(Present Value) What is the present value of an annuity that pays $250,000 in 30 years if interest accumulates at a rate of 7.5% compounded semiannually? (i.e. How do you have to pay NOW for the policy? You make no payments other than your lump sum payment.)

To calculate the present value of the annuity, you will need to use the present value formula for annuities:

PV = PMT × ((1 - (1 + r)^(-n)) / r)

Where:
PV = Present Value
PMT = Payment per period
r = Interest rate per period
n = Number of periods

In this case, the payment per period (PMT) is $250,000, the interest rate per period (r) is 7.5% divided by 2 (since it is compounded semiannually), and the number of periods (n) is 30 years multiplied by 2 (again, because it is compounded semiannually).

Let's plug in the values into the formula:

PMT = $250,000
r = 7.5% / 2 = 0.075 / 2 = 0.0375
n = 30 years × 2 = 60

PV = $250,000 × ((1 - (1 + 0.0375)^(-60)) / 0.0375)

Next, we need to compute the expression within the parentheses:

(1 - (1 + 0.0375)^(-60)) / 0.0375 ≈ 32.5767

Now we can substitute this value back into the formula:

PV ≈ $250,000 × 32.5767 ≈ $8,144,175

Therefore, the present value of the annuity, or the amount you would need to pay today for the policy, is approximately $8,144,175.