3: The Time Value of Money

payments (rather than annual) on consumer loans, we now demonstrate amortisation calculations using monthly

rather than annual payments. First, Equation 3.16a is simply a modified version of Equation 3.16:

Eq. 3.16a (^) PMT
r
r
= n rPn V
[(1+) 1]
(1 +)
−
××
TABLE 3.3 LOAN AMORTISATION SCHEDULE, $25,000 PRINCIPAL, FOR 8% INTEREST, FIVE-YEAR REPAYMENT
PERIOD
End of
year
Loan payment
(1)
Beginning of year
principal (2)
Payments End-of-year
principal
Interest [0.08 × (2)] (3) Principal [(1) − (3)] (4) [(2) − (4)] (5)
1 $6,261.41 $25,000.00 $2,000.00 $4,261.41 $20,738.59
2 6,261.41 20,738.59 1,659.09 4,602.32 16,136.27
3 6,261.41 16,136.27 1,290.90 4,970.51 11,165.76
4 6,261.41 11,165.76 893.26 5,368.15 5,797.61
5 6,261.41 5,797.61 463.80 5,797.61 0
Second, we can generalise this formula to more frequent compounding periods by dividing the interest
rate by m and multiplying the number of compounding periods by m. This changes the equation as follows:
Eq. 3.16b (^) PMT
r
m
r
M
r
m
mn P
mn

• 
  


• 
  

  11
  1
  
  
  
  
  −
  
  
  
  
  
  
  ×
  
  
  
  
  × ×
  ×
  VV
  Use Equation 3.16b to calculate what a monthly car payment will be if you borrow $25,000 for five years at 8% annual
  interest. Once again, PV will be the $25,000 amount borrowed, but the periodic interest rate (r ÷ m) will be 0.00667, or
  0.667% per month (0.08 per year ÷ 12 months per year). There will be m × n = 60 compounding periods (12 months per year
  3 × years = 60 months). Substituting these values into Equation 3.16b yields a car loan payment of just under $507 per month:

  
  

PMT

0.08

12

1

0.08

12

1

1

0.08

12

$25, 000

0.00667

[(1.00667)1]

(1.00667) 25, 000

$506.91

125

125

60

6

=

+







−

















×+







×

=

−

××

=

×

×

The monthly payment is less than one-twelfth the annual payment that we calculated in the previous example
($506.91 × 12 < $6,261.41). The reason for this is that when payments are made more frequently, less interest accrues
between payments, and therefore a lower payment is required to repay the entire loan. (Note that to obtain the precise
figure of $506.91, it is necessary to carry the monthly interest rate out several digits beyond where we have rounded here.)
As a test of your command of the monthly payment formula, see if you can compute the monthly mortgage
payment for a home purchased using a 30-year, $100,000 loan with a fixed 7.5% annual interest rate. Note that
there are 360 compounding periods (12 months per year × 30 years).^12

12 The amount of the mortgage payment is $699.21. To find this solution, just enter the formula ‘=pmt(0.00625,360,100000)’in Excel. The first
argument in this function is the monthly interest rate, 7.5% divided by 12.

example