Mathematics of Investment and Credit. SAMUEL A. BROVERMAN, PHD, ASA. 6th . Edition. Solutions Manual for. ACTEX Publications. Mathematics of Investment and Credit, 5th Edition (ACTEX Academic Series) [ ASA Samuel A. Broverman PhD] on homeranking.info *FREE* shipping on qualifying . Mathematics of Investment and Credit, 5th Edition (ACTEX Academic Series). ASA Samuel A. out of 5 stars Paperback. $ · Mathematics of.

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Library of Congress Cataloging-in -Publication Data. Broverman, Samuel A., Mathematics of investment and credit / Samuel A. Broverman. 5th ed. p. cm . Mathematics of Investment and Credit, 5th Edition - Ebook download as PDF File .pdf) or read book online. Matemática de Investimento e Crédito. Mathematics of Investment. & Credit. Samuel A. Broverman, ph.d, asa Revised edition of the author's Mathematics of investment and credit.

More information about this seller Contact this seller. All rights reserved. What will be the corresponding real. Force of Interest For an investment that grows according to accumulation amount function A t. Occasionally a transaction calls for interest payable in advance. This implicitly suggests that the lender is reinvesting the loan payments as they are being received.

The change is less significant. It can also be seen from Table 1. A nominal rate. Thus when interest is quoted on a nominal annual basis. This is discussed in detail in Section 1. In many situa- tions it is the method required by law. Interest rates and amounts viewed in this way are sometimes referred to as interest payable in arrears payable at the end of an interest period.

In this case the quoted interest rate is applied to obtain an amount of interest which is payable at the start of the interest period. Occasionally a transaction calls for interest payable in advance. Smith receives the loan amount of and must im- mediately pay the lender We see that for d and i to be equivalent rates. One year later he must repay the loan amount of The rate of discount is the rate used to calculate the amount by which the year end value is reduced to determine the present value.

Either measure can be used in the analysis of a financial transaction.

The net effect is that Smith receives and repays one year later. The value at the start of the year is the princip- al amount of. The interest is paid at the time the loan is made. Effective annual interest measures growth on the basis of the in- itially invested amount. In the example just consi - dered we see that an effective annual interest rate of The effective annual rate of discount is another way of describing in- vestment growth in a financial transaction.

On the other hand. Suppose that j is the effective rate of interest for a period of other than one year. The relationships between equivalent interest and discount rates for periods of other than a year are similar. The pricing of US T-Bills is based on simple discount. This underlines the fact that the concepts of discount rate and compound discount form an alternative to the concepts of interest rate and compound interest in describing the behavior of an investment.

From a practical point of view. A 0 in Equation 1. In Section 1.

Simple discount is generally only applied for periods of less than one year. The website provides a brief description of how the various quoted values are related to one another. The relationship between the quoted price and the dis- count rate is based on simple discount in which a fraction of a year is calculated on the basis of a day year. If this is converted to an annual return based on simple interest for a day year not the day year used with the dis-.

Simple Discount With a quoted annual discount rate of d. The quoted discount and in- vestment rates are annual rates. For over 20 years. Calculations for the other T-Bills quoted above are done in the same way. This would be compounded 4 times during the year to an effective annual present value factor of.

The notation d m is taken to mean that discount will have a com- pounding period of -. This annual present value factor could then be described as being equivalent to an effective annual discount rate of 7.

With in effect. This can be explained by noting that interest compounds on amounts increasing in size whereas discount compounds on amounts decreasing in size. It was chosen to facilitate comparison with the table in Example 1.

We see that the nominal annual interest rate convertible conti- nuously from Example 1. The effective annual discount rate used in Example 1. The ex - ercises at the end of the chapter examine in more detail the numerical relationship between equivalent nominal annual interest and discount rates. Note also in Table 1. Jj -year - -year period from interest rate for that period is.

The famous Black-Scholes option pricing model which will be briefly reviewed in Chapter 9 was developed on the basis of stock prices changing continuously as time goes on. Many theoretical financial models are based on events that occur in a continuous time framework.

