Jungle Jim owes three debts:
$500 due in one year plus interest at 6% compounded 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 a focal date of two years.
The focal date is 2 years, that is your reference.
Old debt (owing) = New debt (paying)
Old Debt
First part
500*(1 + 0.06/2)^(1*2) - The future value of the first debt
* (1 + 0.12/4)^(1*4) - To move the debt to the two year focal date. Move focal date + 1 year. Note we use the rate 12% compound quarterly to move it to the focal date as that is what the money is worth now.
Second part
Just leave it a $2000. This debit is already at the focal date. That is why no interest rate is given.
Third part
FV = 1000*(1+0.05/12)^(3*12) - The future value of the third debt
* (1 + 0.12/4)^(-1*4) - To move the debt to the two year focal date. Move focal date - 1 year.
To put the old debt all together:
Old Debt = 500(1+0.06/2)^(1*2)*(1+0.12/4)^(1*4) + 2000 + 1000(1+0.05/12)^(3*12)* (1 + 0.12/4)^(-1*4)
Old Debt = 500(1.03)^2*(1.03)^4 + 2000 + 1000(1.004166667)^36*(1.03)^-4
Old Debt = $3,628.979182
New Debt (Paying) x = payment amount
500(1*0.12/4)^(2*4) - $500 is paid now, add +2 to get it to the focal date
x(1+0.12/4)^(1*4) - A payment is made in 1 year, add +1 to get it to the focal date
x - Just add x. Payment is at focal date
So, we get
New debt = 500(1*0.12/4)^(2*4) + x(1+0.12/4)^(1*4) + x
New Debt = 500(1.03)^8 + x(1.03)^4 + x
Old Debt = New Debt
$3,628.979182 = 500(1.03)^8 + x(1.03)^4 + x
Solving for x gets you x = $1,409.35
So, the size of each payment is $1,409.35
You state different interest rates for the 1st and 3rd debt, but state no rate for the 2nd debt of $2000 due in 2 years.
Other than , choose a time as a reference or focal date
I picked the present time, so
at present, the value of the debt = value of payments
500(1.03)^-2 + 2000(1+i)^-? + 1000(1.0041666..)^-36 = 500 + x(1.03)^-4 + x(1.03)^-8
once you establish what the second interest rate is, evaluate each part, the x terms can be added.
To find the size of the equal payments, we can use the concept of the present value of future cash flows. We will calculate the present values of each debt and then solve for the equal payments.
Let's calculate the present value of each debt:
Debt 1: $500 due in one year plus interest at 6% compounded semi-annually
Using the compound interest formula: PV = FV / (1 + r/n)^(n*t)
PV1 = $500 / (1 + 0.06/2)^(2*1)
PV1 = $500 / (1.03)^2
PV1 = $457.28
Debt 2: $2000 due in two years
Using the present value formula: PV = FV / (1 + r)^t
PV2 = $2000 / (1 + 0.12)^2
PV2 = $2000 / (1.12)^2
PV2 = $1606.01
Debt 3: $1000 due in three years plus interest at 5% compounded monthly
Since interest is compounded monthly, we need to adjust the interest rate.
Effective annual interest rate = (1 + r/n)^(n*t) - 1
Effective annual interest rate = (1 + 0.05/12)^(12*3) - 1
Effective annual interest rate = (1 + 0.004167)^(36) - 1
Effective annual interest rate = 0.05
Using the present value formula: PV = FV / (1 + r)^t
PV3 = $1000 / (1 + 0.12)^3
PV3 = $1000 / (1.12)^3
PV3 = $713.34
Now, let's solve for the equal payments.
Total present value of debts = PV1 + PV2 + PV3 + Equal Payment 1-year PV + Equal Payment 2-year PV
Total present value of debts = $457.28 + $1606.01 + $713.34 + Equal Payment 1-year PV + Equal Payment 2-year PV
Since the focal date is two years, the equal payment 1-year PV will be discounted for two years, and the equal payment 2-year PV will be discounted for one year.
Using the present value formula: PV = FV / (1 + r)^t
Equal Payment 1-year PV = P / (1 + 0.12)^2
Equal Payment 2-year PV = P / (1 + 0.12)^1
Substituting the values into the equation:
Total present value of debts = $457.28 + $1606.01 + $713.34 + P / (1 + 0.12)^2 + P / (1 + 0.12)^1
We know that Total present value of debts = $500 (Initial payment)
$500 = $457.28 + $1606.01 + $713.34 + P / (1 + 0.12)^2 + P / (1 + 0.12)^1
Solving this equation for P, we can find the size of the equal payments.
To find the size of the equal payments, we need to set up equations for each debt and solve them simultaneously.
Let's analyze each debt separately:
1. Debt 1:
- Principal: $500
- Due in one year
- Interest rate: 6% compounded semi-annually
Using the compound interest formula A = P(1 + r/n)^(nt), where:
- A is the future value of the loan (including interest)
- P is the principal amount
- r is the annual interest rate (as a decimal)
- n is the number of times the interest is compounded per year
- t is the number of years
We can calculate A using the given values:
A = 500(1 + 0.06/2)^(2*1)
A = 500(1.03)^2
A = 500 * 1.0609
A ≈ $530.45
2. Debt 2:
- Principal: $2000
- Due in two years
- No interest specified
Since no interest is specified for this debt, we don't need to calculate its future value or interest.
3. Debt 3:
- Principal: $1000
- Due in three years
- Interest rate: 5% compounded monthly
Using the compound interest formula A = P(1 + r/n)^(nt), where:
- A is the future value of the loan (including interest)
- P is the principal amount
- r is the annual interest rate (as a decimal)
- n is the number of times the interest is compounded per year
- t is the number of years
We can calculate A using the given values:
A = 1000(1 + 0.05/12)^(12*3)
A = 1000(1.0042)^36
A = 1000 * 1.16074
A ≈ $1160.74
Now, let's set up the equations for the total outstanding debts using the present worth formula:
PW = FV/[(1 + i/n)^(n*t)]
Where:
- PW is the present worth (total outstanding debts)
- FV is the future value of each debt
- i is the annual interest rate (as a decimal)
- n is the number of compounding periods per year
- t is the number of years
Using the focal date of two years, we are interested in the total outstanding debts in two years.
Equation 1: PW = 530.45/(1 + 0.12/4)^(4*2)
Equation 2: PW = 1160.74/(1 + 0.12/4)^(4*2)
Let's solve these equations simultaneously to find the size of the equal payments.