Jungle Jim owes three debts:
$500 due in one year plus interest at 6% semi-annually, $2000 due in two years, $1000 due in three years plus interest at 5% compounded monthly.
He wishes to discharge these debts by paying $500 now and two equal but unknown payments in one and two years respectively. Find the size of the equal payments if money is, at present, worth 12% compounded quarterly. Use focal date of two years.
I know I need to use the formulas PV=FV(1+i)^-n and FV=PV(1+i)^n but not sure how to approach the question
PV= present value
FV= future value
i=interest rate
n=number of compounding periods
To solve this problem, we can use the formulas you mentioned: PV = FV / (1+i)^n and FV = PV * (1+i)^n.
Let's break down the problem step by step:
Step 1: Calculate the present value (PV) of each debt using the given information.
For the first debt of $500 due in one year, we need to find its present value. The interest rate is 6% semi-annually, which means the interest rate per compounding period is 6% / 2 = 3% = 0.03. The number of compounding periods is 1 year * 2 = 2.
PV1 = $500 / (1+0.03)^2 = $461.77
For the second debt of $2000 due in two years, we don't need to calculate the present value because it is already given.
PV2 = $2000
For the third debt of $1000 due in three years, the interest rate is 5% compounded monthly. In this case, the interest rate per compounding period is 5% / 12 = 0.4167%. The number of compounding periods is 3 years * 12 = 36.
PV3 = $1000 / (1+0.004167)^36 = $764.63
Step 2: Determine the total present value of all debts.
Total PV = PV1 + PV2 + PV3
= $461.77 + $2000 + $764.63
= $3226.40
Step 3: Calculate the equivalent equal payments needed to discharge the debts.
To discharge the debts, Jungle Jim wants to make two equal payments in one and two years. We need to find the size of those equal payments.
The focal date is after two years, and the interest rate is 12% compounded quarterly. So, the interest rate per compounding period is 12% / 4 = 3% = 0.03. The number of compounding periods for the first payment is 1 year * 4 = 4, and for the second payment, it is 2 years * 4 = 8.
Using the formula for future value (FV = PV * (1+i)^n), we can write:
FV = PV1 * (1+0.03)^4 + PV2 * (1+0.03)^8 + PV3
= $461.77 * (1+0.03)^4 + $2000 * (1+0.03)^8 + $764.63
= $508.91 + $2428.45 + $764.63
= $3702.99
Now, we have the future value (FV) and need to find the equal payments. Assuming the equal payments are P, we can write:
FV = P * (1+0.03)^4 + P * (1+0.03)^8
$3702.99 = P * (1+0.03)^4 + P * (1+0.03)^8
Now, we can solve this equation for P.
$3702.99 = P * (1.1255) + P * (1.2925)
$3702.99 = 2.4179P
P = $3702.99 / 2.4179
P ≈ $1529.82
Therefore, the size of the equal payments needed to discharge the debts is approximately $1529.82.