Continuous processes are usually modeled mathematically as limits of discrete time processes. This is how we will approach measuring continuous growth of an investment.

The Jj. In this section we describe a way to measure investment growth in a continuous time framework. If ra is increased. Then S. St decreases as t increases.

Force of Interest For an investment that grows according to accumulated amount function A t. In the case of simple interest. In order for the force of interest to be defined. This was the form of force of interest denoted ear- A t lier as z co. The actuarial notation that is used for the force of interest at time t is usually St instead of z. Continuous investment growth models have been central to the analysis and development of financial models with important practical applications.

In the case of compound interest growth. In the case in which W 8t is constant with value 8 from time n x to time n 2. Using Equation 1. Integrating this equation from time t. Integrating both from time 0 to time n results in j0 A t. This relationship was already seen in Examples 1. Note that both a and b involve 5 year periods. Let us now suppose the force of interest St is constant with value 8 from time 0 to time n.

For m. The interest rate used to settle these one day loans is called the overnight rate. Major financial institutions routinely borrow and lend money among themselves overnight. Overnight Rate Bank A requires an overnight one-day loan of The difference between these amounts of interest is 0.

A widely used measure of inflation is the change in the Con-. Calculate the interest Bank A must pay in this case. This rate is approximately equal to the equivalent force of interest. We are concerned here with analyzing the relationship between interest and inflation in terms of the measurement of return on investments.

It is clear that a high rate of inflation has the effect of rapidly reducing the value pur- chasing power of currency as time goes on. This measure is often seen in financial newspapers or journals.

Alternative measures of inflation might be based on more specialized sectors in the economy. This is made clear in the following example. It is sometimes the case that an economy experiences deflation for a period of time negative inflation. They may be extremely high in some economies and almost insignificant in others. Real Rate of Interest With annual interest rate i and annual inflation rate r. We have used the phrase real return a few times already without being very specific as to its meaning.

Politicians and economists have been involved in numerous debates on the causes and effects of in- flation. It is not surprising then that periods of high inflation are usually accompanied by high interest rates. Inflation rates vary from country to country. The real rate of interest refers to the inflation-adjusted return on an investment. The study of the cause and effect relationship between interest and inflation is the concern of economists.

Investors are also concerned with the level of inflation. As a precise measure of the real growth of an investment. One year later. Smith can buy items. To measure this as a percentage. The dollar value at year end is not the same as that at year beginning.

A closer look. The invested at the beginning of the year is equal in value to after adjusting for inflation at year end. He actually receives interest plus principal for a total of At the time Smith makes the investment.

The US experienced a record peacetime rate of increasing consumer prices of Notice that the lower the inflation rate r. For — instance. One more point to note is that inflation rates are generally quoted as the rate that has been experienced in the year just completed. This pales in compari - son to the inflation rate in Hungary in July of It is usually the case that the rate of interest is greater than inflation. Hyperinflation Hyperinflation refers to the very rapid.

Hyperinflation of the type described in the previous paragraph is usually associated with a wartime economy during which consumer items may become scarce or unavailable. That rate of inflation corresponds to prices doubling every fifteen hours for the entire month.

Between and In Germany between January and November In order to make a meaningful comparison of interest and inflation. A t is the accumu- lated amount function.

It is the factor by which an investment has grown from time 0 to time t. Thus it may be more appropriate to use a projected rate of inflation for the coming year when inflation is considered in conjunction with the interest rate for the coming year. Accumulation Factor and Accumulated Amount Function a t is the accumulated value at time t of an investment of 1 made at time 0. The present value at time 0 of an amount K due at time t is Definition 1. Nominal Annual Rate of Interest.

In actuarial notation the symbol is reserved for denoting a nominal annual rate with interest compounded or convertible m times per year. The nota- tion dis taken to mean that discount will have a compounding period of 1. Force of Interest For an investment that grows according to accumulation amount function A t. The Canadian courts have mostly ruled that either no- minal or effective annual rates satisfy the requirements of Section 4.

