Let’s get happy about PRt+1 = PRt (1+i)
True if all the (e.g. annual) payments are equal, K.
Because at time t+1 the principal paid PRt+1 is the
portion of K remaining after paying the interest It+1
PRt+! = K – It+1
PRt = K - It
Also, because PRt got paid off at t so we no longer
have to pay interest on it. So PR t+1 is bigger than
PRt by an amount i*PRt, hence PRt+1 =PRt (1+i). Or,
algebraically saying the same thing:
It+1 = It - i * PRt
PRt+! = K – It+1
= K – It + i * PRt
= PRt (1+i)
On a mortgage or other loan, the principal paid off is small in the first
few years but grows geometrically. Note that in after-inflation ‘real’
terms, the first few years of a mortgage look pleasanter.
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Level repayments K of a loan of L
Recall the usual formula for level repayments K
of a loan of L is:
L=K a n
Recall the big deal formula for principal repaid
as part of the t+1 th level payment:
PRt+1 = PRt (1+i)
In the first payment K at time 1, the amount of
interest must be:
I1 = i * L
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Last payment K, at time t=n
In the last payment K, at time t=n, we must pay
off the last bit of principal, OBn-1 = PRn , plus
interest on it In = i* OBn-1 = i* PRn.
So the remainder of the last payment is
principal:
OBn-1 = PRn = K – In = K - i* PRn.
Hence
PRn = K/(1+i)
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PRn = K/(1+i)
PRt+1 = PRt (1+i) or PRt = PRt+1 /(1+i)
Combine the above two formulas and we have
ourselves the formulas to put together an
amortization schedule for a loan:
PRt = K /(1+i) n-t+1 =K v n-t+1
It = K - PRt = K(1- v n-t+1)
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Amortization Schedule: Zero Interest
Let’s amortize a $20,000 loan to buy a car, 4
years at zero interest, K=5000=20,000/a4
Time Time t Principal
t
payment outstanding
at t+
0
-20,000 20,000
1
5,000
15,000
2
5,000
10,000
3
5,000
5,000
4
5,000
0
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Interest Principal
portion portion
of K
of K
0
0
0
0
5,000
5,000
5,000
5,000
Amortization Schedule: Non-Zero Interest
Let’s amortize a $20,000 loan to buy a car, 4
years at 10% interest,
K=20,000/a 0.10 4 =6,309.42
Ti
me
t
0
1
Time t
payment
2
6,309.42
3
6,309.42
4
6,309.42
-20,000
6,309.42
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Principal
outstanding at
t+
20,000
=20,0004309.42
=15690.58
(Ka3
=6309.42
*2.4869
=15690.58)
=15690.584740.36
=10950.22
10950.225214.40
=5735.82
5735.825735.82
=0
Interest portion Principal
of K
portion of
K
0
0
=0.10*20,000
6309.42=2,000
2000
=4309.42
=0.10*15690.58 =6309.42=1,569.06
1569.06
=4740.36
0.10*10950.22 =6309.42+1,095.02
1095.02
=5214.40
0.10*5735.82
6309.42=573.58
573.58
=5735.82
Time t
0
1
2
3
4
Payment Prin
outstanding
-20000
20,000
6309.416
15,691
6309.416
10,950
6309.416
5,736
6309.416 0.00000000
Interest rate I=
Annuity 4:
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0.1
3.169865446
Interest
portion
0
2000
1569.058
1095.023
573.5833
Princ Portion
0
4309.416
4740.358
5214.393
5735.833
Q of Class
Above loan just after 1st payment is renegotiated
to be extended to 1+6 years. The interest rate
used is the new market rate of 8% but the
balance of the old loan uses the initial 10%.
Calculate the new annual payment
OB1 =15,691 (at 10%)
Knew= 15,691
/ a6  (at 8%)
= 15,691/4.62287
= 3,394.20
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