Compound interest. Fixed-interest bonds. Equities and real estate. Real returns. Index-linked bonds. Foreign currency investments. Derivative securties. Part II Statistics for Investment. Describing investment data. Modelling investment returns.
Estimating parameters and hypothesis testing. Measuring and testing comovements in returns. Part III Applications. Modern portfolio theory and asset pricing. Market indices. Portfolio Performance Measurement. Bond analysis. Option pricing models. Stochastic investment models. Compound interest tables. Student's t distribution: critical points. Areas in the right-hand tail of the normal distribution. Features Clearly explains the basics of investment mathematics and statistics.
Values of obligations. A financial obligation is a promise to pay, or, an obligation is equivalent to a promissory note. One year after date of note a , what does Y receive on discounting it with a banker B to whom money is worth. Maturity value of obligation 6 is 1. Its value to W, 2 years and 9 months before due, is Under a stipulated rate of interest, the value of an obligation, n years after its maturity date, is the compound amount which would be on hand if the maturity value had been invested for n years at the stipulated interest rate, Example 3.
Note 6 was not paid when due. What should X pay at the end of 5 years to cancel the obligation if money is worth. Maturity value of note is lOOO l. Value at the rate. If a sum is dite with accumulated interest, this fact will be mentioned explicitly. If money is worth. Signed 5. On January 15, , what does Y receive on selling this note to a bank which uses the rate. What would Y receive for the note in problem 3 if he discounted it on July.
What is the value of this obligation two years before it is due to a man to whom money is worth. It was to be repaid on August 15, , with accumulated interest at the rate. No payment was made until August 15, What was due then if money was considered worth. Find the value of the obligation of problem 7 on November 15, , if money is worth.
At the end of one year what should X pay to cancel the obligations if money is worth. X should pay the. By use of Rule 2 of Section 11 find the amount the banker will pay. The value of an obligation depends on when it is due. Hence, to compare two obligations, due on different dates, the values of the obligations must be compared on some common date.
Compare values at the end of 4 years under the rate. The value of a after 4 years is 1. Tho value of b after 4 years is 1. Hence, a is tho more valuable. The value of an obligation on any date, the present for example, is the sum of money which if possessed to-day is as desirable as the payment promised in the obligation, If the present values of two obligations are the same, their values at any future time must likewise be equal, because these future values are the compound amounts of the two equal present values.
Similarly, if the present values are equal, the values at any previous date must have been equal, because these former values would be the results obtained on discounting the two equal present values to the previous date. Hence, any comparison date may be used in comparing the values of two obligations, be- cause if their values are equal on one date they are equal on all other dates, both past and future.
If the value of one obligation is greater than that of another on one date, it will be the greater on all dates. For instance, in Example 4 above, on comparing values at the end of 3 years, the value of a is 1. Hence, as in the original solution, a is seen to be tho more valuable. The comparison date should be selected so as to minimize the computation required. Therefore, the original solution of Example 4 was the most desirable. If money ia worth.
Use 4 years from now as the comparison date. Solve problem 2 with 6 years from now as the comparison date. Which obligation is the more valuable if money is worth. An equation of value is an equation stating that the sum of the values, on a certain comparison date, of one set of obligations equals the sum of the values on this date of another set. Equations of value are the most powerful tools available for solving problems throughout the mathematics of investment.
In writing an equation of value, the comparison date must be explicitly mentioned, and every term in the equation must represent the value of some obligation on this date. To avoid errors, make preliminary lists of the sets of obligations being compared. W wishes to pay in full by making two equal payments at the ends of the 3d and 4th years. W wishes to replace his old obligations by two new ones. Let 4 years from now be the comparison date. This sum must equal the sum of the values of the new obligations given in the right member.
If 5 years from the present were used as the comparison date, the equation would be 1. All obligations were accumulated for one more year in writing, equation 21 as compared with equation List the obliga- tions being compared. W desires to discharge his obligations in problem 1 by two equal payments made at the ends of 1 year and of 1 year and 6 months, respec- tively. Find the payments if money is worth.
What payment made at the end of 2 years will discharge the fol- lowing obligations jf money is worth. What sum, paid at the end of 2 years, will complete payment of the obligations of problem 4 if twice that sum was previously paid at the end pf the first year? Money is worth. What single additional payment should he make at the end of 5 years to cancel his obligations if money is worth.
