john has a loan but doesn't begin to repay his loan for 11 months, at a rate of $500 every month of four month. the interest rate is 8% compounded monthly determin the size of the loan using the following 2 steps
1. calculate the present value, pv1 of annuity payment at the end of the period of deferral
2. calculate the present value, pv2 of the payment at the beginning of the period of deferral
Make a time-line graph, marking months beginning at 0 (now), 1, 2, 3, ...
The way I interpret your question, the first payment will be at month 11 , then payments at 12, 13 and 14
Since the formula for an ordinary annuity assumes the first payment at the end of the first interest period, we would be finding PV1 at month 10
PV1 = 500(1 - 1.006666...^-4)/.0066666..
= 1967.106
2. now we have to "move back" this amount to the present time (now or time spot of 0)
PV2 = 1967.106(1.0066666...)^-10
= 1915.51
If the first payment is at month 12, make the appropriate changes.
an ordinary annuity starting today with eight annual payments of $900
To calculate the size of the loan, we need to find the present value of the annuity payments at the end of the period of deferral (step 1) and the present value of the payment at the beginning of the period of deferral (step 2). Let's calculate each step:
Step 1: Calculate the present value (PV1) of the annuity payments at the end of the period of deferral.
Given:
- Loan repayment starts after 11 months.
- Monthly payment amount = $500.
- Interest rate = 8% compounded monthly.
To find the present value, we will use the formula for the present value of an ordinary annuity:
PV1 = PMT * ((1 - (1 + r)^(-n)) / r)
Where:
- PV1 = Present value of the annuity payments at the end of the period of deferral
- PMT = Periodic payment amount
- r = Interest rate per period
- n = Number of periods
Using the given information:
- PMT = $500
- r = 8% or 0.08 (monthly interest rate)
- n = 4 (number of months)
Substituting the values into the formula:
PV1 = $500 * ((1 - (1 + 0.08)^(-4)) / 0.08)
Now, let's calculate this:
PV1 = $500 * ((1 - (1.08)^(-4)) / 0.08)