UMAP Module 750

UMAPModules inUndergraduateMathematicsand ItsApplications

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Applications of Differential Calculus to Finance

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Module 750Closing In on theInternal Rate of ReturnRobert A. Beezer

48 Tools for Teaching 1996

INTERMODULAR DESCRIPTION SHEET: UMAP Unit 750

TITLE: Closing In on the Internal Rate of Return

AUTHOR: Robert A. BeezerDept. of Mathematics and Computer ScienceUniversity of Puget SoundTacoma, Washington [email protected]

MATHEMATICAL FIELD: Differential calculus

APPLICATION FIELD: Finance

TARGET AUDIENCE: Students in a first-year calculus course

ABSTRACT: The internal rate of return (IRR) is a natural measure ofthe performance of an investment. While its definitionmakes use of the time value of money and is especiallysatisfying, its computation requires solving an equationthat usually cannot be solved with exact methods. Wedescribe three iterative methods for solving equations,motivated by the computation of the IRR. Two of thesemethods are general and well known (bisection, New-ton’s), and the third is peculiar to the computation ofthe IRR. Each method is illustrated by an applicationto an actual investment in stocks, bonds, or a mutualfund. Several other investments are described in theexercises.

PREREQUISITES: Continuity, intermediate value theorem, first and sec-ond derivatives. Optional: L’Hopital’s rule, Newton’smethod, exponential growth.

RELATED UNITS: Unit 640: Internal Rates of Return, by Hiram Paley, Pe-ter F. Colwell, and Roger E. Cannady. This Moduleis more concerned with situations where an invest-ment has more than one IRR. It gives a criteria fordetermining which of several IRRs is more appro-priate for a given investment. Reprinted in UMAPModules: Tools for Teaching 1983, 493–548. Lexington,MA: COMAP, 1984.

Unit 474: Nominal vs. Effective Rates of Interest, by PeterA. Lindstrom. Reprinted in UMAP Modules: Toolsfor Teaching 1987, 21–53. Arlington, MA: COMAP,1988.

Tools for Teaching 1996, 47–78. c©Copyright 1996, 1997 by COMAP, Inc. All rights reserved.

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Closing In on the Internal Rate of Return 49

Closing In onthe Internal Rate of ReturnRobert A. BeezerDept. of Mathematics and Computer ScienceUniversity of Puget SoundTacoma, Washington [email protected]

Table of Contents1. AN INHERITANCE . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Stocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Mutual Funds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2. THE TIME VALUE OF MONEY . . . . . . . . . . . . . . . . . . . . . . 5

3. THE INTERNAL RATE OF RETURN . . . . . . . . . . . . . . . . . . . . 7

4. COMPUTATION OF THE IRR . . . . . . . . . . . . . . . . . . . . . . . 9

5. ITERATIVE SOLUTION METHODS . . . . . . . . . . . . . . . . . . . . . 105.1 The Bisection Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.2 Newton’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.3 Convex Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

6. CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

7. EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

8. SOLUTIONS TO THE EXERCISES . . . . . . . . . . . . . . . . . . . . . . 25

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . 28

ABOUT THE AUTHOR . . . . . . . . . . . . . . . . . . . . . . . . . . 28


MODULES AND MONOGRAPHS IN UNDERGRADUATEMATHEMATICS AND ITS APPLICATIONS (UMAP) PROJECT

The goal of UMAP is to develop, through a community of users and devel-opers, a system of instructional modules in undergraduate mathematics andits applications, to be used to supplement existing courses and from whichcomplete courses may eventually be built.

The Project was guided by a National Advisory Board of mathematicians,scientists, and educators. UMAP was funded by a grant from the NationalScience Foundation and now is supported by the Consortium for Mathemat-ics and Its Applications (COMAP), Inc., a nonprofit corporation engaged inresearch and development in mathematics education.