Determine whether it would be to the creditor's advantage in problem 8 to stipulate that money is worth. Interpolation methods. The usual problem in compound interest, where the rate or the time is the only unknown quantity, may be solved approximately by interpolating in Table V.
The method is the same as that used in finding a number N from a logarithm table when log N is known. Let r be the unknown rate per period. The nominal rate will be 4 r. In finding r by interpola- tion we assume that r is the same proportion of the way from. Since 1. Interest rates per period determined as above are usually in error by not more than 1 J'fr.
Results obtained by interpolation should be computed to one more than the number of decimal places wfiich are ex- pected to be accurate. When interpolating, it is sufficient to use only four decimal places of the entries in Table V. Use of more places does not increase the accuracy of the final results and causes unnecessary computation.
He has verified its truth for- numerous examples distributed over the complete range of T,bi,v. Let k be the necessary number of interest periods. The first and third entries in the table are from Table V. A value of k obtained as above 4 is in error by not more than J of the interest rate J per period. Let k be the number of conversion periods of the rate. With the present as a comparison date, the equation of value for the obligations is 1.
How long will it take for money to double itself if left to accumulate at. Another approximate method is furnished by the follow- ing rule. Rule I. The sum is the time in conversion periods. The error of this approximate result generally is less than a few hundredths of a period. A knowledge of the calculus is necessary in reading this note. In each problem in the table, find the missing quantity. Interest is compounded semi- annually.
How long will it take for money to quadruple itself if invested at. By use of Rule 1, determine how long it takes" for money to double itself under each of the following rates : a. By use of the results of problem 13, determine how long it takes for money to quadruple itself under each of the four rates in problem Logarithmic methods. Problems may arise to which the tables at hand do not apply, or in which more accuracy is desired than is obtainable by interpolation methods. Logarithmic methods are available in such cases.
If Table I were used in obtaining log 1. Let r be the unknown rate per period ; the nominal rate is 4 r. Let k be the necessary number of conversion periods. Use Table II whenever advisable. Solve problems 2 and 5 of Exercise XVI by exact methods.
Solve problems 3 and 4 of Exercise XYI by exact methods. Find the nominal rate which, if converted semi-annually, is equiva- lent to the rate. One dollar is allowed to accumulate at. A second dollar accumulates at. When will the compound amount on the second dollar be three times that on the first? Take the logarithm of both aides of the equation obtained. The equated time. The equated date for a set of obligations is the date on which they could be discharged by a single payment equal to the sum of the maturity values of the obligations.
The time between the present and the equated date is called the equated time, and it is found by solving an equation of value. With the present as the comparison date, the corresponding equation of value is The present was used as the comparison date above to avoid having k appear on both sides of the equation. To obtain the equated time approximately, the following rule is usually used.
Add these products and divide by the sum of the maturity values to obtain the equated time. Rule 1 is always used in finding the equated date for short-term commercial accounts. The equated date for an account is also called the average date and the process of finding the average date is called averaging the account. Since Rule 1 does not involve the interest rate, it is unnecessary to state the rate when asking for the equated date for an account.
Results obtained by use of Rule 1 are always a little too large, so that a debtor is favored by its use. The accuracy of the rule is greater when the interest rate is low than when it is high. The accuracy is greater for short- term than for long-term obligations. Solve by Rule 1. Solve problem 1 by the exact method of Example 1 above. Note 2. I If money is worth. Solve by the exact method II Solve by Rule 1.
Use Rule 1. By use of Rule 1 find the equated time and the equated date for the payment of the notes, considering for convenience that March 9 is the present. Continuously convertible interest ; the force of interest. To prove this we use the theory of limits. Find the effective rate if the nominal rate is. The force of interest, corresponding to a given effective rate i, is the nominal rate which, if converted continuously, will yield the effective rate i. Find the force of interest if the effective rate is.
What has been the annual rate of growth of his capital if the rate is assumed to have been uniform through the 6 years? Solve by two methods. What additional sum will A pay at maturity? Justify your answer in one sentence. Find its present value by the practical rule if money is worth. If he desires to reinvest his money, what is the lowest rate, payable quarterly, which his new securities should yield? A second dollar. When will the compound amount on the 1 second dollar be double the simple interest amount on the first?
