MATH 2040 Introduction to Mathematical Finance

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MATH 2040 Introduction to Mathematical Finance
Instructor Miss Liu Youmei
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Chapter 5 Amortization Schedules and Sinking
Funds
Introduction
Finding the outstanding loan balance
Amortization schedules
Sinking funds
Differing payment periods and interest conversion
periods
Varying series of payments

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Example 3.3

• Compare the total amount of interest that would
  be paid on a 1000 loan over a 10-year period,
  if the effective rate of interest is 9 per
  annum, under the following three repayment
  methods
• (1) The entire loan plus interest is paid in one
  lump-sum at the end of 10 years.
• (2) Interest is paid each year and the principal
  is repaid at the end of 10 years.
• (3) The loan was repaid by level payments over
  the 10 year period.

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Introduction

• There are two methods of paying off a loan
• (a) Amortization Method Borrower makes
  installment payments at periodic intervals.
• (b) Sinking Fund Method Borrower makes
  installment payments
• (i) As the annual interest comes due and pay
  back the original loan as a lump-sum at the end,
• (ii) The lump-sum is built up with periodic
  payments going into a fund called sinking fund.

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Purpose of this chapter

• Besides discussing the two methods of paying off
  a loan, this chapter also discuss how to
  calculate
• (a) the outstanding balance once the repayment
  schedule has begun, and
• (b) what portion of annual payment is made up of
  the interest payment and the principal repayment.

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Finding the Outstanding Balance

• There are two methods for determining the
  outstanding loan balance once the re-payment
  processes begin
• (a) Prospective method
• (b) Retrospective method.

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Prospective method (see the future)

• The original loan at time 0, L0, represents
  the present value of future repayments. If the
  repayments, P, are to be level and payable at
  the end of each year, then the original loan can
  be represented as follows
• The outstanding loan at time t, Lt, represents
  the present value of the remaining future
  repayments
• Note that this assumes that the installments
  prior to time t has been paid on time as
  scheduled.

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Retrospective method (see the past)

• If the repayments, P, are to be level and
  payable at the end of each year, then the
  outstanding loan at time t is equal to the
  accumulated value of the loan at time t, less
  the accumulated value of the repayments made to
  date
• This also assumes that the installments prior to
  time t has been paid on time as scheduled.
  Otherwise, the accumulated value of past payments
  will need to be adjusted accordingly.

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Basic relationship

• A basic relationship is
• Prospective method Retrospective method
• Suppose a loan L is to be repaid with
  end-of-year payments of 1 over the next n
  years.
• Let t be an integer with 0 lt t lt n. The
  value of the n payments at time t is
• Therefore
• So Prospective method Retrospective method

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Comparing two methods

• The prospective method is preferable when the
  size of each level payment and the number of
  remaining payments is known.
• The retrospective method is preferable when the
  number of remaining payments or a final irregular
  payment is unknown.

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Example 5.1

• A loan is being repaid by 10 payments of 2000
  followed by 10 payments of 1000 at the end of
  each half-year. If the nominal rate of interest
  convertible semiannually is 10, find the
  outstanding loan balance immediately after 5
  payments are made by using both the prospective
  method and the retrospective method.

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Example 5.1 Prospective method

• Immediately after 5 payments are made, there
  will be 5 more payments of 2000 followed by
  10 more payments of 1000 at the end of each
  year.
• These payments may be viewed as 15 payments of
  1000 plus 5 payments of 1000 at the end of
  each half-year starting the end of this
  half-year.
• So the present value of these future payments
  are
• To the nearest dollar.

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Example 5.1 Retrospective method

• The original loan amount is the present value of
  20 payments of 1000 plus 10 payments of
  1000 at the end of each half-year starting the
  end of this half-year. So it is
• Retrospectively, the outstanding balance is
• to the nearest dollar.
• Thus the prospective and retrospective methods
  produce the same answer.

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Example 5.2

• A loan is being repaid by 20 payments of 1000
  each. At the time of the 5-th payment, the
  borrower wishes to pay an extra 2000 and then
  repay the balance over 12 years with a revised
  annual payment. If the effective rate of interest
  is 9, find the amount of this revised annual
  payment.

