Www.utstat.toronto.edu



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+1PRt+! = K – It+1PRt = K - ItAlso, 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. Level repayments K of a loan of LRecall the usual formula for level repayments K of a loan of L is:L=K a nRecall 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 * LLast payment K, at time t=nIn 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)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+1It = K - PRt = K(1- v n-t+1)Amortization Schedule: Zero InterestLet’s amortize a $20,000 loan to buy a car, 4 years at zero interest, K=5000=20,000/a4Time tTime t paymentPrincipal outstanding at t+Interest portion of KPrincipal portion of K0-20,00020,00015,00015,00005,00025,00010,00005,00035,0005,00005,00045,000005,000Amortization Schedule: Non-Zero InterestLet’s amortize a $20,000 loan to buy a car, 4 years at 10% interest, K=20,000/a 0.10 4 =6,309.42Time tTime t paymentPrincipal outstanding at t+Interest portion of KPrincipal portion of K0-20,00020,0000016,309.42=20,000-4309.42=15690.58(Ka3 =6309.42 *2.4869=15690.58)=0.10*20,000=2,0006309.42-2000=4309.4226,309.42=15690.58-4740.36=10950.22=0.10*15690.58=1,569.06=6309.42-1569.06=4740.3636,309.4210950.22-5214.40=5735.820.10*10950.22+1,095.02=6309.42-1095.02=5214.4046,309.425735.82-5735.82=00.10*5735.82=573.586309.42-573.58=5735.82Q of ClassAbove 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 paymentOB1 =15,691 (at 10%)Knew= 15,691 / a6 (at 8%) = 15,691/4.62287= 3,394.20 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download