Use either interpolation or logarithmic methods. See problem An annuity is a sequence of periodic pay- ments. An annuity certain is one whose payments extend over a fixed term of years. For instance, the monthly payments made in purchasing a house on the instalment plan, form an annuity certain.
A contingent annuity is one whose payments -last for a period of time which depends on events whose dates of occurrence cannot be accurately foretold. For instance, a sequence of pay- ments such as the premiums on an insurance policy which ends at the death of some individual form a contingent annuity. In Part I of this book we consider only annuities certain. The sum of the payments of an annuity made in one year is called the antwl rent. The time between successive payment dates is the payment interval.
The time between the beginning of the first 'payment interval and the end of the last, is called the term of the annuity. Unless otherwise stated, all payments of an annuity are equal, and they are due at the ends of the.
Under a specified rate of interest, the present value of an annuity is the sum of the. The amount of an annuity is the sum of the compound amounts that would be on hand at the end of the term if all payments should accumulate at interest until then from the dates on which they are due. NOOTB 1. We obtain the present value A of this annuity by adding the 2d column in the table below, and the amount S by adding the 3d column.
This relation is verified to hold ; A 1. The examples below 1 illustrate methods used later to obtain fundamental annuity formulas. The entries in the 2d and 4th columns below are verified by the principles of compound interest. The bracket contains a geometrical progression of 30 terms for which the ratio is w - 1. By the formula for the sum of a geometrical progression, 2.
The geometrical progression, in. We verify that A 1. A is the sum of the entries in the 2d row below. Present value of payment 1. Compute the formulas for A and and verify as in equation 26 that A is the present value of S, due at tt end of the term. Compute A and S and veril that A is the present value of S, due at the end of the term. It is important to realize that formulas 27 and 28 may be used whenever the payment interval of the annuity equals the con- version period of the interest rate, In deriving the formulas, the interest period was called 1 year, merely for concreteness.
Find the present values and the amounts of the annuities below. The cash price is the sum of the present values of all payments. The remaining 12 payments come at the ends of the payment intervals and hence form a standard annuity. The man of problem 11 has just paid the installment due at the end of 4 years and 6 months. What additional payment, if made immediately, would cancel his remaining indebtedness if money is worth.
His remaining indebtedness at any time, or the principal out- standing, is the present value of all remaining payments. See the table of Note 2 of Section Further annuity formulas. Consider the annuity whose annual. P' P etc. Also see heading of Table XI. The fact that O'n at t is the nominal rate which, if converted p times per year, yields the effective rate -i, is of importance in the applications of equation We use j f merely as a convenient abbreviation for its complicated algebraic expression.
The word year was used in this statement and in the proof of formulas 29 to 32 for the sake of concreteness. All of the reasoning remains valid if the word year is changed throughout to interest period. Jp Example 1. Payments occur twice in each interest period. Hence, use formulas 33 with the data listed below. Solution, Use formulas 33, because the payments occur three times in each interest period. A ogai. Thus, formulas 30 and 32 express the present value and the amount of an annuity payable p tunes per interest period in terms of the present value and the amount of an annuity payable once per interest period.
Complete the division. Just after the 48th payment to the fund has been made, how much is in the fund if it accumulates at. What is a fair valuation for the project if money is worth. Assume that the crossing will be used for 50 years. Thus - is the sum which, if paid at the end of 1 year, is equivalent to p payments of made at equal intervals during the year. To find the present value and the amount of this annuity when money is worth the nominal rate j, compounded m times per year, we might first compute the corresponding effective rate i and then use formulas 29 and For an annuity under Case 1 below we usually may compute the present value A and the amount 8 by means of our tables.
For an annuity under Case 2, the explicit formulas for A and S must be computed with much less aid from the tables. Case 1. The annuity is payable p times per interest period where p is an integer. The method of Section 22 applies, with additional simplification when p 1. Case 2. The annuity is not payable an integral number of times per interest period. Other sim- plifying formulas could be derived but they would not be of sufficiently gen- eral application to justify their consideration.
Find the present value A if money is worth. The annuity comes under Case 2. First disregard the cash payment. What sum is to his credit at the end of 4 years if interest is accumulating at the rate. The amount on hand is the amount of an annuity which comes under Case 1. Casel n 8 int. Formula 29 2[ 1. The answer is not stated to five digits because the numerator In every problem where the present value or amount of an annuity is to be computed, first list the case and the elements of the annuity as in the examples above.