Paul J. CampbellSolomon Garfunkel

EditorExecutive Director, COMAP


1. An InheritanceLet’s suppose that while you were a sophomore in college, your Great Aunt

Ina died, and her estate provided you with a substantial sum of money. Yourrelatives all counseled you to invest the money wisely so you could use itto pay as much of your graduate school tuition as possible. Ina’s husband,Adolph, suggested that you place the money in the stocks of Ina’s three fa-vorite companies—J.C. Penney, Walt Disney, and Sizzler International. Yourconservative grandfather suggested that you purchase a U.S. Treasury Note,whereas your daring older sister coaxed you to buy shares of an aggressivegrowth mutual fund. Which investment would be the best? This would be adifficult question to answer when looking ahead to the future, but with hind-sight we can analyze the performance of each of these alternative investments.So let’s assume that you took the advice of one of your relatives and it is nowtime to graduate and spend your inheritance on your graduate-school tuition.How can we analyze the three alternatives (stocks, bond, or mutual fund), tosee if the decision that you made as a sophomore was the most profitable one?

To answer this question, we describe a measure of the performance of aninvestment, known as the internal rate of return (IRR). While it has an especiallysatisfying definition, it is usually difficult to compute exactly. We present threedifferent iterative methods for computing the IRR. Together, their formulationsrequire several of the basic ideas of differential calculus—continuity, the inter-mediate value theorem, and first and second derivatives.

But first, let’s describe your inheritance more carefully, and the nature ofthe three proposed investments. Let’s assume that Ina’s bequest was in theamount of $20,000 and became available to you on February 22, 1993. Yourfinal analysis was undertaken after the last final exam of your senior year, onMay 15, 1995.

1.1 StocksStock represents partial ownership of a company and is bought so that the

purchaser can receive a portion of the profits of the company. In practice, aprofitable company does two things with its profits. A portion is given to theowners of the stock as cash—this is known as a dividend. (Companies that arenot profitable may still pay a dividend by distributing some of their excesscash.) Profits not distributed as a dividend are retained by the company so thatit can expand and, presumably, make ever greater profit. Often, small, newcompanies keep most of their profits for expansion, whereas large, establishedcompanies pay out most of their profits as dividends. The price for a share of acompany is thus dependent on how profitable the company is (or will be) andfluctuates as expectations for the company’s profits fluctuate. For a successfulor promising company, the price of a share increases as expectations for profitsincrease; conversely, for an unsuccessful company, the price of a share decreasesas expectations for profits decrease (or vanish altogether).

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Great Uncle Adolph had suggested the stocks of three companies, J.C. Pen-ney (department stores), Walt Disney (movies and theme parks), and SizzlerInternational (steakhouse restaurants). We need to know several pieces of in-formation about each stock in order to analyze its performance. In Tables 1–3,we begin with the number of shares purchased, the price per share, and thetotal cost on February 22, 1993, together with the commission to a stockbrokerto handle the purchase. Dividends paid follow next. Notice that it often takesseveral weeks or months for a dividend to be paid. For example, you are due adividend from Disney for the months of January, February, and March of 1995,yet Disney did not send it to you until May 20, 1995—five days after you soldthe stock. Finally, the table concludes with the price per share and total valueon May 15, 1995, and the commission paid to a stockbroker to handle the sale.

Including the cost of commissions, you would have invested $19,972.76.Dividends received would total $1,734.40. Proceeds from the sale, after payingthe commissions, would be $21,935.70.

Table 1.

J.C. Penney, 200 shares.

Date Event Per Share Amount

Feb 22 ’93 Purchase $37.00 $7,400.00Feb 22 ’93 Commission 74.40May 1 ’93 Dividend 0.33 66.00Aug 1 ’93 Dividend 0.36 72.00Nov 1 ’93 Dividend 0.36 72.00Feb 1 ’94 Dividend 0.36 72.00May 1 ’94 Dividend 0.36 72.00Aug 1 ’94 Dividend 0.42 84.00Nov 1 ’94 Dividend 0.42 84.00Feb 1 ’95 Dividend 0.42 84.00May 1 ’95 Dividend 0.42 84.00May 15 ’95 Sale 46.875 9,375.00May 15 ’95 Commission 86.25

1.2 BondsA U.S. Treasury Note is a type of bond, essentially a loan to the U.S. Govern-

ment for a specified time period. During the life of the bond, the U.S. Treasurypays interest to the bondholder every 6 months at a fixed interest rate (thecoupon) that is specified when the bond is first issued. At the end of the life ofthe bond, the original amount of the loan (its face value) is returned to the bond-holder. Although the time period can be as long as 30 years, there are marketswhere these bonds can be bought and sold during the intervening years. Thus,you do not have to buy a bond when they are issued by the Treasury, nor do youhave to hold it until the end of the time period (its maturity). The price at whicha bond is bought or sold depends upon how current interest rates compare to

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Table 2.

