Introduction to the Mathematics of Finance 2ed  - Ebook download as PDF File .pdf), Text File .txt) or read book online. book is necessarily mathematical, but I hope not too mathematical. Finance by J.J. McCutcheon and W.F. Scott. mathematics of finance. We calculate an accumulated amount of some special im - mediate annuities by solving the special type of non-homogenous linear. An Introduction to the Mathematics of Finance: A Deterministic Approach P.D.F. This revision of the McCutcheon-Scott classic follows the core.
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By J. J. McCutcheon and W. F. Scott. | An Introduction to the Mathematics of Finance. By McCutcheon J. J. and Scott W. F.. - Volume Issue 3 - Simon Carne. This book is a revision of the original An Introduction to the Mathematics of. Finance by J.J. McCutcheon and W.F. Scott. The subject of financial. for the subject CT1 (Financial Mathematics) of the Actuarial Profession. The .. in financial mathematics the profitability of an investment for a short period of time of  J.J. McCutcheon, W.F. Scott. An Introduction to the Mathematics of Fi-.
What is the value of this series at time 0 on the basis of an interest rate i per unit time? This is easily seen as follows. Some securities have varying coupon rates D or varying redemption prices R. Assuming that the interest basis is unaltered. Our last solution makes use of a technique known as the indirect valuation of the capital.
It is a common assumption on consistent markets. The force of interest which. Since the basis of taxation of capital gains is usually different from that of interest income.
The theory developed in the preceding sections is unaltered if we replace the term interest by interest and capital gains less any income and capital gains taxes. Some investments. Many other securities provide both interest income and capital gains. Both income and capital gains tax are considered more fully in Chapters 7 and 8. The nominal rate of interest converted pthly. Theory of Interest Rates loss being used for a negative capital gain. If the investment is subject to an effective rate of compound interest i.
Note how these values tend to the force of interest at the appropriate time. Measuring time in years from the start of the given year and assuming that over the year the force of interest per annum was a linear function of time. Find the value at the start of the year of the nominal rate of interest per annum on transactions of term a 3 months.
Find also the corresponding values midway through the year. At the start of a given year. Find the lump sum payment under option i and the amount of the annual annuity under option ii. In return. As part of a review of his future commitments the borrower now offers either a To discharge his liability for these three debts by making an appropriate single payment 5 years from now.
Exercises ii At what constant force of interest per annum does this series of payments have the same present value as that found in i? Reproduced and republished with permission from the CFA Institute.. CFA Institute.. Closed-form expressions to evaluate the present values and accumulations of various types of annuity are derived. In this situation we assume that.. Present Values and Accumulations.
Throughout this chapter. Published by Elsevier Ltd.. The concept of an equation of value is discussed. The sum of 1 e d may be considered as a loan of 1 to be repaid after 1 unit of time on which interest of amount d is payable in advance. Eqs 3.
For this reason. But what sum paid continuously at a constant rate over the time interval [0, 1] has the same value as either of these payments? Let the required amount be s such that the amount paid in time increment dt is sdt.
This establishes the important fact that a payment of d made continuously over the period [0,1] has the same value as a payment of d at time 0 or a payment of i at time 1. Each of the three payments may be regarded as alternative methods of paying interest on a unit loan over the period. In certain situations, it may be natural to regard the force of interest as the basic parameter, with implied values for i, v, and d.
In other cases, it may be preferable to assume a certain value for i or d or v and to calculate, if necessary, the values implied for the other three parameters. Note that standard compound interest tables e.
It is left as a simple, but important, exercise for the reader to verify the relationships summarized here. Value Of. When i is small, approximate formulae for d and d in terms of i may be obtained from well-known series by neglecting the remainder after a small number of terms. We note that if i is small, then i, d, and d are all of the same order of magnitude.
Similar expressions can be derived, which give approximate relations between any of combination of d, d, and i. In most situations, only one of atr and btr will be non-zero.
At what force or rate of interest does the series of outlays have the same present value as the series of receipts? At force of interest d, the two series are of equal present value if and only if n X.
Equation 3. The latter form is known as the equation of value for the rate of interest or the yield equation. Alternatively, the equation may be written as n X. The equation of value, corresponding to Eq. For any given transaction, Eq.