A great deal of information on financial theory and practice can be found on the internet. These include usury laws limiting the level of interest rates and statutes specifying interest rate disclosure and interest calculation. Governments at all levels federal. Vaguely worded statutes regulating interest rates can result in legal dis- putes as to their interpretation.. Exercises with a asterisk can be regarded as supplementary exercises which cover topics in more depth.

If a withdrawal is made during the first five and one- half years. Carl with- draws K at the end of each of years 4. Calculate K. Find the accumulated value of the investment 10 years after it is made for each of the following rates: The balance in the account at the end of year 10 is At the end of 10 years. Tina will make the same two deposits.

Calculate n. Find the 5-year average annual compound rates of return for the period January What is the actual average annual compound growth rate over the past five years? What annual simple in- terest rate is implied? In order for there to be no advantage in redeeming early and reinvesting at the higher rate.

To what annual simple interest rate does this correspond? What is the larg- est simple interest rate. What is the accumulated value of the account on the day he would have received his tax refund check? Find the minimum an - nual rate needed for a 6-month deposit at the end of the first 6-month period so that Jones accumulates at least the same amount with two successive 6-month deposits as she would with the one-year deposit.

The terms of payment allow for a discount of 2. After days. Jones can also invest in a 6-month GIC at annual rate 7. Smith does not have the cash on hand 7 days later. Calculate i. Calculate j. If Smith sells after 6 months. The remainder is used to purchase units in the fund. Also offered is a two-year subscription at a cost of What is the effective annual interest rate that makes the two-year subscription equivalent to two suc- cessive one-year subscriptions?

Show that the graph of compound interest growth over time with the vertical axis transformed in this way is linear. Find the payment needed July 1. Each payment is to be replaced by two payments of Find the 2-month rate of interest implied by this proposal if the new payment scheme is financially equivalent to the old one.

If the machine has a lifetime of 4 years and interest is at a monthly rate of. The employees each earn What is the implied effective annual interest rate if the re- placement payment is accepted as equivalent to the original debts?

This shows that the yield rate quote to. P that Smith paid for the T-bill. He proposes to repay them with a single payment of one year from now. Show that any price from Are j and k equal? If not. The plan is to prohibit fishing for two years on the lake. On July 1. Smith sells to Brown the rights to the remaining payments for Find the original number. Smith wants the payments to start in 2 months rather than now.

The first deposit is on December The account pays an annual in-? Find the balance in the account on March Var- ious deposits and withdrawals are made during the period. Assume that t. Use the differential to approximate the change in the price of the T-bill if the yield rate changes to Mike deposits 2 X into a different savings account at time 0.

What minimum nominal annual rate must River Bank pay in order to be as attractive as Mountain Bank? River Bank pays interest compounded daily. What is the effective annual after-commission rate that Smith earns? Smith pays on November 9 and cashes in the bond on the following January 1.

The bond can be cashed in anytime after January 1 of the following year. The current interest rate on the yen deposits is a nominal annual rate of 3. The gov- ernment allows purchasers to pay for their bonds as late as No- vember 9. Smith does not convert the yen to dollars. He finds a bank that will issue such a term deposit.

Smith decides to roll over the deposit every 3 months. Algebraically the definition follows the relationship in Equation 1. What is his equivalent effective annual rate of interest for his transaction?

Rank ' the values in increasing size. Smith buys a. To keep his yen available. Find X under each of the following interest calculation methods. Immediately upon receiving an interest payment. What is the smallest whole number of times per year that Bank B must compound its interest in order that the rate at Bank B be at least as attractive as that at Bank A on an effective annual basis? Show that after one year Smith has the same total that she would have had if she had deposited the full X in the account and left it on deposit for the year.

In - terest is credited at a nominal discount rate of d compounded quarterly for the first 10 years. What annual simple discount rate should be charged on a loan for 1 year? Bruce deposits into his bank account. The accumulated bal- ance in the fund at the end of 30 years is Verify the relationships between the quoted discount rate.

Each account earns an effective annual discount rate of d. In other words. What percent reduction in the retail price will result?