To find A and S for an annuity we could always proceed as under Case 2, even though the annuity comes under Case 1. The only difference in method is that, under Case 2, the fundamental time unit is the year, whereas under Case 1 it is the interest period. The classification of annuity computations under two cases would not be advisable if we were always to compute A and S by the explicit formulas, as is necessary in Example 3.
But, if we used the general formulas of Case 2, with the year as a time unit, in problems under Case 1 to which Tables VII, VIII, and XII apply, unnecessary computational confusion would result and other inconvenient auxiliary formulas would have to be derived. Hence, use the method of Case 1 whenever possible. Use Table II when it is an aid to accuracy. How much will be in the fund just after the 30th deposit?
He should pay the present value of the coupons plus the present value of the redemption price. What is an equivalent cash price for the farm if money is worth. What does the debtor pay? Find the present value of the remaining payments. What sum did he borrow if the creditor's interest rate is. In problem 36, at the end of 4 years, X desires to make a single payment which will cancel his liability due to his previous failure to pay, and also will discharge the liability of the payments due in the future.
Another statement of this result would 'be that "A is the present value of S, due at the end of the term of the annuity," Use the result of problem The accumulation of the savings account, and the bonds themselves, are to be given to the boy on his 25th birthday. Find the value of the property received by him on that date. If you can invest money at. Assume in both cases that you would have to pay the upkeep.
Annuities due. The payments of the standard annuities considered previously were made at the ends of the payment intervals. An annuity due is one whose payments occur at the beginning of each, interval, so that the first payment is due im- mediately. The definitions of the amount and of the present value of an annuity as given in Section 19 apply without change of word- ing to an annuity due. It must be noticed, however, that the last payment of an annuity due occurs at the beginning of the last interval, whereas the end of the term is the end of this interval.
Hence, the amount of an annuity due is the sum of the compound amounts of the payments one interval after the last payment is ,made. The amount of this annuity is the sum of the compound amounts of the payments at the end of 6 years, the end of the term. For the treatment of annuities due and for other purposes in the future, it is essential to recognize that, regardless of when a sequence of periodic payments start, they will form an ordinary annuity if judged from a date one payment interval before the first payment.
Moreover, the sum of the accumulated values of the payments on the last payment date is the amount of this ordinary annuity. Q is 3 months before the present. Considered from Q the payments form an ordinary annuity whose term ends at L and whose present value A' and amount S' are Case 1 12 int. Since S' is the sum of the accumulated values of the payments at L, we ac- cumulate S' for 3 months to find S, which is the sum of the values at time T.
Second solution. Hence, the first solution was less complicated numerically. In some problems, however, the second solution would be the least complicated. Two rules may be stated corresponding, respectively, to the two methods of solution considered above. Rule 1. To find A and S for an annuity due, first find the present value A' and the amount S' of an ordinary annuity having the same term, a.
Then : a A is the compound amount on A' after one payment interval. To find A for an annuity due, first find A', the 'present value of all payments, omitting the first. To find S first obtain S 1 , the amount of the ordinary annuity having a payment at the end of tho term in addition to the payments of the annuity due. Then, 8-S'-W. It is customary in actuarial textbooks to use black roman type to indicate amounts and present values of annuities duo.
Find A and S for each annuity due in the table, by use of the specified rule. Carry through, the solution of problem 2 by Rule 1 far enough to be able to state why it is inconvenient. Deferred annuities. A deferred annuity is one whose term does not begin until the expiration of a certain length of time. Consider the time scale in Figure 4. The payments form an ordinary annuity when judged from B, 6 months before the first payment.
S' is the sum of accumulated values at time T. Since A' is the sum of the discounted values at B, we must discount A' for 4 years to obtain the present value A. A - Second solution for A. The new payment dates are indicated by circles in Figure 4. The present value A of the deferred annuity equals the present value A ' of the new annuity over the whole 10 years minus the present value A" of the payments over the first 4 years, which are not to be received.
Both A. From Example 1 it is clear that the amount of a deferred annuity equals the amount of an ordinary annuity having the same term. Corresponding to the two methods used above in obtaining A, we state the two rules below. To obtain A for an annuity whose term is deferred w years, first find A', the present value of the ordinary annuity having the same term.
