Vanna has just financed the purchase of a home for $200 000. She agreed to repay the loan by making equal monthly blended payments of $3000 each at 4%/a, compounded monthly.
How much would Vanna have saved if she had obtained a loan 3%/a, compounded monthly?
I will calculate how long it would take her to pay off the loan at 4%
i = .04/12 = .003333...
n = ? months
PV = 200000
paym = 3000
200000 = 3000(1 - 1.003333..^-n)/.003333...
.2222... = 1 - 1.00333..^-n
1.00333...^-n = .7777...
take logs of both sides and follow log rules
-n log 1.003333... = log .7777...
-n = log .77777/log 1.003333 = appr -75.5
n = 75.5 months
Now repeat the calculation for 3%. It will obviously be a shorter period since the rate is lower.
Take the difference in times and that will be the monthly payments she would save.
To calculate the amount Vanna would have saved if she had obtained a loan with a lower interest rate, we can use the formula for the present value of an ordinary annuity.
The formula for the present value of an ordinary annuity is:
PV = PMT × [(1 - (1 + r)^(-n)) / r]
Where:
PV = Present value (amount of the loan)
PMT = Equal monthly payment
r = Interest rate per period
n = Number of periods
In this case, Vanna is making equal monthly payments (PMT) of $3000 and the interest rate (r) is 4% per year compounded monthly. The loan term (n) is not specified, so we'll assume it is a 30-year mortgage.
Calculating the present value (PV) for the original loan:
PV = $3000 × [(1 - (1 + 0.04/12)^(-30*12)) / (0.04/12)]
PV ≈ $386,637.15
Now, let's calculate the new loan amount using the lower interest rate of 3% per year compounded monthly:
PV_new = $3000 × [(1 - (1 + 0.03/12)^(-30*12)) / (0.03/12)]
PV_new ≈ $395,097.49
To determine how much Vanna would have saved, we subtract the new loan amount from the original loan amount:
Savings = PV - PV_new
Savings ≈ $386,637.15 - $395,097.49
Savings ≈ -$8,460.34
Based on these calculations, Vanna would have saved approximately -$8,460.34 if she had obtained a loan with a lower interest rate of 3% compounded monthly. This means that taking the loan with the higher interest rate resulted in her paying less overall compared to the hypothetical lower-interest loan.