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It should be noted diat in terms of die time diagram, diis is equivalent to bringing the entire amount of interest / onto the time diagram at time t instead of time 1. The expressions + (1 -0/is often called die "exposure associated widiS,."

19. Derive formula (5.20).

20. Find an expression for die total net contributions during die n periods in formula (5.18).

5.6 Time-weighted rates of interest

21. In Example 5.9 assume diat May 1 is changed to June 1 and November 1 is changed to October 1.

a) Would the yield rate change when computed by die dollar-weighted mediod?

b) Would the yield rate change when computed by die time-weighted method?

22. In Example 5.9 assume diat everydiing is unchanged except diat an additional $5000 is wididrawn on July 1.

a) Recompute die yield rate by die dollar-weighted mediod.

b) The yield rate by die time-weighted mediod cannot be calculated. Explain why not.

23. Deposits of $1000 are made into an investment fiind at time 0 and time 1. The fiind i balance is $1200 at time 1 and $2200 at time 2.

a) Compute the annual effective yield rate computed by a doUar-weighted calculation.

b) Cwnpute die annual effective yield rale which is equivalent to diat produced by ] a time-weighted calculation.

24. Let A be die fiind balance on January 1, die balance on June 30, and die balance on December 31.

25. 26.


28. 29.

I 30.

I 31.

investment expenses of $5000. Find the net effective rate of interest yielded by the funds.

16. A fund earning 4% effective has a balance of $1000 at the beginning of the year. If $200 is added to the fund at the end of three months and if $300 is withdrawn from the fund at the end of nine months, find the ending balance under the assumption

i-,, = (i -0.

17. a) Under the assumption that j ,(, = (1 - f)/, find an expression for (ig. b) Under the assumption that ,/o = expression for , ,/,.

18. Let the exposure associated with / (i.e. the denominator of formula (5.13)) be denoted by E. By using formula (5.17) show diat die force of interest at any point is given by

a) If there are no deposits or withdrawals, show that yield rates computed by die dollar-weighted mediod and die time-weighted method are bodi equal to (C - A)IA.

b) If there was a single deposit of D immediately after the June 30 balance was calculated, find expressions for the dollar-weighted and time-weighted yield rates.

c) Rework ) if die deposit occurred immediately before die June 30 balance was calculated.

d) Verbally interpret die fact that die dollar-weighted yield rates in ) and (c) are equal.

e) Show that the time-weighted yield rate in ) is greater than in (c).

5.7 Portfolio methods and investment year methods

Findijj from the data in Table 5.2 assuming the first payment is made in calendar year + 3.

ft is known diat 1 + if = (1.08 -I- .0050* "" for integral 1 < < 5, and integral y,0 <y < 10. If $1000 is invested for three years beginning in year = 5, find die equivalent level effective rate of interest.

An investment year mediod is defined by creating an accumulation function which is a function of two variables. Let a{s, t) be the accumulated value at time t of an original investment of one unit at time s, where 0 < s < t.

a) Express 5,in terms oia{s, t).

b) Expressais, t) in terms of5

c) Express a{s, t) in terms of a{s) and a{t) for die portfoUo mediod.

d) Find a(0, t) assuming a level effective rate of interest i.

e) Find a{t, t).

5.8 Capital budgeting

If an investors interest preference rate is 12%, should the investment in Exercise 4 be accepted or rejected?

A used car can be purchased for $5000 cash or for $2400 down and $1500 at die end of each of die next two years. Should a purchaser widi an interest preference rate of 10% pay cash or finance die car?

d) Draw a graph of P(i) for Example 5.4.

b) What can you conclude about the graph of P{i) when all yield rates are imaginary?

5.9 More general borrowing/lending models

An investor considering whedier or not to make die investment given in die illustration at the beginning of Section 5.3 is confused diat die yield rate is eqiial to


2000 -4000

3000 -4000


If = 15% and/= 10%, find b5.

Miscellaneous problems

34. An investment account is established on which it is estimated that 8 % can be earned over die next 20 years. If die interest each year is subject to income tax at a 25% tax rate, find die percentage reduction in die accumulated interest at die end of 20 years.