We consider only real roots as d has a physical meaning. Alternative terms for the yield are the internal rate of return and the moneyweighted rate of return for the transaction, as discussed in Chapter 6. Although for certain investments the yield does not exist since the equation of value 3.
This is described in Theorem 3. The proof of Theorem 3. There is one easily described class of transaction for which the yield equation always has precisely one positive root, and this is described in Theorem 3.
Some of the fcti g will be positive and some negative, according to the convention described previously. Then the yield equation has exactly one positive root. It should be noted that Theorem 3. We omit the proof of the theorem as it is beyond the scope of the book. The existence of the yield for this type of transaction has been established by Theorem 3. The analysis of the equation of value for a given transaction may be somewhat complex. See Appendix 2 for possible methods of solution.
In this case, the root is unique and lies between i1 and i2. Demonstrate that a unique positive yield exists for this transaction and verify that it is The equation of value for the transaction is. This slightly more general form may be called the equation of value at time t0. It is, of course, directly equivalent to the original equation which is now seen to be the equation of value at time 0 , as expected from Eq. Find the yield for the transaction. Solution Choose 1 year as the unit of time.
Since f 0. A first approximation for the yield, obtained by linear interpolation, is 0: If the yield were required to a greater degree of accuracy, one might evaluate f 0.
The yield to four decimal places is, in fact, 8. Calculate a The rate of interest per annum, b The rate of discount per annum, c The force of interest per annum for the transaction.
Note that, for illustrative purposes, we have found each of i, d, and d from first principles. It would also be possible find just one value, say i, and compute the other values from Eq. Assuming that the lender agrees to the request, and that the calculation is made on the original interest basis, find the amount of the second payment under the revised transaction. Then, applying the equation of. The interpretation of the yield in Example 3.
If the investor expects to be able to make deposits over the next 5 years at a greater rate of interest than 8. Note that an equivalent approach to Example 3. The deferred payment of. Solution Using the definition of the rate of commercial discount from Section 1. Calculate the annual rate of discount and the effective rate of interest implied in each case.
It should be noted that the longer loan has the greater effective annual rate of discount and interest. Such a sequence of payments is illustrated in Figure 3.
For the series of payments illustrated in Figure 3.
In general, the quantity an is the present value at the start of any period of length n of a series of n payments, each of unit amount, to be made in arrears at unit time intervals over the period. It is common to refer to such a series of. When there is no possibility of confusion with a life annuity. It is common to refer to such a series of payments. The other three equations may be similarly interpreted. The accumulated value of the series of payments at the time the last payment is made is denoted by sn.
Equations 3. The reader should be able to write down these four expressions of Eq. The Basic Compound Interest Functions The reader should verify these relationships algebraically and by general reasoning. Confirm Eqs 3. Solution Standard tables give values of a10 and s10 equal to those calculated at each i. In view of the relationship 3. Present Values and Accumulations P As the rate of interest i increases.
It is convenient to have standard tables of annuity and accumulation values at various rates of interest. Such tables can be found later in this book.
Considering the quantity an as the value of an n-year payment stream made in arrears at time t. Using Eqs. We note that 8. Find the annual yield for this transaction. Since the right side of the equation is a monotonic function of i.
Since 3. Find the amount of each annual repayment. We then deduct from the accumulated account. Solution Two alternative solutions are considered: Four years later. Our remarks in Section 3. We estimate i by linear interpolation as i z 0: On 15 November i. This is illustrated in Figure 3.
Example 3. This can be evaluated as 3. At this stage it is perhaps worth pointing out that the Eq. During an increment of length dt at. Use two different approaches. In this case. The Basic Compound Interest Functions Either of these two equations may be used to determine the value of a deferred immediate annuity.
The value at time 0 of an annuity payable continuously between time 0 and time n. Formulae like Eq. In Chapter 4 we shall show that. Since Eq. Note similar methods to those used in Example 3.
If m is a non-negative number. For an annuity in which the payments are not all of an equal amount. A simple way of recalling Eq. The two sides of the equation represent the value at the start of the transaction of the payments made by the lender and the borrower. Solution From first principles. If the annuities are payable for n time units. For the latter. For the former. Provided that the reader has a good grasp of the underlying principles.
