suppose that your bank pays 10% interest, compounded semiannually. Use table 12-2 of your text to find how much should be deposited now to yield an annuity payment of $400 at the end of every six months, for 4 years.
To find out how much should be deposited now to yield an annuity payment of $400 at the end of every six months for 4 years, you can use the present value of an annuity formula.
The formula to calculate the present value of an annuity is:
PV = A * ((1 - (1 + r)^(-n)) / r)
Where:
PV = Present Value
A = Annuity Payment
r = Interest Rate per compounding period
n = Number of compounding periods
Given:
Annuity Payment (A) = $400
Interest Rate (r) = 10% (which is 0.1)
Number of compounding periods (n) = 4 years * 2 (since compounding is done semiannually)
Now let's calculate the present value using the formula:
PV = $400 * ((1 - (1 + 0.1/2)^(-4*2)) / (0.1/2))
First, let's simplify the exponent part:
(1 + 0.1/2)^(-4*2) = (1 + 0.05)^(-8)
Now, let's calculate the present value:
PV = $400 * ((1 - (1.05)^(-8)) / (0.05))
Using a financial calculator or spreadsheet, you can find that (1.05)^(-8) is approximately 0.681
PV = $400 * ((1 - 0.681) / 0.05)
PV = $400 * (0.319 / 0.05)
PV = $400 * 6.38
PV = $2,552
So, in order to yield an annuity payment of $400 at the end of every six months for 4 years, a deposit of approximately $2,552 should be made now.