Then, A equals the value of A' discounted for w years. The present values and the amounts of deferred annuities are indicated in actuarial writings by the symbols for ordinary annuities with a number prefixed showing the time for which the term is deferred. Find the present value of each deferred annuity in the table, by use of the specified rule. Carry through the solution of problem 4 by Rule 2 until you are able to state why it is inconvenient. Solve problem 3 if money is worth.
Assum- ing that he will live to receive all payments, find the present value of his expectation if money is worth. Find the present value of all future upkeep if money is worth. Hence ,. Use a continuous annuity as an appropriation. Under the rate. If the conversion period of an interest rate is not stated, assume it to be 1 year. The binomial theorem can be used in interest computations to which the tables do not apply.
As a special case of the binomial theorem, 2 we have 21 o! If n is a negative integer or a fraction, the series contains infinitely many terms. In this case, if x lies 1 A knowledge of the binomial theorem ifl needed in this section. The proof of this statement is too difficult for an elementary treatment. From formula 31, o. The next term in series 38, beyond the last one computed, is negligible in the 8th decimal place, and hence our result is accurate to the 7th decimal place with only a slight doubt as to the 8th.
Hence, "" The final 9 is not dependable because the final in the denominator was doubtful. Compute 3, at. If the deposits accumulate at the rate. Obtain the result correct to four significant figures. In problem 2, what is the amount at the end of 10 years? A man W has occupied a farm for 5 years and, pending decision of a case in court, has paid no rent to B to whom the farm is finally awarded.
If the man in problem 7 should place his surplus earnings in a bank, what will he have at the end of his working life if his savings earn interest at the rate. What is the present value of his agreement if money is worth. At the end of 3 years, the man of problem 9 decides to pay off his obligation to the government immediately. What should he pay if money is worth. What is the present value of the contract if the future liabilities are discounted at.
The purchaser desires to change to annual payments. What should he pay at the end of each year if money is con- sidered worth. Would the result be any different if the annuity were payable annually? In every problem below, the payment interval of the annuity and the conversion period of the interest rate will be given or, in other words, the p and m of equations I and III of Section 24 will always be known.
For an annuity under Case 1 there remain for consideration the five quantities A, S, R, i, ri. If an annuity comes under Case 2, similar remarks apply to the four quantities S, R, j, ri and to A, R, j, ri. Determination of the payment. Therefore, from at.
On doing this sub- traction mentally, we are able to read the result. Casel n 82 int. Exam-pie 3. Recognize that the solutions above wjere arranged so as to avoid computing quotients, except for the easy division by. If money is worth 5. Determination of the term.
If the term of an annuity is unknown, interpolation methods furnish the solution of the problem with sufficient accuracy for practical purposes. Let the unknown term be k interest periods. Exampk 2. Interest is at the rate. OCoff at. Q3 OB ol. Q , 8. Hence, for an annuity whose term is 4. Determination of the interest. Note 6. Lot r be the unknown rate per period. We are justified only in saying that the rate is approximately.
Supplementary Example 2. It is probable that? Since our tables do not use the rate. Since By logarithms, o at. S07 T We could obtain the solution of Example 1 with any desired degree of accuracy by successive computations as in Example 2. Our accuracy would be limited only by the extent of the logarithm tables at our disposal.
In more complicated examples, the solution may be obtained by first considering a new problem of the simple type met in Example 1. Let the unknown nominal rate be j. The rate j, compounded semi-annually, must be equivalent to the rate. Hence, the effective rates corresponding to these two rates must be the same. By use of the result of problem 11, find the effective rate of interest paid by the association of problem At what effective rate does the trust company credit interest on the fund?
First find the equivalent nominal rate, payable quarterly. Difficult cases and. When the formulas of Case 2 apply to an annuity, it is necessary to use the explicit formulas III in finding the term if it is unknown, Example 1.
What is the term if money is worth. The same preliminary work should be used for both of problems 7 and 8. First determine the nominal rate, converted quarterly, under which. Case 2 applies. Hence, let the unknown term be k years. From Table V, the denominator is 2.
The solutions of problems, treated by interpolation in Section 31, may be obtained by solving exponential equations, as in solution 6 above. In these exact solutions it is always necessary to use the explicit alge- braic expressions for the present values and the amounts of the annui- ties concerned.
