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Chapter 6 – Amortization Schedules And Sinking Funds 6.1 (page 166) Introduction 2 general methods of repaying loans amortization installment payments at periodic intervals (usually level) e.g. biweekly, monthly, quarterly, semi-annually, annually e.g. mortgages, car loans, furniture loans sinking funds lump sum payment at the end of the term of the loan periodic interest payments PLUS payments into a sinking fund that is to accumulate to the amount of the loan to be repaid payments may be into a trust for security reasons payments into the sinking fund may be a requirement of the issue of the original loan amount sometimes the sinking fund balance is used to retire part of the loan on the open market (usually because the after tax interest being earned on the sinking fund is substantially less than the cost of the interest payments on the loan – may be interest penalties to pay in order to retire in this fashion) 6.2 (page 167) Finding The Outstanding Balance Amortization installment payments are in the form of an annuity synonymous terms outstanding loan balance outstanding principal unpaid balance remaining loan indebtedness prospective method look into the future f30cbc20-1d53-4c1f-99b8-b355058e178d.doc 1 October 31, 2000 what is the present value of the remaining payments when calculated at the loan rate retrospective method original loan accumulated to the current time less payments accumulated to current time (both at the loan rate) retrospective = prospective payments are first applied to pay interest on the previous outstanding loan balance remainder used to reduce the outstanding loan balance Lender In Flow: P P P P Time: 0 1 2 n-1 n Loan: L Equation Of Value: @ i, L = PV(Out Flow) = PV(In Flow) = P ∙ an|i therefore, P = L / an|i Loan Balance Immediately After A Payment: Bk = Outstanding balance just after kth payment = P ∙ ank|i (“prospective” method – present value of remaining payments) = L ∙ (1+i)k – P ∙ sk|i (“retrospective” method – accumulated value at time k of L less accumulated value of past k payments) Note: Bk+t = Bk ∙ (1+i)t for 0 < t < 1 BtP balance from prospective viewpoint = P ∙ ank|i BtR balance from retrospective viewpoint = L ∙ (1+i)k – P ∙ sk|i We want to show they are equivalent P ∙ ank|i = L ∙ (1+i)k – P ∙ sk|i which can be rewritten as P ∙ ank|i = P ∙ an|i ∙ (1+i)k – P ∙ sk|i f30cbc20-1d53-4c1f-99b8-b355058e178d.doc 2 October 31, 2000 The payments cancel out and you are left with 1 vn (1 i ) k 1 ank|i = ∙ (1+i)k – i i (1 i ) k v nk (1 i ) k 1 1 v nk = ank|i i i 6.3 (page 169) Amortization Schedules each payment is part interest and part principal split varies with each payment interest portion is larger in early payments and smaller later Pk = principal portion of kth payment Ik = interest portion of kth payment by definition, P = Pk + Ik Ik = Bk-1 ∙ i (interest on previous Outstanding Loan Balance) Pk = P ∙ v n+1-k (P1, P2, ..., Pn is a geometric sequence which is linear if i = 0) Example Of Amortization Of 3 Year Loan Of $10,000 @ 7% Time Payment Interest Principal Outstanding Portion Portion Balance 0 10,000.00 1 3,810.52 700.00 3,110.52 6,889.48 2 3,810.52 482.26 3,328.26 3,561.22 3 3,810.51 249.29 3,561.22 0.00 Using The Calculator: i) Finding The Payment 3 N 10,000 PV 7 %I CPT PMT ==>> 3810.5167 ii) Finding O/S Balance @ End Of Period Enter 1, 2, or 3 to find balance and press BAL Note: does the calculation without rounding to pennies 2 BAL ==>> 3561.2305 iii) Finding Interest Portion Of Payment At End Of Period 3 I/P ==>> 249.2861 iv) Finding Principal Portion – Use The X Y Key f30cbc20-1d53-4c1f-99b8-b355058e178d.doc 3 October 31, 2000 Table 6.1 Loan of a n|i By definition, payment of 1 at end of each period Repaid over n periods at interest rate i Period Payment Interest Paid Principal Repaid Outstanding Loan 0 an|i 1 1 i an|i 1 v n vn an|i vn an 1|i 2 1 i an 1|i 1 v n 1 v n 1 an 1|i vn 1 an 2|i ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ t 1 i at |i 1 vt vt an t 1|i v n t 1 an t |i ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ n-1 1 i a2|i 1 v2 v2 a2|i v 2 a1|i n 1 i a1|i 1 v v a1|i v 0 Total n n an|i an|i f30cbc20-1d53-4c1f-99b8-b355058e178d.doc 4 November 5, 2000 Another view of the amortization an|i is the loan value at time 0 one period later, you have accumulated it with one period’s interest and taken out a payment of 1 (1 i ) v n 1 1 v n 1 B1 an|i (1 i ) 1 1 i i An additional view of the amortization At time zero, you have a series of payments of 1 that have a value of an|i : v v2 v3 ... vn-3 vn-2 vn-1 vn 0 1 2 3 ... n-3 n-2 n-1 n One year later, their value has grown with one years interest and the payment at time 1 has been paid and so is no longer outstanding so the outstanding loan balance is now the sum of: 1 v v2 . . . vn-4 vn-3 vn-2 vn-1 0 1 2 3 ... n-3 n-2 n-1 n which equals an 1|i . Assumptions To Date constant i payment period and interest conversion period are the same payments are level f30cbc20-1d53-4c1f-99b8-b355058e178d.doc 5 November 5, 2000 Example A loan of $100,000 is to be repaid by 40 annual payments with the first payment one year from now. Assuming interest is at 6%, a) What is the annual payment needed to amortize the loan? P = $100,000 / a40|.06 = $100,000 / 15.0463 = $6,646.15 or 100000 PV 0 FV 6 %I 40 N CPT PMT = 6646.1536 b) What is the Outstanding Loan Balance at t = 25? Assuming you already have the above input into your calculator, input 25 2nd BAL = $64,549.08 otherwise, B25 = $6,646.15 ∙ a15|.06 = $6,646.15 ∙ 9.7122 = $64,548.74 Note: using the interest tables loses accuracy in about the 5th digit c) What is the interest portion of the 15th payment? Once again assuming you have the inputs in a) still in the calculator, input 15 2nd INT = $5,185.26 otherwise, i∙B14 =.06∙$6,646.15∙ a14|.06 =.06∙$6,646.15∙13.0032 = $5,185.27 6.4 (page 175) Sinking Funds sometimes lender insists that sinking fund be built up to reduce the amount lost on default we will deal with regular contributions to the sinking fund the borrower is paying regular interest on the outstanding balance of the loan plus making regular contributions to a sinking fund to retire the loan at the end of the term net loan outstanding is the loan less the sinking fund balance f30cbc20-1d53-4c1f-99b8-b355058e178d.doc 6 November 5, 2000 if interest earned on the sinking fund equals interest being paid on the loan, the two methods are equivalent you will remember our formula from chapter 3 1 1 i s n i a n i amount to interest amount needed to accumulate + on the = amortize the loan to the loan loan even though the interest paid on the loan is level, the net interest is declining as the sinking fund earns interest net interest is same under the two methods IF sinking fund is earning the same rate as is being paid on the loan Sinking Fund For A $10,000 5 Year Loan At 10% Interest Year Interest Sinking Interest Sinking Net Loan Paid On Fund Earned Fund Loan Deposit On Sinking Fund 0 10,000.00 1 1,000.00 1,637.97 1,637.97 8,362.03 2 1,000.00 1,637.97 163.80 3,439.75 6,560.25 3 1,000.00 1,637.97 343.97 5,421.70 4,578.30 4 1,000.00 1,637.97 542.17 7,601.84 2,398.16 5 1,000.00 1,637.97 760.18 