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MATH 2040 Introduction to Mathematical Finance

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MATH 2040 Introduction to Mathematical Finance Instructor: Miss Liu Youmei Chapter 5 Amortization Schedules and Sinking Funds Example 3.3 Compare the total amount of ... – PowerPoint PPT presentation

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Title: MATH 2040 Introduction to Mathematical Finance


1
MATH 2040 Introduction to Mathematical Finance
Instructor Miss Liu Youmei
2
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

3
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.

4
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.

5
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.

6
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.

7
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.

8
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.

9
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

10
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.

11
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.

12
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.

13
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.

14
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.

15
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

16
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.

17
Example 5.3
  • Solution

18
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.

19
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.

20
An Amortization Schedule
21
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

22
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.

23
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

24
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

25
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

26
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

27
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

28
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

29
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.

30
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

31
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

32
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?

33
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

34
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

35
Example 5.9(1)
  • And must equal the present value of the
    future payments. Thus, we have
  • Which gives

36
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

37
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.

38
Example 5.10
  • Solution

39
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.

40
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.

41
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.

42
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.

43
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.

44
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.

45
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

46
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

47
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?

48
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

49
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

50
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

51
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

52
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

53
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

54
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

55
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

56
Example 5.15
  • We have

57
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.

58
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.

59
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.

60
Example 5.17
  • Solution

61
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.

62
Example 5.18
  • Solution
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