Walt Disney, 200 shares.


Feb 22 ’93 Purchase $45.00 $9,000.00Feb 22 ’93 Commission 84.00May 20 ’93 Dividend 0.39 78.00Aug 20 ’93 Dividend 0.48 96.00Nov 20 ’93 Dividend 0.26 52.00Feb 20 ’94 Dividend 0.68 136.00May 20 ’94 Dividend 0.45 90.00Aug 20 ’94 Dividend 0.49 98.00Nov 20 ’94 Dividend 0.42 84.00Feb 20 ’95 Dividend 0.84 168.00May 20 ’95 Dividend 0.60 120.00May 15 ’95 Sale 54.375 10,875.00May 15 ’95 Commission 95.25

Table 3.

Sizzler International, 340 shares.


Feb 22 ’93 Purchase $9.875 $3,357.50Feb 22 ’93 Commission 56.86Apr 15 ’93 Dividend 0.04 13.60Jul 15 ’93 Dividend 0.04 13.60Oct 15 ’93 Dividend 0.04 13.60Jan 15 ’94 Dividend 0.04 13.60Apr 15 ’94 Dividend 0.04 13.60Jul 15 ’94 Dividend 0.04 13.60Oct 15 ’94 Dividend 0.04 13.60Jan 15 ’95 Dividend 0.04 13.60Apr 15 ’95 Dividend 0.04 13.60May 15 ’95 Sale 5.625 1,912.50May 15 ’95 Commission 45.30

the fixed interest rate of the bond. If the bond has a higher interest rate thancurrently available otherwise, then it costs more to buy, and vice versa.

Your grandfather suggested that you purchase a U.S. Treasury Note thatmatures on May 15, 1995, and which pays interest at an 11.25% annual rate. InFebruary of 1993, most savings accounts were paying interest at a rate of only3.00% or 4.00%, so to obtain a bond with such a high interest rate you must pay ahigh price (a premium). For every $100 of face value, it costs $115 6

32 . Since bondstypically have face values in $1,000 increments, your $20,000 could purchase abond with a face value of $17,000 at a total cost of $19,581.87 (170 × $115 6

32 ).The bond then pays semiannual interest of $956.25 ($17,000 × 0.1125 × 1

2 ) onthe 15th of the months May 1993, November 1993, May 1994, November 1994,and May 1995. Finally, also on May 15, 1995, the bond will mature and you willreceive its $17,000 face value back from the U.S. Treasury. So you would haveinvested $19,631.87 and received back interest and principal of $21,781.25.

3


1.3 Mutual FundsA mutual fund is a company that does nothing but hold the stocks of other

companies together with the bonds of governments and corporations. For thisreason, a more descriptive name for a mutual fund is the infrequently used“investment company.” Many different individuals purchase shares of theinvestment company, which in turn pools this money together and buys stocksand bonds. If these stocks and bonds go up in price, then so does the price ofthe shares of the investment company. As the stocks pay dividends, and thebonds pay interest, these amounts are rolled up into one annual dividend to theshareholders of the investment company. Similarly, if the investment companysells stocks at a higher price than the purchase price, these amounts are collectedtogether and paid out to the shareholders of the investment company as “capitalgains.” Mutual funds are convenient for small investors for several reasons:

• Investors can place their money in one investment and let an experiencedinvestor make all the decisions about which stocks or bonds to buy or sell.

• Unlike the direct purchase of stock, many mutual funds do not charge com-missions.

• Mutual funds sell fractional shares, so all of an investor’s money can be putto work and any dividends received can also be reinvested easily.

• There is a wide variety of funds, specializing exclusively in different typesof investments (all stocks, all bonds, etc.) and with different investing styles.

Your daring sister promoted the Twentieth Century Growth Fund, whichis characterized as an “aggressive growth” fund—meaning it invests in stockswhose prices can vary widely, both up and down (but hopefully more up thandown). We suppose that you have chosen to establish your account in this fundso that whenever the fund pays out a dividend, it is automatically reinvested bypurchasing a few new shares. Table 4 illustrates how your investment wouldhave fared.

