1. The payment for a loan repaid with equal payments is the annuity payment with the loan value as the PV of the annuity. So, the 30-year loan payment will be:
PVA = C({1 – [1/(1 + r)] t } / r)
$35,000,000 = C{[1 – 1 / (1 + .061/12) 360] / (.061/12)}
C = $212,098.17
And the monthly payments for the 20-year loan will be:
PVA = C({1 – [1/(1 + r)] t } / r)
$35,000,000 = C{[1 – 1 / (1 + .061/12) 240] / (.061/12)}
C = $252,774.22
2. The interest payment is the beginning balance times the interest rate for the period, and the principal payment is the total payment minus the interest payment. The ending balance is the beginning balance minus the principal payment. The ending balance for a period is the beginning balance for the next period. The amortization table for an equal payment is:
Year Beginning Total Interest Principal Ending
Balance Payment Payment Payment Balance
1 $35,000,000.00 $212,098.17 $177,916.67 $34,181.51 $34,965,818.49
2 34,965,818.49 212,098.17 177,742.91 34,355.26 34,931,463.23
3 34,931,463.23 212,098.17 177,568.27 34,529.90 34,896,933.32
4 34,896,933.32 212,098.17 177,392.74 34,705.43 34,862,227.89
5 34,862,227.89 212,098.17 177,216.33 34,881.85 34,827,346.04
6 34,827,346.04 212,098.17 177,039.01 35,059.17 34,792,286.88
3. The bi-weekly payment is one-half of the 30-year
traditional mortgage payment, or:
Bi-weekly payment = $212,098.17 / 2
Bi-weekly payment = $106,049.09
Now we have the present value of an annuity, the interest
rate, and the number of payments. We
need to find the number of periods of the annuity payments.
Note that if you make a payment every
two weeks, you will make 52/2 = 26 payments per year. So, we
can solve the present value of an
annuity equation for the number of periods as follows:
PVA = C({1 – [1/(1 + r)]t} / r)
$35,000,000 = $106,049.09{[1 – 1 / (1 + .061/26)t] /
(.061/26)}
Now we solve for t:
1/1.00235t= 1 – [($35,000,000)(.00235) / ($106,049.09)]
1.00235t= 1/.2384 = 4.1950
t = ln 4.1950 / ln 1.00235
t = 635.24 periods
Since they are 26 bi-weekly periods in a year, the time
necessary to pay of the bi-weekly mortgage will be:
Bi-weekly payoff = 635.24 / 26
Bi-weekly payoff = 24.43 years
The bi-weekly payments pay off the loan quicker for two
reasons. First, one-half of the payment gets
to the bank quicker each month, which reduces the interest
that accrues each month. Second, the
company is actually making 13 full payments each year (26
bi-weekly periods amounts to 13
monthly payments).
The traditional answer on how much the company saves is as
follows: The total payments under the
30-year traditional mortgage will be:
30-year total payments = 360 × $212,098.17
30-year total payments = $76,355,342.98
And the total payments on the bi-weekly mortgage will be:
Bi-weekly total payments = 635.24 × ($106,049.09/2)
Bi-weekly total payments = $67,366,136.74
So, the traditional answer for how much the bi-weekly
mortgage saves is the difference between
these two answers. Unfortunately, this calculation is very
misleading. This is actually a “pseudo
interest” savings, which is caused by the different
maturities of the loans. If the actual interest rate is
6.1 percent, the present value of the two cash flows is
still $35 million. More interest accrues in the
30-year traditional mortgage because of the longer length,
but the present value is still the same as
the present value of the bi-weekly mortgage, so the two
mortgage cash flow streams are equivalent.
In actual fact, the bi-weekly mortgage is more expensive. We
can see this by examining the EAR for
the two loans. The EAR of the monthly mortgage is:
EAR = [1 + (APR / m)]m– 1
EAR = [1 + (.061/12)]12– 1
EAR = .0627 or 6.27%
And the EAR of the bi-weekly mortgage is:
EAR = [1 + (APR / m)]m– 1
EAR = [1 + (.061/26)]26– 1
EAR = .0628 or 6.28%
The bi-weekly mortgage is actually more expensive with the
same APR because there are more N N compounding periods in a year under this option.
4. The loan payments for the first 59 months are the same as
the traditional 30-year mortgage, which is
$212,098.17. This mortgage payment will be made in the 60th
month as well, but the company will
also make the bullet payment. The bullet payment can be
found by using an amortization table, but
the easier method is to find the present value of the
remaining loan payments. The present value of
the remaining loan payments in month 60 will be:
PVA = C({1 – [1/(1 + r)]t} / r)
PVA = $212,098.17{[1 – 1 / (1 + .061/12)300] / (.061/12)}
PVA = $32,609,016.11
So, the total payment in month 60 will be:
Month 60 payment = $212,098.17 + 32,609,016.11
Month 60 payment = $32,821,114.28
5. The interest-only loan requires only interest payments
each month, which will be:
Monthly interest payments = $35,000,000(.035/12)
Monthly interest payments = $102,083.33
The company will make these payments for the first 119
months, and then repay the principal and interest on the 120th payment. So, the
120th payment will be:
Last payment = $35,000,000 + 102,083.33
Last payment = $35,102,083.33
6. The best loan is the interest-only loan because it has
the lowest interest rate. One risk of the loan is
that the company may not pay off the principal before
maturity, which could mean it may refinance
at a higher rate in the future. Of course, the rate in the
future could be the same, or even lower, but
there is still a refinancing risk. One way to show that the
interest-only loan is the better option is to
consider what happens if the company makes the same payments
as it would if took out the
traditional 30-year mortgage. If the company makes these
payments, it would pay off the interestonly
loan in:
PVA = C({1 – [1/(1 + r)]t} / r)
$35,000,000 = $212,098.17{[1 – 1 / (1 + .035/12)t] /
(.035/12)}
Now we solve for t:
1/1.00292t= 1 – [($35,000,000)(.00292) / ($212,098.17)]
1.00292t= 1/.5737 = 1.7431
t = ln 1.7431 / ln 1.00292
t = 225.39 months
or:
225.39 / 12 = 18.78 years