If the rate is. Compare this with the result by interpolation 2 in Example 2 of Section Solve by both an inter- polation and an exact method. Solve illustrative Example 1 of Section 31 by the exact method.
Solve problem 9 of Exercise XXX by the exact method. Solve by the exact method. What interest rate, compounded sew-annually, is being used in the transaction? Under the rate 3. What effective rate of interest is being charged?
First find the nominal rate, compounded monthly. If interest is at the rate. Find the effective rate of interest by use of the relation 39 of Section He made no payments of either interest or principal for 4 years.
At that time, he agreed to discharge all liability in connection with the debt by making equal payments at the end of each 3 months for 3 years. Find the quarterly payment. Amortization of a debt. A debt, whose present value is A, is said to be amortized under a given rate of interest, if all lia- bilities as to principal and interest are discharged by a sequence of periodic payments.
When the payments are equal, as is usually the case, they form an annuity whose present value must equal A, the original liability. Hence, most problems in the amortization of debts involve the present value formulas for annuities. Many amortization problems have been solved in previous chapters.
The debt is to be paid, interest as due and original principal included, by equal installments at the end of each year for 5 years, a Find the annual payment. The present value of the payment annuity, at the rate. The schedule shows that the payments satisfy the creditor's demands for interest and likewise return his principal in installments. Notice that tho repayments of principal increase from year to year, while tho interest, payments decrease.
Amortization schedules are very useful in the bookkeeping of both debtor and creditor because the exact outstanding liability at every interest date is clearly shown. The outstanding principal, or liability at any date, is sometimes called the book value of the debt at that time. NOTE 2. A numerical veri- fication of this fact is obtained in the amortization table above if we merely alter the titles of tho columns, as below, leaving the rest of the table unchanged.
By the end of 6 years, the fund reduces to zero. In Example 1, without using the amortization schedule, determine the principal outstanding at the beginning of the third year. The outstanding principal, or liability, is the present value of all payments remaining to bo made. These form an annuity whose term is three years. The outstanding principal is This is the third entry of the first column of tho amortization schedule. Exampk 3. In problem' 1, without using the amortization table, find the prin- cipal unpaid at the end of 1 year and 6 months, just after the payment due has been made.
The fund is to provide equal payments at the end of each year for 5 years, at the end of which time the fund is to bo exhausted, a Find the annual payment to three decimal places. See Note 2, Section 34; think of the trust company as the debtor. In problem 3, without using the table, find the amount remaining in the fund at the end of 2 years, after the payment due has been made.
He wishes to discharge his debt, principal and interest included, by twelve equal semi-annual installments, the first due after 6 months. Find the necessary semi- annual payment. What part of the assessment will remain unpaid 8-t the beginning of the 4th year, after the payment due has been made?
What part of the debt will remain unpaid at the beginning of the 6th year, after the payment due has been made? In problem S, what part of the llth payment is interest and what part is repayment of principal? The debtor failed to make the first four payments. What payment at the end of 5 years would bring the debtor up to date on liis contract?
What rate of interest is being used by the insurance company? Amortization of a bonded debt. In amortizing a debt which is in the form of a bond issue, the periodic payments cannot be exactly equal. The first payment is due at the end of 1 year. At the end of the next year ,91 - Construct a schedule for the retirement of the debt, principal and interest included, by five annual payments as nearly equal as possible, the first payment due at the end of 1 year.
Construct a schedule for the amortization of the debt by 10 annual payments as nearly equal as possible. In the schedule,, make a separate column for each class of bonds. Problems in which the periodic payment is known. If the present liability of a debt, the interest rate, and the size and frequency of the amortization payments are known, the term of the payment annuity can be found as in Section Bxampk 1.
Let k be the time in interest periods necessary to amortize it with interest at the rate. A partial payment will be necessary at the next payment date. These conclusions are verified in the schedule below. Without using the amortization table, find the principal still unpaid in Example 1 at the end of 2 years, after the payment due has been made. To find M, write an equation of value, under the rate. NOTE 1. Recognize that 1. Hence, equation 41 shows that M is the difference between what the creditor should have and what he actually has.