Table 4.

Twentieth Century Growth Fund.

Date Event Per Share Total Value Shares Shares Owned

Feb 22 ’93 Purchase $22.99 $20,000.00 869.943 869.943Dec 20 ’93 Dividend 2.83 2,461.93 869.943Dec 20 ’93 Purchase 22.09 2,461.93 111.500 981.443Dec 19 ’94 Dividend 3.27 3,209.32 981.443Dec 19 ’94 Purchase 18.43 3,209.32 174.136 1,155.579May 15 ’95 Sale 21.14 24,428.94 1,155.579 0.000

While the per-share price has decreased over the two years, the dividendsthat were reinvested in the purchase of additional shares have contributed to again of $4,428.94.

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1.4 ComparisonIt should be clear that these three investments each behave somewhat dif-

ferently. Would it be enough simply to compare the total amount of money inhand on May 15, 1995, to determine which investment is best? Although themutual fund has provided a gain of $4,428.94 and the stocks have provided asmaller gain of $3,697.34, the stocks have paid dividends throughout the life ofthe investment; the gains from the mutual fund were reinvested and thereforewere not available until the end of the investment. Similarly, the bond has aneven smaller gain, $2,149.38, but it has paid out these gains early on in the formof sizeable interest payments (at a cost of a return of principal that is smallerthan our initial investment).

As we will see in the next section, the effect of inflation means that a dollartoday is not worth the same as a dollar a year from now. So when we receiveour money can be as important as how much we receive. After we analyze thisidea carefully, we can formulate an exact comparison of your three alternativeinvestments using the IRR.

2. The Time Value of MoneyWould you rather receive $1,000 today or $1,050 a year from today? The

answer to that question depends in large measure on what you could purchasewith these two amounts. If the inflation rate is 8%, then what costs $1,000 todaywill cost $1,080 in one year. In this case, we would prefer to have the $1,000now. However, if inflation is 3%, then what costs $1,000 today will cost $1,030in one year, and we might be content to wait a year to receive the $1,050. This iswhat is referred to as the “time value of money.” Presuming inflation persists,the later we receive an amount of money, the less it is worth.

We place money in savings accounts or other investments in hope that themoney grows faster than the inflation of prices, so that we have even morepurchasing power in the future. Let’s consider how money grows in a savingsaccount. Suppose that we invest $100 in a savings account with a 5% annualinterest rate, and that the bank credits interest quarterly (4 times a year). Noticethat we earn interest on our interest. Table 5 illustrates how the balance in thesavings account grows.

In general, the balance after m years is $100(1 + .05/4)4m. What if we wereto earn interest every month? Then we would have $100(1 + .05/12)12m in ouraccount afterm years. It should be clear that if we placeP dollars into a savingsaccount with an annual interest rate of r, and if interest is credited n times ayear, then after m years the balance is

P (1 + r/n)nm.

What happens if the interest is credited “continuously”? This suggests thatwe let the number of times interest is paid per year, n, become arbitrarily large.

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Table 5.

Growth of savings under quarterly compounding.

Quarter Interest Paid Balance

0 1001 100× (0.05/4) 100(1 + 0.05/4) = 101.252 100(1 + 0.05/4)× (0.05/4) 100(1 + 0.05/4)2 = 102.523 100(1 + 0.05/4)2 × (0.05/4) 100(1 + 0.05/4)3 = 103.784 100(1 + 0.05/4)3 × (0.05/4) 100(1 + 0.05/4)4 = 105.095 100(1 + 0.05/4)4 × (0.05/4) 100(1 + 0.05/4)5 = 106.416 100(1 + 0.05/4)5 × (0.05/4) 100(1 + 0.05/4)6 = 107.747 100(1 + 0.05/4)6 × (0.05/4) 100(1 + 0.05/4)7 = 109.098 100(1 + 0.05/4)7 × (0.05/4) 100(1 + 0.05/4)8 = 110.44

......

So we are interested in

limn→∞

P (1 + r/n)nm = P ( limn→∞

(1 + r/n)n)m. (1)

One of the remarkable surprises of differential calculus is that an applicationof L’Hopital’s Rule gives

limn→∞

(1 + r/n)n = er. (2)

Applying (2) in (1) shows thatP dollars compounded continuously at an annualrate of r yields Perm dollars after m years.