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Example 5.2

• The balance after five years, prospectively is
• If the borrow makes an additional payment of
  2000, then the loan balance becomes 6060.70.
• So to repay this balance by 12 more payments, the
  equation of value is
• Solving for X, we get

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Example 5.3

• A 20,000 mortgage is being repaid with 20
  annual installments at the end of each year. The
  borrower makes five payments and then is
  temporarily unable to make payments for the next
  two years. Find an expression for the revised
  payment to start at the end of the 8th year if
  the loan is still to be repaid at the end of the
  original 20 years.

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Example 5.3

• Solution

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Amortization Methods

• If a loan is repaid by the amortization method,
  each payment consists of interest and principal.
• Determining the amount of interest and principal
  is important for both the lender and the
  borrower.
• For example, interest and principal are generally
  treated differently for income tax purposes.

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Amortization Schedules

• An amortization schedule is a table which shows
  the division of each payment into principal and
  interest,
• Together with the outstanding balance after each
  payment.
• Suppose a loan requires repayment by n payments
  of 1 at the end of year. Then the initial loan
  is
• Let It and Pt be the amount of interest and
  principal included in the t-th payment.

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An Amortization Schedule
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Remarks (1)

• The total of all interest payments is represented
  by the total of all amortization payments less
  the original loan
• The total of all the principal payments must
  equal to the original loan

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Remarks (2)

• Note that the outstanding loan at t n is
  equal to 0.
• The whole point of amortizing is to reducing the
  loan to 0 within n years.
• Principal repayments increase by a factor of (1
  i) in each period. This is because in each
  period, the outstanding balance is decreasing,
  and as a result the interest charged is also
  decreasing. So more principal is paid in each
  subsequent payment.

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Remarks (3)

• If the installment payment at the end of each
  period is R, then we have the relationship
• which represents the recursion method.
• the outstanding loan balance at the end of
  tth period
• the amount of interest paid in the tth
  installment
• the amount of principal repaid in the same
  installment

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Example 5.4

• Ron is repaying a loan with payments of 1 at the
  end of each year for n years. The amount of
  interest paid in period t plus the amount of
  principal repaid in period t 1 equals X.
• Calculate X.
• Solution

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Example 5.5

• A 1000 loan is being repaid by payments of 100
  at the end of each quarter for as long as
  necessary, plus a smaller final payment. If the
  nominal rate of interest convertible quarterly is
  16, find the amount of principal and interest in
  the fourth payment

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Example 5.5

• The outstanding load balance at the beginning of
  the fourth quarter, i.e. the end of the third
  quarter, is
• The interest contained in the fourth payment is
• The principle contained in the fourth payment is

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Example 5.6

• A loan is being repaid with quarterly
  installments of 1000 at the end of each quarter
  for five years at 12 convertible quarterly. Find
  the amount of principal and interest in the sixth
  installment.
• Solution

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Example 5.7

• A loan is being repaid with a series of payments
  at the end of each quarter for five years. If the
  amount of principal in the third payment is 100,
  find the amount of principal in the last five
  payments. Interest is at the rate of 10
  convertible quarterly.
• Solution

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Example 5.8

• A borrows 10,000 from B and agrees to repay it
  with equal quarterly installments of principal
  and interest at 8 convertible quarterly over six
  years. At the end of two years B sells the right
  to receive future payments to C at a price that
  will yield C 10 convertible quarterly. Find the
  total amount of interest received
• By C.
• By B.

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Example 5.8 (1)

• The quarterly installment paid by A is
• (1)The price C pays is the present value of the
  remaining payments at a rate of interest equal to
  2.5 per quarter, i.e.
• The total payments made by A over the last four
  years is
• (16)(528.71)8459.36
• The total interest received by C is
• 8459.36-6902.311557.05

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Example 5.8 (2)

• (2) There are methods to calculate the total
  interest received by B.
• a. The outstanding loan balance on Bs original
  amortization schedule at the end of two years is
• The total principal repaid by A over the first
  two years is
• 10,000-7178.672821.33
• The total payments made by A over this period are
• (8)(528.71)4229.68
• Thus, the total interest received by B apparently
  is
• 4229.68-2821.331408.35

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Example 5.8 (2)

• b. By lending out 10,000, B gets
  (8)(528.71)6902.3111131.99 in return.
• So the total interest received by B is
• 11131.99 -10,0001131.99
• Note that a and b result in different answers,
  which one is more reasonable?