A bank will buy the note from Smith using a simple discount rate d. Calculate d. What is the equivalent simple interest rate i earned by the bank over the period? What happens to i as n increases? Find the actual rate of return day interest rate that Smith earned during the time he held the T-Bill. Bob has no money. At the end of the second year Bob pays When the year is up. What is i in the first year?

He has the following two investment options. If options a and b result in the same accumulated amount on December The period from January 1 to July 1 is regarded as exactly! What nominal rate of interest. At the same time. Find the accumulated value at time 1 and at time 2.

Fabio deposits into a different bank account. Determine Z. At the end of 5 years. Determine k. What is the present value if the effective annual discount rate is cut in half? Peter deposits into a separate account. The amount of inter- est earned from time 3 to time 6 is X. If the force of interest is 8 during the first year and 1. If the force of.

After 7! Calculate X. His account is credited interest at a nominal rate convertible semiannually.

Find an expression for the accumulation A t function A t. If Smith closes the account during the year. According to the income tax structure. This demand by borrowers is a factor in the inevitable correction that leads to the interest rate becoming larger than the inflation rate. Thus Smith paid What will be the corresponding real.

What is his net gain on this transaction? This illustrates that during times when inflation rates exceed interest rates. One year from now Smith sells the items at the inflated price.

If both of these alternatives require the same amount of currency today. The investor pays tax only on the real interest paid [i. Alternative- ly Smith can now buy Canadian dollars at the exchange rate of 0. If inflation is at rate r. What effective annual rate must an investor earn on Britargs in Falkvinas in order that his real rate of interest match the real rate earned by an investor in Canadian dollars?

What rate of interest would an inves- tor have to earn on a standard term deposit in order to have the same after-inflation. The generic term used to describe a series of periodic payments is annuity. It is often the case as in the loan and bond examples that the payments are made at reg- ularly scheduled intervals of time. Yogi Berra Many financial transactions involve a series of payments. In this chapter we will de- velop methods for valuing a series of payments.

Prior to the availability of sophisticated calculators and computer spread- sheet programs. In the examples in Chapter 1 that dealt with transactions involving more than one payment. The calculations in many of the examples presented here can be done in an efficient way using a financial calculator or computer spreadsheet program. The presentation here emphasizes understanding the underlying prin- ciples and algebraic relationships involved in annuity valuation.

When a transaction involves a number of payments made in a systematic way. In a life insurance context. Since this book deals almost entirely with annuities-certain. The more precise term for a series of payments that are not contingent on the occurrence of any specified events is annuity-certain an annuity whose payments will definitely be made. Many of the methods developed in the past are no longer important for calculation purposes.

Smith deposits the payments in a bank account on the last day of each month.. If the first payment is deposited on May The balance in the account on June The accumulated value of the series of payments.

We can see from the diagram that since the valuation point is the time that the nth deposit is made. By applying the geometric series formula. Let us consider a series of n payments or deposits of amount 1 each. Therefore the first deposit has grown with compound interest for n.

This is the sum of the accumulated values of the individual deposits. If there is no possibility of confusion with other interest rates in a particular situation. Definition 2. The number of payments in the series is called the term of the annuity.

It should be emphasized that the notation can be used to express the accumulated value of an annuity provided the following conditions are met: The conventional interpretation of this phrase is to regard the valuation as an accumulated annuity-immediate. Figure 2. This series of payments is referred to in actuarial terminology as an ac- cumulated annuity-immediate.

If the interest rate is 0. This is allowed to continue for n periods. Suppose further that each interest payment is reinvested and continues to earn interest at rate i. Accumulated value of an annuity-immediate What level amount must be deposited on May 1 and November 1 each year from to Then the accumulation of the rein- vested interest.

We can interpret this expression in the following way. If the level amount deposited every 2. As a second step. There are a total of 16 deposits 2 per year for each of the 8 years from to — inclusive and they occur every year. Smith makes the nd deposit into the account. This can be also be represented in the following way. What is the balance in the account at that time?

The accumu- lated value at that time is 30[ This is at the end of April. What is the accumulated value of the account on December Annuity accumulation with non-level interest rates Suppose that in Example 2. This accumulated value is 1.

