ADVANCED PROGRAMME MATHEMATICS 

SECTION: FINANCE

LOANS EXAMPLE 1 A home loan of R600 000 is amortized over a period of 20 years @ 10,5%  p.a. compounded monthly. Determine (a) the monthly payment(b) the value of the loan plus interest after 20 years(c) the value of the repayments plus interest after 20 years

ANSWER: 

240T0T

R 600 000i(12) = 10,5% p.j.

n

v

x[1 (1 i) ]Lo n : Pa

i

2400,105x [1 (1 ) ]

12600 0000,105

12

240

Monthly p :

0,105600 000

12x0,105

1 (1 )12

aym

R5 990,28

ent

(a)

2400,105600 000 (1 )

12R4 855 150,60

(b)

(c)

2400,1055 990,28 [(1 ) 1]

120,105

12R4 855 150,60

240

n

v

At T :

x [(1 i) 1]Payments plus interest: F

i

240T0T

R 600 000 i(12) = 10,5% p.j.

240

n

At T :

Loan plus interest : F P(1 i)

Note:

At any stage:

n

n

Balance outstanding Loan plus interest Payments plus interest

x (1 i) 1P(1 i)

i

When outstanding balance 0 (i.o.w. when loan is amortised/paid off )

Loan plus interest Payments plus interest

Loan plus interest Payments plus interest 0

x x x

Lo : R6an 00 000

Payments :

Balance = ? Balance = 0

(12)i 10,5% p.j.

71T 72T 240T

ANSWER

We determine the future value (F) of the loan after 6 years and subtract the future value (Fv) of the first 6 years' 72 paid payments.

EXAMPLE 2 Determine the outstanding balance of the house loan mentioned above after 6 years.

nn x[(1 i) 1]

P(1 i)i

72At T :

Balance outstanding Loan plus interest Repayments plus interest

72

72

0,1055 990,28 [(1 ) 1]0,105 12600 000(1 )

0,1051212

R1123 483,47 R597 297,63

R526 185,84

(Only small amount paid off after 6 years!)

ALTERNATIVE METHOD

 

Calculate the outstanding balance after 6 years by not using a future value but a present value annuity.

(A shorter way!)

 

x (12)i 10,5% p.j.

71T 72T

x x

Lo : R6an 00 000

Balance = ? Balance = 0

240T

We regard the 6th year as the present (the “now”) and then determine the present value of the following 14 years’ 168 unpaid instalments. (I.o.w. what must still be paid at this moment if it can be paid in cash. The future interest gets discounted.)

Payments :

x (12)i 10,5% p.j.

71T 72T

x x

Lo : R6an 00 000

Balance = ? Balance = 0

240T

Payments :

We regard the 6th year as the present (the “now”) and then determine the present value of the following 14 years’ 168 unpaid instalments.

1680,1055 990,28 [1 (1 ) ]

120,105

12R526 185,97

(Few cents different in Example 2.)

n

v

x[1 (1 i) ]P

i

THEREFORE ALTERNATIVE METHOD

Balance outstanding Present value of outstanding amount

CONCLUSION

 

To calculate the outstanding balance, we therefore calculate the present value of the unpaid instalments of the loan.

Value of payments Loan plus interest Outstanding balance

Outstanding balance Present value of outstanding payments

SUMMARY:

Outstanding balance Loan plus interest Repayments plus interest

PROOF THAT the 2 methods produce the same result.

kv vP(1 i) F of k payment P of (n k) payments s

Where k = the number of payments already made

n = the total number of payments

therefore n – k = the number of outstanding paymentsk

k x[(1 i) 1]P(1 i)

i

n kkx [1 (1 i) ] x [(1 i) 1]

. (1 i)i i

n k kx{ [1 (1 i) ] (1 i) [(1 i) 1] }

i

k n k k

n k

(n k )

x[(1 i) (1 i) (1 i) 1]

ix

[1 (1 i) ]ix [1 (1 i) ]

i

...And this is the present value of the

unpaid / outstanding payments!!

In more detail:

Balance outstanding

Loan plus C.I. over k periods k payments plus C.I.

Present value of the loan's (n k) outstanding periods of payments

where C.I. compound interest

Do Exercise 5.1 p. 170

Enrichment 

The notation a n┐i ( read: a angle n at i )

n1 (1 i)

i

is used for the factor to be multiplied to a regular payment x to calculate the present value of the annuity.