By the method of Example 2 we can find the final installment in Example 1 without computing the amortization table. Notice that, since the exact number of the re- maining payments is known, part 6 should be done like illustrative Example 2 of Section 34 ; it would be clumsy to use the method of illus- trative Example 2 of the present section.
Sinking fund method. A sinking fund is a fund formed in order to pay an obligation falling due at some future date. In the following section, unless otherwise stated, it is assumed that the sinking funds involved are created by investing equal periodic payments. Then, the amount in a sinking fund at any time is the amount of the annuity formed by the payments, and examples involving sulking funds can be solved by use of the formulas for the amount of an annuity.
Recognize that the sinking fund is a private affair of the debtor. Usually, the desire for absolute safety for the fund would compel the debtor to invest it at a lower rate than he himself pays on his debt. The book value of the debtor's indebtedness, or his net indebted- ness, at any time may be defined as the difference between what he owes and what he has in his sinking fund.
Thus, at the end of 2 years, the book value of the debt is - The amount in the sinking fund at any time is the amount of the payment annuity up to that date and can be found without forming the table 6. Thus, the amount in the fund at the end of 2 years is Find the expense of the debt if the fund accumulates at the rate.
How much is in the sinking fund in problem 4 at the beginning of the 7th year? Find the quarterly payment if interest is at the rate. Comparison of the amortization and the sinking fund methods. Hence, the amortization method may be considered as a special case of the sinking fund method, where the creditor is custodian of the sinking fund and invests it at the rate i.
The conclusions of the last paragraph are obvious without the use of any formulas. If the debtor is able to invest his fund at the rate r, greater than i, his expense will be less than under the amortization method because, under the latter, he is investing a sinking fund with his creditor at the smaller rate i. Equation 42 is sometimes called the amortization equation, and equation 44 is called the sulking fund equation. Is it better to amortize the debt in 6 years by equal semi-annual in- stallments, or to pay interest when due and to retire the principal in one installment at the end of 6 years by the accumulation of a sinking fund by semi-annual payments, invested at.
By how much, per dollar of assessed valuation, will the annual taxes of the county be raised on account of the expense of the debt? At the end of 4 years, what payment, in addition to the one due, would cancel the remaining liability if the creditor should permit the future payments to be discounted, under the rate. The answer obtained is the rate at which the debtor could afford to amortize his debt, instead of using the sinking fund method described in the problem.
At the end of that time it was decided to amortize the remaining indebtedness by equal payments at the end of each year for 8 years. Find the annual payment. Then, the value of the "Old Obligation" below must equal the sum of the values of the "New Obligations " on whatever comparison date is selected. The end of 7 years is the most convenient comparison date. A certain state provided for the sale of farms to war veterans under the agreement that a interest should be computed at the rate.
Determine both the nominal rate, compounded semi-annually, and the effective rate. What is the size of the fund at the end of 16 years? It is advisable, first, to find the amount at the end of: 16 years due to the payments during the first 10 years. What effective rate of interest did the debtor pay? Write an equation of value at the end of 6 years. Transpose all quantities in the equation to one side and solve by interpolation. Sec Appen- dix, Note 3, Example 2.
Funds invested with building and loan associations. A building and loan association is a cooperative enterprise whose main purpose is to provide funds from which loans may be made to members of the association desiring to build homes. Some members are investors only, and do not borrow from the as- sociation.
Others are simultaneously borrowers and investors. Each share is paid for by equal periodic installments called dues, payable at the beginning of each month. Profits of the association arise from investing the money received as clues.
Members share in the profits in proportion to the amount they have paid on their shares of stock, and then- profits are credited as payments on their stock. The owner may then withdraw its value or may allow it to remain invested with the association. Over moderate periods of time, the interest rate received by an association on its invested funds is approximately constant.
The amount to the credit of a member, who has been making periodic payments on a share, is the amount of the annuity formed by his payments, -with interest at the rate being earned by the association. What is to the credit of the member just after the. The payments form an ordinary annuity of 20 payment inter- vals, if the term is considered to begin 1 month before the first payment is made.
If the association is earning at the rate. By the beginning of the 82d month, this book value has earned. Hence, no payment is necessary at the begin- ning of the 82d month. The nominal rate is What is to the mem- ber's credit just after the 50th payment? What is the effective rate earned on the shares? At what rate, com- pounded monthly, did his money increase during the 80 months? Retirement of loans made by building and loan associa- tions.