35. A borrower needs $800. The funds can be obtained in two ways:

(0 By promising to pay $900 at die end of die period. (h) By borrowing $1000 and repaying $1120 at die end of die period. If die interest preference rate for die period is 10%, which option should be chosen?

36. The proceeds from a hfe insurance poUcy are left on deposit, widi interest credited at die end of each year. The beneficiary makes withdrawals from die fiind at die end of each yearr, for r = 1,2,.. . , 10. At die minimum interest rate of 3% guaranteed in die poUcy, die equal annual withdrawal would be $1000. However, the insurer credits interest at die rate of 4% for die first four years and 5% for die next six years. The actual amount withdrawn at the end of year t is


where F, is die amount of die fund, including interest, prior to die wididrawal. Calculate Wjq .

37. An investment fund had a balance on January 1 of $273,000 and a balance on December 31 of $372,000. The amount of interest eamed during die year was

either 10% or 20%. In order to resolve the confusion, die investor decides to "pre-fiind" die contribution at the end of two years in a fund earning 12%.

a) Compute the yield rate on the transaction under these conditions.

b) Is the yield rate in (a) unique?

32. Again considering the same transaction as in Exercise 31, find the project financing rate / corresponding to a project retiirn rate r = 15%.

33. The cash flows for the first five years of a ten-year investment period are as follows:

t C,

$18,000 and die computed yield rate on die fund was 6%. What was die average date for conuibutions to and wididrawals from die fund?

An invesmient fimd is started widi an initial deposit of 1 at time 0. New deposits are made continuously at die annual rate 1 + J at time t over die next n years. The force of interest at time t is given by5, = (1 + 0 Rnd die accumulated value die fund at die end of n years.

Amortization schedules and sinking funds


In Chapter 6 various methods of repaying a loan are analyzed in more depth than in previous chapters. In particular, two methods of repaying a loan are discussed:

The amortization method. In this method the borrower repays the lender by means of installment payments at periodic intervals. This process is called "amortization" of the loan.

The sinking fund method. In this method the borrower repays the lender by means of one lump-sum payment at the end of the term of the loan. The borrower pays interest on the loan in installments over this period. It is also assumed that the borrower makes periodic payments into a fund, called a "sinking fund," which will accumulate to the amount of the loan to be repaid at the end of the term of the loan.

Chapter 6 also considers the following questions which are related to the repayment methods mentioned above:

1. How can the outstanding loan balance at any given point in time be determined?

2. How can any payments made by the borrower be divided into repayment of principal and payment of interest?

Sections 6.2 and 6.3 consider the amortization method, while Section 6.4 considers the sinking fund method. Succeeding sections develop extensions and generalization of both methods.


If a loan is being repaid by the amortization method, the installment payments form an annuity whose present value is equal to the original amount of the loan. Section 6.2 is concerned with determining the amount of the outstanding loan balance any time after the inception date of the loan. The reader may encounter a variety of terms synonymous with "outstanding loan balance" in common usage. Among these are "outstanding principal," "unpaid balance," and "remaining loan indebtedness."

Determining the amount of the outstanding loan balance is of great significance in practice. For example, if a family is buying a home with a 30-year mortgage, after making mortgage payments for 12 years, how much would they have to pay in one lump sum in order to completely pay off the mortgage?

These are two approaches used in finding the amount of the outstanding loan balance; namely, the prospective method and the retrospective method. The names chosen are appropriate, since the prospective method calculates the outstanding loan balance looking into the future, while the retrospective method calculates the outstanding loan balance looking into the past.

According to the prospective method, the outstanding loan balance at any point in time is equal to the present value at that date of the remaining payments. According to the retrospective method, the outstanding loan balance at any point in time is equal to the original amount of the loan accumulated to that date less the accumulated value at that date of all payments previously made.

It is possible to show that, in general, the prospective and retrospective methods are equivalent. At the inception date of the loan we have the following equality:

Present Value of Payments = Amount of Loan.

We now accumulate each side of this equation to the date at which the outstanding loan balance is desired, obtaining:

Current Value of Payments = Accumulated Value of Loan.

However, the payments can be divided into past and future payments giving:

Accumulated Value of Past Payments + Present Value of Future Payments = Accumulated Value of Loan.

Now, rearranging, we obtain:

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