This is illustrated in the following modification of Example 2. SOLUTION ] In a situation in which the interest rate is at one level for a period of time and changes to another level for a subsequent period of time. The We first calculate the accumulated value in the account on December The accumulated value on December The accumu- lated value of the annuity at the time of the final payment can be found in the following way.

This me- 9 thod can be extended to situations in which the interest rate changes more than once during the term of the annuity.

The following example illustrates this point. We now consider the present value of an annuity. Note in Figure 2. There is an alternative way of approaching this situation. Time 0 i 2. The value of the final 14 payments. The total accumulated value at time 24 is Using the same technique as in Example 2. If we track the ac- count balance after each withdrawal.

Balance after 4'1' withdrawal: Suppose that the amount of the initial deposit is X. Determine the amount of the deposit Smith makes today. Balance after T' withdrawal: The present value of an annuity of payments is the value of the payments at the time.

The present value of the series of payments. It is often the case that. Consider again a se- ries of n payments of amount 1 each. There is a specific actuarial symbol that represents the present value of a such an annuity.. A typical situation in which the present value of an annuity-immediate arises is the repayment of a loan. In financial practice. The present value is calculated using the loan interest rate.

The car dealer offers Brown two alternatives on the loan: The first graph is an illustration of the fact that as interest rates increase. The following two graphs illustrate the present value of an annuity- immediate first as a function the rate of interest.

The second graph shows that the present value of a payment made far in the future is small. Since payments begin one month one payment period after the loan. Assum- ing interest accrues from the time of the car purchase. Valuation of an annuity some time before payments begin Suppose that in Example 2. Then the equation of value for option a is With a derivation similar to that for Equation 2. Equation 2. Level Interest Rates Just as Equation 2. Then it follows that.

This again illustrates the inverse relationship between the present value of an income stream and the interest rate used for valuation. Furthermore it was noted earlier in this section that increases as n increases. Since the valua- tion of the perpetuity occurs one period before the first payment, it would be referred to as a perpetuity-immediate. This notion of perpetuity can be considered from another point of view.

Suppose that X is the amount that must be invested at interest rate i per period in order to generate a perpetuity of 1 per period. In order to generate a payment of 1 without taking anything away from the existing principal amount X , the payment of 1 must be generated by interest alone. This can go on indefinitely, as long as the only amount withdrawn at the end of each year is the interest generated for that year. Valuation of a perpetuity A perpetuity-immediate pays X per year.

Brian receives the first n pay- ments, Colleen receives the next n payments, and Jeff receives the re- maining payments. To make a valuation of a level series of equally spaced payments, the information needed is 1 the number of the payments, 2 the valuation point, and 3 the interest rate per payment period.

There are a few particular valuation points that arise frequently in practice, and there is actuarial notation and terminology to represent those valuations. For a level series of payments of 1 each, we have already seen s- and an- nuity-immediate valuations. In general, if we are told that an annuity has n payments made at the end of each period, this is interpreted as the payments being made at times 1 end of Ist period , 2 end of 2nd period , Another standard annuity form is that of the annuity -due.

This form occurs most frequently in the context of life annuities, but can also be defined in the case of annuities-certain. In the case of present value, an annuity-due refers to the valuation of the annuity at the time of and including the first payment. In the case of accumulated value, annuity-due refers to the valuation of the annuity one payment period after the final payment. If an annuity is described as having payments occurring at the beginning of each period, the implication is that annuity-due valuation is intended.

There would be payments at time 0 beginning of the Ist period , time 1 begin- ning of 2nd period , The present value of the annuity would be found at time 0 and the accumulated value would be found at time n. Note that for both the present value and accumulated value of the annui- ty-due, the valuation point is one period after the valuation point for the corresponding annuity-immediate. This leads to the relationships. The first deposit occurred on January 1, Jim became unemployed and missed making deposits 60 through He then continued making monthly deposits of How much did Jim accumulate in his fund, including interest on December 31, , assuming payments continued through to December 1, ?

