Question

a few years back, dave and jana bought a new home. they borrowed $230,415 at an annual fixed rate of 5.49% (15 - year term) with monthly payments of $1,881.46. they just made their 35th payment, and the current balance on the loan is $199,089.44.
interest rates are at an all - time low, and dave and jana are thinking of refinancing to a new 15 - year fixed loan. their bank has made the following offer: 15 - year term, 3.0%, plus out - of - pocket costs of $2,937. the out - of - pocket costs must be paid in full at the time of refinancing.
build a spreadsheet model to evaluate this offer. the excel function:
= pmt(rate, nper, pv, fv, type)
calculates the payment for a loan based on constant payments and a constant interest rate. the arguments of this function are:
rate = the interest rate for the loan
nper = the total number of payments
pv = present value (the amount borrowed)
fv = future value the desired cash balance after the last payment (usually 0)
type = payment type (0 = end of period, 1 = beginning of the period)
for example, for dave and janas original loan, there will be 180 payments (12*15 = 180), so we would use = pmt(0.0549/12, 180, 230415, 0, 0) = $1,881.46. note that because payments are made monthly, the annual interest rate must be expressed as a monthly rate. also, for payment calculations, we assume that the payment is made at the end of the month.
the savings from refinancing occur over time, and therefore need to be discounted back to current dollars. the formula for converting k dollars received at the beginning of the month and received t months hence to current dollars is:
\\(\frac{k}{(1 + r)^{t - 1}}\\)
where r is the monthly inflation rate. alternatively, the present value of dollars received at the end of the month is:
\\(\frac{k}{(1 + r)^{t}}\\)
assume that r = 0.002 and that dave and jana make their payment at the end of each month,
use your model to calculate the savings in current dollars associated with the refinanced loan versus staying with the original loan.
if required, round your answer to the nearest whole dollar amount. if your answer is negative use \minus sign\.

Explanation:

Step1: Calculate new monthly payment

Using the PMT function. The rate for the new loan is \(3.0\%/12 = 0.0025\), \(nper=12\times15 = 180\), \(pv = 199089.44\), \(fv = 0\), \(type=0\).
\(PMT(0.0025,180,199089.44,0,0)\)

$$=\frac{0.0025\times199089.44\times(1 + 0.0025)^{180}}{(1 + 0.0025)^{180}-1}$$

Let \(x=(1 + 0.0025)^{180}\approx1.56748\)

$$=\frac{0.0025\times199089.44\times1.56748}{1.56748 - 1}$$

$$=\frac{0.0025\times199089.44\times1.56748}{0.56748}\approx1398.93$$

Step2: Calculate monthly savings

Original payment is \(1881.46\), new payment is \(1398.93\). Monthly savings \(S=1881.46-1398.93 = 482.53\)

Step3: Calculate present - value of savings

Using the present - value of an ordinary annuity formula \(PV=\sum_{t = 1}^{180}\frac{S}{(1 + r)^{t}}\) where \(r = 0.002\) and \(S=482.53\)
The formula for the present value of an ordinary annuity is \(PV = S\times\frac{1-(1 + r)^{-n}}{r}\)
Substitute \(S = 482.53\), \(r=0.002\), \(n = 180\)

$$PV=482.53\times\frac{1-(1 + 0.002)^{-180}}{0.002}$$

Let \(y=(1 + 0.002)^{-180}\approx0.70469\)

$$PV=482.53\times\frac{1 - 0.70469}{0.002}=482.53\times\frac{0.29531}{0.002}$$

$$PV=482.53\times147.655\approx71264$$

Step4: Subtract out - of - pocket costs

Out - of - pocket costs \(C = 2937\)
Net savings \(=71264-2937=68327\)

Answer:

\(68327\)