It is quite possible that the reader may prefer yet another method. From first principles. There are 12 annuity payments. Because of the increasing effect of discount with time. Solution As before. By using an increased rate of interest.
At this stage it is worth emphasizing that the payments either discrete or continuous need not be certain for the equation of value to be formed and used.
We now demonstrate how these two approaches can be considered equivalent. In order to avoid ambiguity. With this notation. When there is no ambiguity as to the value of i. The Basic Compound Interest Functions approximation understates the true value.
If it is desired to emphasize the rate of interest. The equation of value or yield equation as stated in Eq. Calculate the fair price for the investment on this basis. Doing this allows Eq. The corresponding addition to the rate of interest. We can express the discounting factors in terms of the force of interest d and introduce a new quantity. This form is consistent with the probability of payment reducing with time.
Solution Let the fair price of the investment be P in each case. You expect her to earn two-thirds of her tuition payment in scholarship money. Standard notation exists for all such accumulations.
Annuities can be immediate or deferred. Closed-form expressions can be derived to evaluate the present values of all such annuities. Discounting is expressed in terms of d. The equivalent expression expressed in terms of i is called the yield equation. Development begins on 1 January To estimate whether you have set aside enough money. How much should you set aside now to cover these payments? It is assumed that.
The income is received for 25 years. At the end of 20 years the investor will receive the accumulated amount of his payments. Find the effective annual rate of discount offered by the manufacturer to retailers who pay cash. Credit for 6 months will no longer be available. Does this new arrangement offer a greater or lower effective annual rate of discount to cash purchasers? The terms for cash payment will be unaltered. Find also his yield per annum on the complete transaction. Express this as an effective annual interest rate charged to those retailers who accept the credit terms.
Let i and d be the corresponding rates of interest and discount Since i p is the total interest paid.. The material is related to the nominal rate of interest.. How much interest should he pay? This question motivates what follows. Reproduced and republished with permission from the CFA Institute In Section 3.
In particular Each of these payments may be regarded as the interest for the period [ Further Compound Interest Functions or. Equations 4. Eqs 4. It is usual to include values of i p and d p.
It is essential to appreciate that. If we choose to regard i p or d p as the basic quantity. As has been mentioned in Chapter 2. It is customary to refer to i p and d p as nominal rates of interest and discount convertible pthly. The general rule to be used in conjunction with nominal rates is very simple. Express i m in terms of l. Further Compound Interest Functions Various approximations for i p and d p in terms of p and i. By proportion. We shall refer to an annuity which is payable p times per unit time as payable pthly.
Show that. The annuity payments. Consider now that annuity for which the present value is an: This approach was discussed in Section 3. In light of Eqs 4. An annuity payable pthly in arrears. If the payments are in advance.
When the rate of payment is constant and equal to 1 per unit time. If a p denotes the present value of this second annuity. Further Compound Interest Functions The present values of an immediate annuity and an annuity-due. What is the value of this series at time 0 on the basis of an interest rate i per unit time?
The situation is illustrated in Figure 4. This technique of equivalent payments may be used to value a series of payments of constant amount payable at intervals of time length r. Then the original series of payments of X. Convertible quarterly. Convertible monthly. The value is then Effective. Find the purchase price. Convertible half-yearly. This technique is illustrated in Example 4. The reader should make no attempt to memorize this last result.
For certain non-integral values of n. Further Compound Interest Functions 4. For our present purpose. Until now. Note that this is not equal to the value obtained from Eq. As before. The reader should verify that. Note that the last value should be calculated from Eq. Calculate the annual effective rate of interest obtained from the bill.
When n is not an integer multiple of p. Exercises n n When n is an integer multiple of p. Find the accumulated amount of the account immediately before the sixth payment is made. On 1 January the investor will be paid the accumulated amount of the account. Find the initial annual amount of the annuity. Immediately after receiving the monthly payment due on 1 November An annuity is payable for 20 years.
An annuity is payable in arrears for 15 years. The annual amount of the annuity is doubled after each 5-year period. On 1 November a man was in receipt of the following three annuities. In return the company will receive a level monthly annuity for 15 years. Exercises Find the amount of the revised annuity. Find to the nearest pound the amount of the monthly annuity payment.
