the annuity:
To determine the current value of the annuity, we need to discount the value of the annuity three years from now to the present value. For this, we can use the present value formula again:
PV = FV / (1 + r)^n
Where:
PV = Present Value
FV = Future Value
r = Interest Rate per Period
n = Number of Periods
In this case, the future value (FV) is the value of the annuity three years from now that you calculated in step 2. The interest rate (r) is still 5.5% per semester, and the number of periods (n) is 3 semesters.
Now plug in these values into the formula:
PV = FV / (1 + 5.5%)^3
After calculating this equation, you will get the current value of the annuity.