He must pay monthly interest usually in advance on the principal of his loan and dues on his stock. This method of retiring a debt is essentially a sinking fund plan, where the debtor's fund is invested in stock of the association. If shares in the association mature at the end of 82 months, at what nominal rate, compounded monthly, may the bor- rower consider that he is amortizing his debt? Let r be the unknown rate per period of 1 month. The nominal rateis If the shares mature at the end of 80 months, without a payment at that time, at what effective rate does a borrower amortize his debt?
If the shares mature at the end of months, without a payment at that time, what is the effective rate paid by a borrower? Depreciation ;. Fixed assets, such as buildings and machinery, diminish in value through use. Deprecia- tion is denned as that part of their loss in value which cannot be remedied by current repairs.
In every business enterprise, the effects of depreciation should be foreseen and funds should be accumulated whose object is to supply the money needed for the replacement of assets when worn out. The deposits in these depreciation funds are called depreciation charges, 1 and are de- ducted periodically, under the heading of expense, from the cur- rent revenues of the business.
The replacement cost of an asset equals its cost when new minus its salvage or scrap value when worn out. The replacement cost is also called the wearing value ; it is the value which is lost through wear during the life of the asset. A depreciation fund is essentially a sinking fund whose amount at the end of the life of the asset equals the replacement cost. Many different methods are in use for estimating the proper depreciation charge.
Under the sinking fund method, the periodic depreciation charges are equal and are invested at compound inter- est at a specified rate. A good depreciation plan is in harmony with the funda- mental principle of economics that capital invested in an enterprise should be kept intact. The book value of the machine, - In Example 1, find the condition per cent at the end of 6 years.
In illustrative Example 1 above, find the amount in the deprecia- tion fund at the end of 4 years, without using the depreciation table. Under the sinking fund plan, determine the depreciation charge which should be made at the end of each 6 months if the fund accumulates at. Its depreciation is to be covered by deposits in a fund at the end of each 3 months. Find the quarterly deposit if the fund earns.
Straight line method. Sf year by -, and hence the book value decreases each year by - n n This method is called the straight line method, because we obtain straight lines as the graphs of the book value and of the amount in the fund see problem 1 below. An essential characteristic of the straight line method is that, under it, all of the annual decreases in book value are equal to -th of the total n wearing value S. It is usually stated, under this method, that is vrritten off n the book value each year for depreciation.
Recall, from the table in Example 1 of Section 41, that the annual decreases in book value are not equal under the sinking fund plan ; they increase as the asset grows old. Hereafter, any general reference to the use of the sinking fund method should be understood to include the straight line plan as one possibility.
In supplementary Section 48 below, another depreciation plan is considered which applies well to assets whose depreciation in early life is large compared to that in old age. The student is referred to textbooks on valuation and on accounting for many other special methods which are in use. Usually, each of these is particularly desirable for a certain type of assets.
Composite life. If the plant of an enterprise consists of several parts whose lives are of different lengths, it is useful to have a definition for the average or composite life of the plant as a whole. Under the sinking fund method let i be the effective rate earned on the depreciation fund.
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About Michael C. Michael C. Books by Michael C. Tami Charles is a former teacher and the author of picture books, middle grade and young adult novels, and nonfiction. As a teacher, she made Read more Adams , Philip M. Booth , David C. Bowie , Della S. Request permission to reuse content from this site. Undetected location. NO YES. Investment Mathematics. Selected type: Paperback. Added to Your Shopping Cart. Print on Demand. This is a dummy description. Investment Mathematics provides an introductory analysis of investments from a quantitative viewpoint, drawing together many of the tools and techniques required by investment professionals.
Using these techniques, the authors provide simple analyses of a number of securities including fixed interest bonds, equities, index-linked bonds, foreign currency and derivatives. The book concludes with coverage of other applications, including modern portfolio theory, portfolio performance measurement and stochastic investment models. He has studied financial markets for over thirty years, as a practitioner in the City of London and as an academic.
His research interests focus mainly on investment trust pricing and risk. He has a long experience of teaching and researching in the fields of investment and social insurance and is author or co-author of a number of books and papers in these fields. Permissions Request permission to reuse content from this site. Table of contents Preface. Part I Security Analysis. Compound interest. Fixed-interest bonds. Equities and real estate.
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