Chapter 6 - Amortization and Sinking Funds

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 BalanceAmortization 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

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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 nLoan: L

Equation Of Value: @ i, L = PV(Out Flow) = PV(In Flow) = P ∙

therefore, P = L / Loan Balance Immediately After A Payment:Bk = Outstanding balance just after kth payment= P ∙ (“prospective” method – present value of remaining payments)= L ∙ (1+i)k – P ∙ (“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

balance from prospective viewpoint = P ∙ balance from retrospective viewpoint = L ∙ (1+i)k – P ∙

We want to show they are equivalentP ∙ = L ∙ (1+i)k – P ∙ which can be rewritten asP ∙ = P ∙ ∙ (1+i)k – P ∙ The payments cancel out and you are left with

= ∙ (1+i)k –

=


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 ∙ vn+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

PortionPrincipal Portion

Outstanding Balance

0 10,000.001 3,810.52 700.00 3,110.52 6,889.482 3,810.52 482.26 3,328.26 3,561.223 3,810.51 249.29 3,561.22 0.00Using The Calculator:i) Finding The Payment3 N 10,000 PV 7 %I CPT PMT ==>> 3810.5167ii) Finding O/S Balance @ End Of PeriodEnter 1, 2, or 3 to find balance and press BAL Note: does the calculation without rounding to pennies2 BAL ==>> 3561.2305iii) Finding Interest Portion Of Payment At End Of Period3 I/P ==>> 249.2861iv) Finding Principal Portion – Use The X Y Key


Table 6.1 Loan of 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 Loan01 12 1∙ ∙ ∙ ∙ ∙∙ ∙ ∙ ∙ ∙∙ ∙ ∙ ∙ ∙t 1∙ ∙ ∙ ∙ ∙∙ ∙ ∙ ∙ ∙∙ ∙ ∙ ∙ ∙

n-1 1n 1

Total n

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Another view of the amortization 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

An additional view of the amortizationAt time zero, you have a series of payments of 1 that have a value of :

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 .

Assumptions To Date constant i payment period and interest conversion period are the same payments are levelExampleA 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 / = $100,000 / 15.0463 = $6,646.15 or100000 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 ∙ = $6,646.15 ∙ 9.7122 = $64,548.74Note: 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∙ =.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 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

amount to accumulate +

interest on the =

amount needed to 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% InterestYear Interest

Paid On Loan

Sinking Fund Deposit

Interest Earned On Sinking Fund

Sinking Fund

Net Loan

0 10,000.001 1,000.00 1,637.97 1,637.97 8,362.032 1,000.00 1,637.97 163.80 3,439.75 6,560.253 1,000.00 1,637.97 343.97 5,421.70 4,578.304 1,000.00 1,637.97 542.17 7,601.84 2,398.165 1,000.00 1,637.97 760.18 10,000.00 0.00

Check that equivalent to amortization: $10,000 / = $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

In practice, rate earned on sinking fund (j) usually < i total payment then becomes

which can be reexpressed as

or


which equals when i = j

Sinking Fund For A $10,000 5 Year Loan At 10% Interest But Sinking Fund Earning 8%

Year Interest Paid On Loan


Interest Earned On Sinking Fund

Sinking Fund

Net Loan

0 10,000.001 1,000.00 1,704.56 1,704.56 8,295.442 1,000.00 1,704.56 136.37 3,545.49 6,454.513 1,000.00 1,704.56 283.64 5,533.70 4,466.304 1,000.00 1,704.56 442.70 7,680.96 2,319.045 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)

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 RateSinking

Fund Rate

4% 6% 8% 10% 12%

4% 5,052.35 7,052.35 9,052.35 11,052.35 13,052.356% 4,646.15 6,646.15 8,646.15 10,646.15 12,646.158% 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 12,225.9412% 4,130.36 6,130.36 8,130.36 10,130.36 12,130.36


6.5 (page 181) Differing Payment Periods And Interest PeriodsPayments 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)

use = j as your interest rate

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)

use = j as your interest rate and


ExampleCreate 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 / = $767.28

t Interest Paid


Interest Earned on Sinking Fund

Amount In Sinking Fund

Net Loan

0.00 10,000.000.25 767.28 767.28 9,232.720.50 400.00 767.28 11.42 1,545.98 8,454.020.75 767.28 23.02 2,336.27 7,663.731.00 400.00 767.28 34.79 3,138.33 6,861.671.25 767.28 46.73 3,952.33 6,047.671.50 400.00 767.28 58.85 4,778.45 5,221.551.75 767.28 71.15 5,616.88 4,383.122.00 400.00 767.28 83.63 6,467.78 3,532.222.25 767.28 96.30 7,331.36 2,668.642.50 400.00 767.28 109.16 8,207.79 1,792.212.75 767.28 122.21 9,097.27 902.733.00 400.00 767.28 135.45 10,000.00 0.00

Note: sinking fund interest is shown as it accrues – in actual fact, it is only being credited semi-annually.


Chapter 6 - Amortization and Sinking Funds

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Transcript of Chapter 6 - Amortization and Sinking Funds