It is desirable to consider this schedule in greater detail In particular.. The division of each payment into interest and capital is frequently necessary for taxation purposes.. The amount lent is simply the present value In this situation the investor may draw up a schedule which shows the amount of interest contained in each payment and also the amount of the loan outstanding immediately after each payment has been received In many situations ir may not depend on r.
In addition. The amount of loan repaid at time t is simply the amount by which the payment then made. An alternative expression for Ft is obtained by multiplying Eq. Loan Repayment Schedules to know the amount of loan outstanding at the time of default.
This is referred to as the retrospective approach for calculating the loan outstanding at time t. Letting ft denote the amount of loan repaid at time t. In the notation of Section 5.
This is particularly important where the loan outstanding is required in order to compute a new loan schedule following changes to the term or interest rate. The lender may construct a schedule showing the division of each payment into capital and interest.
Immediately after the tth repayment has been made. Equation 5. Capital Repaid Loan Outstanding 1 1. Immediately after the third repayment was made. The equation of value at the time the loan was issued is Xa5 0: The loan outstanding immediately after each repayment is given by the final column of the following table: Payment Interest Content The first line here.
Loan Repayment Schedules Table 5. Use a repayment schedule to calculate the value of the lump sum required to repay the loan at this time. Solution We require the annual repayment amount. For a loan repayable by a level annuity payable pthly in arrears over n time units and based on an interest rate i per unit time. Solution Both methods require the annual repayment amount.
The accumulated values of these are b Under the prospective approach. Since we are dealing with a level annuity. This is Because of rounding errors. Find a The monthly repayment. This may be obtained from one of the forms Alternatively. We need to know when this is first greater than one-half of the total monthly payment. The loan outstanding at the end of the first year i. The total of the first 12 capital payments is therefore Derive expressions for the capital and interest content of the tth payment and for the loan outstanding after the tth payment has been made.
Then The loan outstanding after t payments have been received is therefore Both expressions are equal to The investor draws up a schedule showing the division of each repayment into capital and interest. Since the tth payment is 1.
This is simply the value of the outstanding payments from the prospective method. The reader should verify that this equals the expression above for the outstanding loan at time t. Calculate the revised monthly repayment amount after this change. Solution We begin by calculating the initial repayment amounts. Working in years. When we use the retrospective approach. The schedule is easily drawn up as follows: The total charge for credit and the APR have to be disclosed in advertisements and in quotations for consumer credit agreements.
Note that the time unit used to specify F need not be the time interval between repayments. Regulations made under powers introduced in the UK Consumer Credit Act lay down what items should be treated as entering into the total charge for credit and how the rate of charge for credit should be calculated.
Consider a loan of L0. The APR is therefore closely associated with the internal rate of return for the loan that we will cover in Chapter 6. Which is to be repayable over a certain period by n level installments.
Flat Rates of Interest In many situations in which a loan is to be repaid by level installments at regular intervals. This amount includes the total capital and interest paid. For this reason it is only useful for comparing loans of equal term. Assuming that the payments are in arrears. Solution a The period of the loan is 2 years.. The mortgage is a level-payment. Loan A: The level loan repayment can be calculated with knowledge of the interest rate.
Under the retrospective method. Loan B: Freeloans is a company that offers loans at a constant effective interest rate for all terms between 3 years and 10 years. After 2 years. Each repayment amount covers the interest owed on the capital outstanding in the prior period and a contribution towards the capital repayment. The mortgage rate is 7. Under the prospective method..
The loans are typically of short duration and to high-risk consumers.
Loan Repayment Schedules student is approached by a representative of Freeloans who offers the student a year loan on the capital outstanding. These costs are still incurred even if the payment is not made by the consumer. Repayments are collected in person by representatives of the bank making the loan. Campaigners on behalf of the consumers and campaigners on behalf of the banks granting the loans are disputing one particular type of loan.
This new loan is used to pay off the original loans and will have repayments equal to half the original repayments. Assuming that the interest basis is unaltered. Assuming that the lender agrees to the request and carries out his calculations on the original interest basis.