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Example 5.9

• An amount is invested at an annual effective rate
  of interest i which is just sufficient to pay 1
  at the end of each year for n years. In the first
  year the fund actually earns rate i and 1 is paid
  at the end of the year. However, in the second
  year the fund earns rate j where jgti. Find the
  revised payment which could be made at the ends
  of years 2 through n
• (1) Assuming the rate earned reverts back to i
  again after this one year.
• (2)Assuming the rate earned remains at j for the
  rest of the n-year period

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Example 5.9(1)

• The initial investment is and the
  account balance at the end of the first year is
  
• .Let X be the revised payment. We can get the
  followings

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Example 5.9(1)

• And must equal the present value of the
  future payments. Thus, we have
• Which gives

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Example 5.9 (2)

• 2. The development is identical to case 1 above,
  except that the present value of the future
  payments, which equals , is computed at rate
  j instead of i. Thus, we have
• Which gives

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Example 5.10

• A loan is being repaid with installments of
  1 at the end of each year for 20 years. Interest
  is at effective rate i for the first 10 years and
  effective rate j for the second 10 years. Find
  expressions for
• The amount of interest paid in the 5th
  installment.
• The amount of principal repaid in the 15th
  installment.

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Example 5.10

• Solution

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Sinking Funds

• Suppose of a loan of is repaid with single
  lump-sum at t n. If annual end-of-year
  interest payment of are being met
  each year, then that lump-sum required is .
• Suppose the lump-sum required at time n is to
  be built up in a sinking fund, and this fund is
  credited with effective interest rate i.

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Sinking Fund Payments

• If the lump-sum is to be built up with annual
  end-of-year payments for the next n years, then
  the sinking fund payment is
• Then the total annual payment made by the
  borrower is the annual interest due on the loan
  plus the sinking fund payment, i.e.

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Net amount of loan (1)

• The accumulated value of the sinking fund at time
  t, denoted by SFt, is the accumulated value
  of the sinking fund payments made to date and is
  calculated as follows
• The loan itself will not grow if the annual
  interest is paid at the end of each year.
• We shall call the amount of the loan over the
  accumulated amount of sinking fund the net amount
  of loan.

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Net amount of loan (2)

• The net amount of loan can be calculated as
  follows
• Net Loant Loan ? SFt
• In other words, the net amount of the loan under
  the sinking fund method is the same as the
  outstanding loan under the amortization method.

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Net amount of interest

• Each year, the amount of interest the borrower
  pays interest to the lender is .
• Each year the borrower also earns interest from
  the sinking fund to the amount of i ? SFt-1.
• So the actual interest paid by the borrower in
  year t, called the net amount of interest, is
• So we see that the net amount of interest paid
  under the sinking fund method is the same as the
  interest payment under the amortization method.

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Sinking Fund Increase

• The sinking fund grows each year by the amount of
  interest it earns and by the end-of-year
  contribution that it receives.
• In other words, the annual increase in sinking
  fund is the same as the principal repayment under
  the amortization method.
• Both methods are aiming at paying back the
  principal. In amortization method, that was done
  every year. In the sinking fund method that was
  done at the very end, and at the same time, an
  amount was set aside to accumulate to the final
  payment.

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Sinking fund with different interest rate (1)

• Usually, the interest rate on borrowing, i, is
  greater than the interest rate offered by
  investing in a fund, j.
• The total payment under sinking fund approach is
  then
• We wish now to determine the interest rate i',
  for which the amortization method would provide
  for the same level of payment

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Sinking with different interest rate (2)

• Therefore, the amortization payment, using this
  mixed interest rate, will cover the smaller
  amortization payment at rate j and the interest
  rate shortfall, i ? j, that the smaller payment
  does not recognize.
• The mixed interest rate can be approximated by
  the formula

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Example 5.11

• John wants to borrow 1000.
• HSBC offers a loan in which the principal is to
  be repaid at the end of four years. In the
  meantime, 10 effective is to be paid on the loan
  and John is to accumulate the amount necessary to
  repay the loan by means of annual deposits in a
  sinking fund earning 8 effective.
• HS Bank offers a loan for four years in which
  John repays the loan by amortization method.
• What is the rate charged by HS Bank if the two
  offers make no difference to John?