We note that 13 deposits will be missed, the 60th to the 12 nd , inclusive. The 60 th deposit would have occurred on December 1, , and the 12 nd deposit would have occurred on December 1 , The valuation point is December 31, , which is one month after the final deposit on December 1 , If none of the deposits had been. The interest rate is. The actual accumulated value is 58, The value on December 31 , of the missed payments is. Note that the accumulated value of the missed payments could also be for- mulated as x 1.

The value of the deposits made is 58, A series of payments can be valued at any time. Annuity-immediate and annuity due, accumulated and present value, are based on the most fre- quently used valuation points.

We have seen that annuities can be valued some time after they end or some time before they begin. Valuations can also be done within the term of the annuity, so that we would find the accumulated value of payments already made combined with the present value of payments yet to be made.

The general term to refer to the value of an annuity at any point in time is current value. In the annuities considered in Section 2. It may often be the case that the quoted interest rate has a com- pounding period that does not coincide with the annuity payment period.

For the purpose of a numerical evaluation of the annuity, we focus on the annuity payment period and determine and use the interest rate per pay- ment period that is equivalent to the quoted interest rate.

The first deposit is March 31, and the final one is December 31, A calendar quarter is regarded as exactly -year. The accumulated value of the annuity is then. The accumulated value of the annuity is. This is the way in which such a situation is dealt with in practice. Most mortgage loans are set up to have monthly payments, but the interest rate may not be quoted as a monthly rate.

Round-off error can occur if an approximate interest rate is used. Note that in Example 2. Calculators generally have at least 8 digits of accuracy, which is suffi- cient for all practical purposes. The tax incentive in the plan is that when the child begins post- secondary studies and begins to withdraw funds from the RESP ac- count, any amounts withdrawn above the original deposits made by the parents are regarded as income to the child.

It is anticipated that the child will be in a low tax bracket while pursuing post-secondary stu- dies, and will pay less tax than the parents otherwise would have paid on any earned interest. At the current time this book is being written, the Bank of Nova Scotia is advertising an annual effective interest rate of 2.

When the quoted interest rate is an effective annual rate of interest and the payments are made more frequently than once per year, the actuarial concept of an mthly payable annuity can be applied. Part b of Example 2. The total paid per year is 4, for 16 years.

The actuarial notation. In the exercises it is shown that. The general form of an accumulated mthly payable annuity-immediate is KsQV , which is interpreted as follows. The effective annual interest rate is i and payments of amount each occur at the end of every yjy -year period total amount paid per year is K.

There is a similar notation for the present value of an mthly payable annuity. For the same set of payments just described, denotes the present value of the series one payment period or. In the exercises at the end of this chap- ter it is shown that. This mthly payable annuity notation arises in a life-annuity context, where it is more likely to be used.

The annuities considered up to now all have specified individual payments at specified points in time. They are discrete annuities and frequently occur in practical situations. For theoretical purposes and for modeling complex situations, it is sometimes useful to consider continuous annuities, those which have payments made continuously over a period of time. In part b of Example 2.

The exercises at the end of the chapter consider a generalization of this situation in which payments are made every of a year. As m becomes larger the time between succes- sive payments becomes smaller. Then the amount paid dur- ing the interval from time tx to time t 2 measured using the period as the unit of time is equal to t2 - t ]. Suppose the payment continues for n pe- riods, measured from time 0 to time n.

In order to find the accumulated value of the n periods of payment, it is not possible to add up the accumu- lated values of individual payments as was done for the discrete annuities considered earlier. But we can determine the accumulated value at time n of the infinitesimal amount paid between time tx and time t 2 using diffe- rential calculus.

Continuous annuity In and Smith deposits 12 every day into an account and in he deposits 15 every day into the account. The account earns inter- est from the exact time of the deposit, with interest quoted as an effective annual rate. Find the amount in the account, including interest, on December 31 a exactly based on the daily deposits, and b using the approximation that deposits are made continuously.

Using the approach illustrated in part b of Ex- ample 2. The accumulated value would be 1. Suppose a general accumulation function is in effect.