Exercises 5. The terms of the loan provided that at any time the lender could alter the rate of interest. There was no further change to the rate of interest.
Find the revised amount of the level annual repayment. Loan Repayment Schedules 5. The borrower was given the option of either increasing the amount of his level monthly repayment or extending the term of the loan the monthly repayment remaining unchanged. Find the revised term of the loan and. Find n. The lender had the right to alter the conditions of the loan at any time and.
An investor who is not liable to tax will in fact receive the gross amount of each annuity payment i. The annuity decreases in such a way that. Exercises c Immediately after paying the 33rd quarterly installment. Find the amount of the revised quarterly repayment.
The request was granted. CFA Institute The chapter also introduces the concept of real returns that will be revisited in later chapters.
The investment or project will normally require an initial outlay and possibly other outlays in future. More complicated techniques using statistical theory CHAPTER 6 Project Appraisal and Investment Performance This chapter is largely concerned with a number of applications of compound interest theory to the assessment of investments and business ventures.
Some writers use terminology and symbols which differ from those usually employed by actuaries It is often prudent to perform calculations on more than one set of assumptions. Recall from Section These matters are. Measuring time in years. Project Appraisal and Investment Performance If any payments may be regarded as continuous. Conditions under which the yield exists.
In economics and accountancy. The latter term is By the time the project ends at time T. A more accurate value is 2. Under these conditions. We assume that the borrowing powers of the investor are not limited. Since i0 z 2. It might be thought that the investor should always select the project with the higher yield. Determine whether or not the business venture of Example 6. There may even be more than one cross-over point. If this rate of interest were 5.
Example 6. Although iA is larger than iB. In order to determine this time t1. We should therefore advise him that an investment in either loan would be profitable..
Now NPVA 0. The difference j1 e j2 between these rates of interest depends on various factors. In many practical problems. The concepts of net present value and yield are. Would you advise him to invest in either loan. Solution We first consider Investment A: The net cash flow of the opencast project to the mining company is as follows: On the assumption that the price of ore is such that this project will just break even. We reconsider the mining venture of Example 6. Solution Let P be the break-even price per tonne of ore.
When the mining company has funds to invest. By trial and interpolation. Eliminating k from these equations. The company has insufficient funds to finance this venture. This will occur when 1. It is estimated that the opencast site will produce As is shown in Examples 6. If the project is viable i. Let P 0 be the break-even price per tonne of ore under these conditions.
If this time t1 exists. If interest is ignored in Eq.. Suppose that the project ends at time T. As is clear by general reasoning.. Note that the annual rate of payment of loan interest is 1. What is the minimum price per tonne of ore which would make the project viable? Project Appraisal and Investment Performance The discounted payback period is often employed when considering a single investment of C. It is usual in many.
The accumulated profit after 25 years is from Eqs 6. Solution If interest is ignored. This is much less than the true discounted period from Example 6. In certain economic conditions. The resulting analysis may be easily understood and interpreted by those responsible for making investment decisions. It follows that. Project Appraisal and Investment Performance If e is not too large.. According to his calculations which are all based on prices. Solution a With no inflation.
Assuming that all prices and costs escalate at a compound annual rate e. The fencing may be assumed to last for 20 years. The initial costs associated with increase of size of the flock are as follows: Purchase price of sheep 4.
Since the term of the loan is n years. The payments are used to pay interest at rate i on the outstanding loan and. Under these schemes. The building society makes no allowance for expenses or for taxation. It is. Note that in Example 6. Suppose that the period in question is from time t0 to time tn where time is measured in years. Consider now an investment of 1 at time t0.
This is not always the case. Using appropriate methods. In practice these subintervals are often periods of length 1 year. The time-weighted rate of return does not depend upon any particular subdivision of the period. It also depends on the particular subdivision of the entire period.. A third index. To allow for changes in the size of the fund with time. This yield is generally referred to as the money-weighted rate of return per annum on the fund for the period.
It does not depend upon any particular subdivision of the entire period and generally gives greater weight to the yields pertaining to the times when the fund is largest. Let the given period be [t0. The money-weighted rate of return and the time-weighted rate of return are best understood in the context of an example.