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Example 5.11

• Under both method, John has to make 4 annual
  payments. So if the two offers make no difference
  to John, the annual payments must be equal.
• Annual payment under HSBC plan is
• Suppose the amortization offer is i effective,
  then
• or
• By iteration method, we can determine i
  10.94.
• Note that the approximation method gives
  10 0.5(10?8) 11,
• Which is close to the above result of 10.94

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Differing payment periods and interest conversion
periods

• Find the rate of interest, convertible at the
  same frequency as payments are made, that is
  equivalent to the given rate of interest
• Using this new rate of interest, construct the
  amortization schedule

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Example 5.12

• A debt is being amortized by means of monthly
  payments at an annual effective rate of interest
  of 11. If the amount of principal in the third
  payment is 1000, find the amount of principal in
  the 33rd payment
• The principle repaid will be a geometric
  progression with common ration
• The interval from the 3rd payment to the 33rd
  payment is
• (33-3)/122.5 years.
• Thus the principal in the 33rd payment is
  

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Example 5.13

• A borrows 10,000 for five years at 12
  convertible semiannually. A replaces the
  principal by means of deposits at the end of
  every year for five years into a sinking fund
  which earns 8 effective. Find the total dollar
  amount which A must pay over the five year period
  to completely repay the loan.
• Solution

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Example 5.14

• A borrower takes out a loan of 2000 for two
  years. Find the sinking fund deposit if the
  lender receives 10 effective on the loan and if
  the borrower replaces the amount of the loan with
  semiannual deposits in a sinking fund earning 8
  convertible quarterly.
• All three frequencies differ
• Interest payments on the loan are made annually
• Sinking fund deposits are made semiannually
• Interest on the sinking fund is convertible
  quarterly

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Example 5.14

• The interest payments on the loan are 200 at the
  end of each year. Let the sinking fund deposit be
  D. Then
• or

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Varying Series of Payments

• Assume that the varying payments by the borrower
  are R1,R2., Rn and that . Let the
  amount of the loan be denoted by L. Then the
  sinking fund deposit for the t-th period is
  Rt-iL.
• The accumulated value of the sinking fund at the
  end of n periods must be L, we have
• Or

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Example 5.15

• A borrower is repaying a loan at 5 effective
  with payments at the end of each year for 10
  years, such that the payment the first year is
  200, the second year 190, and so forth, until
  the 10th year it is 110. Find (1) the amount of
  the loan (2)the principal and interest in the
  fifth payment.
• The amount of the loan is

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Example 5.15

• We have

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Example 5.16

• A borrows 20,000 from B and agree to repay it
  with 20 equal annual installments of principal
  plus interest on the unpaid balance at 3
  effective. After 10 years B sells the right to
  future payments to C, at a price that yields C 5
  effective over the remaining 10 years. Find the
  price which C should pay to the nearest dollar.

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Example 5.16

• Each year A pays 1000 principal plus interest on
  the unpaid balance at 3. The price to C at the
  end of the 10th year is the present value of the
  remaining payments, i.e.
• The answer must be less than the outstanding loan
  balance of 10,000, since C has a yield rate in
  excess of 3.

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Example 5.17

• A loan is amortized over five years with monthly
  payments at a nominal interest rate of 9
  compounded monthly. The first payment is 1000 and
  is to be paid one month from the date of the
  loan. Each succeeding monthly payment will be 2
  lower than the prior payment. Calculate the
  outstanding loan balance immediately after the
  40th payment is made.

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Example 5.17

• Solution

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Example 5.18

• A loan is repaid with payments which start at
  200 the first year and increase by 50 per year
  until a payment of 1000 is made, at which time
  payments cease. If interest is 4 effective, find
  the amount of principal in the fourth payment.

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Example 5.18

• Solution