The present value. We can solve for the unknown time factor n algebraically as follows: If accumula- tion is based on force of interest 8r. In exam- ples considered so far. Calculator functions also allow the distinction between an- '. The following examples illustrate these ideas. Most financial calculators have functions that solve for the fourth variable if any three of M. In either case. The integer part will be the number of full periodic pay- ments required.

The same comments apply to the present value of a level payment annuity-immediate. Solving for the unknown time will usually result in a value for n that is not an integer.

Find the number of regular depo- sits required and the additional fractional deposit in each of the follow- ing two cases: In this case an additional fractional deposit also called a balloon payment of amount The accumulated amount on deposit at the time of. In situations not so elementary as those in Example 2. At the Thus 14 deposits of the full amount of 50 are required. If the account is allowed to accumulate another half-year.

In that case some sort of approximation technique must be applied. The relationship TABLE 2. The first deposit is made on the last day of January Continuing in this way we obtain the results shown in Table 2. In which month does the accumulated value of the fund become greater than the total gross contribution to that point? From an inspection of the inequality. Note that Table 2. An excerpt from the website is below. IRA Advantage: An IRA is a deposit account in which funds accumulate tax-deferred until withdrawn at the time of retirement.

Upon closer inspec- 0. This example assumes deductible con- l mm tributions. Earnings grow tax-deferred warn wMtm until withdrawn at the end of the pe- Tax-Deferred riod. There may also be some income tax reduction at the time of each deposit.

The before-tax. Individual Retirement Account. The fully taxable accumulated value at the end of 40 years is 1. We can find the effective annual interest rate i which results in the stated accumulated value: The equivalent effective annual after-tax rate of interest is 1. This is not explicitly stated on the webpage. This means that any interest earned on the de- posits will be taxed at that rate. Q Unknown Interest Verify the numerical values shown in Figure 2.

We will consider the determination of the unknown interest rate. In order to value the annuity if the payment amounts do not follow any uniform pattern. Another situation would be where the in- vestment consists of a series of varying cashflows. Occasionally a complicated situation may arise in which it is not clear whether there is any solution for z. These considerations will be addressed in Chapter 5.

If the valuation rate of interest is i per payment period. Solving for the interest rate in a more general financial transaction whose payments are not level can lead to significant complications. This can be done in a straightforward way. Both functions which compute accumulated and present values of a series of up to 20 different payment amounts. For many standard financial transactions. There are a few points to keep in mind when considering a situation in- volving an unknown rate of interest.

Algebrai- cally. Many insurance companies sell in- dexed annuities with a fixed index rate r. The following example illustrates this idea..

This means that the annuity payment is adjusted periodically usually annually to account for inflation. Inflation is unlikely to be constant from year to year..

The series of payments is There is a basic geometric payment annuity valuation formula that can be applied in most such situations. At a rate i per payment period.

The present value of the series at the time of the first payment is. Smith receives monthly family allowance payments on the last day of each month. In most practical situations i would be larger than r. The accumulated value on January 1. In such a situation it is usually necessary to modify the payment period to coincide with the geometric increase period. Immediately upon receipt of a payment.

Monthly payments are constant during each ca- lendar year at 25 each month in The change in payment amount occurs once each year. This shows that the present value of an annuity-due whose payments form a geometric progression can be formulated as an annuity with level payments valued at a modified rate of interest an inflation-adjusted rate of interest. We can then apply one of the expressions just given for present and accumulated values. A way of simplifying this sum is to first group the deposits on an annual basis..

Ol We make the following assumptions: A basic form of this model assumes a constant rate of increase in the amount of the dividend paid. Using this model. This present value can be formulated as. What he receives is the 10 years of dividends and the sale price P at the end of 10 years. John sells the stock for a price of P. The risk-adjusted rate would be larger than interest rates on essentially risk- less government securities.

Calculate P. In this derivation. It would be prudent for an inves- tor to assume that there is some risk as to whether or not the anticipated fh- ture dividends will actually be paid. The fifth edition includes expanded coverage of forwards, futures, swaps and options. Convert currency. Add to Basket.

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