deposits $1,000 now, $2,000 2 years from now, and $5,000 5 years from now into a fund paying 12% compounded semiannually.
a. What sum of money will be accumulated at the end of sixth year?
b. What equal deposits of size A, made every 6 months (with the first deposit at t=0 and the last deposit at the end of the sixth year), are equivalent to the three deposits?
To find the sum of money that will be accumulated at the end of the sixth year, we need to calculate the future value of each deposit and then sum them up.
a. Calculating the future value of each deposit:
For the initial deposit of $1,000 made now, we don't need to calculate its future value since it is already the present value.
For the deposit of $2,000 made 2 years from now, we need to compound it for 4 semiannual periods (6 years - 2 years = 4 semiannual periods), assuming a semiannual compounding interest rate of 12%/2 = 6%. The formula to calculate the future value of a single deposit is FV = PV * (1 + r)^n, where FV is the future value, PV is the present value, r is the interest rate per period, and n is the number of periods. Plugging in the values: FV = $2,000 * (1 + 0.06)^4 = $2,000 * (1.06)^4 = $2,000 * 1.262476 = $2,524.95.
For the deposit of $5,000 made 5 years from now, we need to compound it for 2 semiannual periods (6 years - 5 years = 1 semiannual period). Plugging in the values: FV = $5,000 * (1 + 0.06)^1 = $5,000 * (1.06)^1 = $5,000 * 1.06 = $5,300.
b. To find the equal deposits of size A that are equivalent to the three deposits combined, we need to find the present value of these future value amounts. Since all deposits are made semiannually, we need to use the semiannual compounding interest rate of 6%.
Using the formula for the present value of a single deposit, PV = FV / (1 + r)^n, where PV is the present value, FV is the future value, r is the interest rate per period, and n is the number of periods:
For the future value of $1,000, the present value is equal to the deposit amount itself, which is $1,000.
For the future value of $2,524.95, the present value is calculated as: PV = $2,524.95 / (1 + 0.06)^(4*2) = $2,524.95 / (1 + 0.06)^8 = $2,524.95 / 1.593852 = $1,582.12.
For the future value of $5,300, the present value is calculated as: PV = $5,300 / (1 + 0.06)^(1*2) = $5,300 / (1 + 0.06)^2 = $5,300 / 1.1236 = $4,720.98.
Now, we can calculate the total present value by summing these individual present values: $1,000 + $1,582.12 + $4,720.98 = $7,303.10.
Therefore, the sum of money that will be accumulated at the end of the sixth year is $7,303.10.
To find the equal deposits of size A, made every 6 months, we need to find the payment amount that will yield the same future value as the accumulated amount in part (a).
Using the formula for the future value of a series of equal deposits, FV = A * [(1 + r)^n - 1] / r, where FV is the future value, A is the payment amount, r is the interest rate per period, and n is the number of periods:
Plugging in the values: $7,303.10 = A * [(1 + 0.06)^(6*2) - 1] / 0.06.
Now, we can solve this equation to find the value of A. Simplifying the equation:
$7,303.10 * 0.06 = A * [(1 + 0.06)^12 - 1].
$438.19 = A * (1.06^12 - 1).
$438.19 = A * (1.790847 - 1).
$438.19 = A * 0.790847.
A = $438.19 / 0.790847.
A ≈ $554.04.
Therefore, equal deposits of size $554.04, made every 6 months, are equivalent to the three deposits.