10,000.00 0.00 Check that equivalent to amortization: $10,000 / a5|.10 = $2,637.97 IF i on loan = i on sinking fund: total payment is same annual net interest is same annual net repayment is same f30cbc20-1d53-4c1f-99b8-b355058e178d.doc 7 November 5, 2000 In practice, rate earned on sinking fund (j) usually < i total payment then becomes 1 1 i s n j a n i& j which can be reexpressed as 1 1 (i j ) an j a n i& j or an j a n i& j 1 (i j ) a n j which equals a n i when i = j Sinking Fund For A $10,000 5 Year Loan At 10% Interest But Sinking Fund Earning 8% Year Interest Sinking Interest Sinking Net Loan Paid On Fund Earned Fund Loan Deposit On Sinking Fund 0 10,000.00 1 1,000.00 1,704.56 1,704.56 8,295.44 2 1,000.00 1,704.56 136.37 3,545.49 6,454.51 3 1,000.00 1,704.56 283.64 5,533.70 4,466.30 4 1,000.00 1,704.56 442.70 7,680.96 2,319.04 5 1,000.00 1,704.56 614.48 10,000.00 0.00 This is (i-j) ∙ loan higher than the payment needed for a 5 year amortization at 8% ($10,000 / 3.9927 = $2,504.56) f30cbc20-1d53-4c1f-99b8-b355058e178d.doc 8 November 5, 2000 That is, the schedule above is the same as if the loan and sinking fund had been both at 8% but the additional amount of (.10 - .08) times the loan ($200 in this case) is being paid in loan interest Total Payment $100,000 40 Year Loan Loan Rate Sinking 4% 6% 8% 10% 12% Fund Rate 4% 5,052.35 7,052.35 9,052.35 11,052.35 13,052.35 6% 4,646.15 6,646.15 8,646.15 10,646.15 12,646.15 8% 4,386.02 6,386.02 8,386.02 10,386.02 12,386.02 10% 4,225.94 6,225.94 8,225.94 10,225.94 12,225.94 12% 4,130.36 6,130.36 8,130.36 10,130.36 12,130.36 6.5 (page 181) Differing Payment Periods And Interest Periods Payments Less Frequent Than Interest Convertible payment every kth period interest compounds k times in that period n is the number of measurement periods over which the loan is being amortized (m / k) ∙ n payments to amortize the loan (or build sinking fund) i (m) k use (1 ) 1 = j as your interest rate m Payments More Frequent Than Interest Convertible k payments before interest compounds m is the number of times interest compounds in the normal measurement period n is the number of measurement periods over which the loan is being amortized k ∙ m ∙ n payments to amortize the loan (or build sinking fund) f30cbc20-1d53-4c1f-99b8-b355058e178d.doc 9 November 5, 2000 i ( m ) 1/ k use (1 m ) 1 = j as your interest rate and f30cbc20-1d53-4c1f-99b8-b355058e178d.doc 10 November 5, 2000 Example Create the sinking fund table for the following: a 3 year loan of $10,000, with interest payable semiannually at the nominal interest rate of 8.00% is to be retired by a sinking fund funded by quarterly deposits earning an effective semi-annual interest rate of 3.00% Interest of $400 (.08/2 ∙ $10,000) is paid on the loan The quarterly rate on the sinking fund is 1.4889% (1.03.5-1) $10,000 / s12|.014889 = $767.28 t Interest Sinking Interest Amount Net Loan Paid Fund Earned In Sinking Deposit on Fund Sinking Fund 0.00 10,000.00 0.25 767.28 767.28 9,232.72 0.50 400.00 767.28 11.42 1,545.98 8,454.02 0.75 767.28 23.02 2,336.27 7,663.73 1.00 400.00 767.28 34.79 3,138.33 6,861.67 1.25 767.28 46.73 3,952.33 6,047.67 1.50 400.00 767.28 58.85 4,778.45 5,221.55 1.75 767.28 71.15 5,616.88 4,383.12 2.00 400.00 767.28 83.63 6,467.78 3,532.22 2.25 767.28 96.30 7,331.36 2,668.64 2.50 400.00 767.28 109.16 8,207.79 1,792.21 2.75 767.28 122.21 9,097.27 902.73 3.00 400.00 767.28 135.45 10,000.00 0.00 Note: sinking fund interest is shown as it accrues – in actual fact, it is only being credited semi-annually. f30cbc20-1d53-4c1f-99b8-b355058e178d.doc 11 November 5, 2000