Project Appraisal and Investment Performance This value of i is the time-weighted rate of return per annum for the period [t0. The second subinterval is the second year.
In some cases the linked internal rate of return with reference to a particular subdivision. A further disadvantage of the money-weighted rate of return is that the equation may not have a unique solution. The time value of money is not considered in the calculation. In many situations. The money-weighted rate of return of an investment fund over a period of time is obtained from solving the equation of value for the yield.
IRRs can be used to compare the return per unit investment achieved by different projects. This disadvantage motivates the use of the linked internal rate of return with reference to subdivisions that coincide with standard reporting periods of the fund. The internal rate of return IRR is the value of i that solves the equation of value for a project.
Summary Note that the time-weighed rate of return in Example 6. Also on 1 January For both projects. On 1 January Justify your answer. Assume that this dividend is not reinvested. After analysis.
Two other investment-counseling companies. From 1 January A rate between A rate between 0. A property developer. The development would then be complete. What is the appropriate investment decision? The developer has three possible project strategies.
She believes that she can sell the completed housing: Invest in both projects. Invest in Project 2 because it has higher NPV. Invest in Project 1 because it has higher IRR. Exercises The two projects are mutually exclusive. Calculate the sale price that the developer believes that she can receive. Project Appraisal and Investment Performance The developer also believes that she can obtain a rental income from the housing between the time that the development is completed and the time of sale.
For each project. Should he undertake the venture and. The projects are described in the following table: Sheep rearing Goat breeding Forestry Initial cost: Project Appraisal and Investment Performance a Calculate the internal rates of return of each of these projects to the nearest 0.
He may also lend money for any desired term at this same rate. If you have any money to invest after paying bank interest. Should you require further loans. Sales of compound: Outlays Income Initial outlay cost of building plant: Partial repayment of the loan will be allowed at any time.
The costs of disposal of chemical waste and demolition of the plant will be met from this account. Disposal of waste and demolition of plant: Manufacturing costs: Comment on your answers.
You should assume that investors may buy fractional parts of a unit. Ignore expenses and taxation. Find his yield on the completed transaction if i He bought the same number of units on each date.
Project Appraisal and Investment Performance b Suppose that the bank loans may be repaid partially. You should assume that investors may purchase fractional parts of a unit. On this basis. Exercises ii The yield obtained by an investor who purchased units on 1 April in each year from to inclusive and who sold his holding on 1 April This chapter is necessarily long CHAPTER 7 The Valuation of Securities One of the most important areas of practical application of compound interest theory is in the valuation of stock market securities and the determination of their yields Fixed-interest securities normally include in their title the rate of interest payable The annual interest payable to each holder Public Boards Where the interest payments are known in monetary terms in advance i.
The terms of the issue are set out by the borrower In the present chapter we begin by introducing many of the securities in question.. This chapter also introduces the complications of incorporating taxation into calculations.. Note that the terms security and loan are used interchangeably The stock is redeemable at par on 7 December The Valuation of Securities 7. It was redeemable at par on any interest date between 26 January and 26 July inclusive at the option of the government.
We consider as illustrations the following British government stocks.. Some bonds have variable redemption dates.. In fact.
It is distinct from capital gains tax. This stock.. More Complicated Examples. This stock is redeemable at par on any interest date the government chooses Some securities have varying coupon rates D or varying redemption prices R.
Some banks allow the interest and redemption proceeds to be bought and sold separately This stock was issued in as a conversion of an earlier stock. If an investor is liable to income tax.. The redemption payment is a return of the amount initially lent and is not subject to income tax. The redemption date is the date on which the redemption money is due to be paid. Particular examples include those discussed in the following sections.
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Institutional Subscription. Online Companion Materials. Instructor Ancillary Support Materials. Free Shipping Free global shipping No minimum order. Chapter 1. Introduction 1. Theory of Interest Rates 2. The Basic Compound Interest Functions 3. Further Compound Interest Functions 4. Present Values and Accumulations 4. Loan Repayment Schedules 5. Project Appraisal and Investment Performance 6. The Valuation of Securities 7. More Complicated Examples 7. Capital Gains Tax 8. Term Structures and Immunization 9.
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