To illustrate the slight advantage of compounding interest continuously,suppose that we deposit $100 into a savings account that pays 5% interestcompounded continuously. What is the balance two years later? The finalbalance is $100(e0.05)2 = $110.52, just 8 cents more than the amount receivedfrom the similar account above that only compounds quarterly.

Suppose now that inflation is running at a constant 4%. Would you ratherhave $1,050 two years from now, or $1,120 three and a half years from now? Tocompare these two different amounts at two different times, we can computetheir “present values.”

The present value of an amount M , at a rate r at a time t from now, is theamount that needs to be invested at an interest rate of r for a time periodt so that the final balance is M .

For example, the present value of $1,050 two years from now at a rate of 4%,is an amount P that satisfies P (e0.04)2 = 1050, or P = 1050 e−.08 = $969.27.From this example, it should be clear that the present value is given in generalby

P = M(e−r)t = M e−rt. (3)

Then the present value of $1,120 at a rate of 4%, three and a half years fromnow, is 1120 e−0.04(3.5) = $973.68. Having placed the two amounts on equal

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footing by considering their values at the same time (the present), we can seethat $1,120 three and a half years from now has the greater purchasing power(assuming constant inflation of 4%).

3. The Internal Rate of ReturnQuestions about the growth of investments often take the form of problems

like, “If we deposit $300 in a savings account that has an interest rate of 5%,what will our balance be four years later?”, or “How much money should wedeposit in a savings account that has an interest rate of 7% so that our balancewill be $900 in two years time?” In each case, the interest rate is one of thegiven quantities. Let’s consider a question where the interest rate is what wewish to discover.

Suppose that $600 is deposited in a savings account, where interest is com-pounded continuously. After three years the balance is $729.19. What is theinterest rate on this account? We desire to solve the equation

600e3r = 729.19,

where the two sides represent equivalent expressions for the final balance whenthe interest rate is denoted by r. Using logarithms to solve for r, we find thatthe unique solution is r = 0.065 = 6.5%.

Complicating matters somewhat, suppose now that $600 is deposited in asavings account, where interest is compounded continuosly. At the end of thefirst year, we deposit an additional $100, and at the end of the second year wewithdraw $150. The balance at the end of three years is $657.31. What interestrate is in effect for this account? We denote the unknown rate as r and computean expression for the final balance. Table 6 illustrates how the account grows.

Table 6.

Growth of an account at interest with deposits and withdrawals.

Year Event Balance

0 Initial Deposit 6001 Interest Paid 600er

1 Deposit 600er + 1002 Interest Paid (600er + 100)er = 600e2r + 100er

2 Withdrawal 600e2r + 100er − 1503 Interest Paid (600e2r + 100er − 150)er = 600e3r + 100e2r − 150er

Thus, equating two expressions for the final balance, we find that we needto solve the equation

600e3r + 100e2r − 150er = 657.31. (4)

The presence of r in three exponentials, each with a different multiplier, makesthis a much more difficult equation to solve than in the previous example,

7


where a logarithm worked nicely. It is the difficulty in solving an equation ofthis form that is precisely the rationale for the techniques that we develop inlater sections. (Note that in this example, since the additional deposit and thewithdrawal occurred exactly at the end of a year, we can view the equation as acubic in er. With a solution to the resultant cubic equation, we could again uselogarithms to solve for r. Exercise 19 further describes this route to a solution.However, this approach does not work as easily if the time periods are notexact multiples of a year. Furthermore, the resulting polynomial equationscould turn out to have very high degree.)

As motivation for the definition of the internal rate of return, we massage (4):

600e3r + 100e2r − 150er = 657.31,

−600e3r − 100e2r + 150er + 657.31 = 0,

−600 + (−100)e−r + 150e−2r + 657.31e−3r = 0.

We can now regard the coefficients of the exponentials as “cash flows”—movements of money in and out of the savings account, where a negativecoefficient is cash that flows away from us (deposits) and a positive coefficientis cash that flows in to us (withdrawals). Note that we are treating the finalbalance as if we were withdrawing it and closing the account. As describedby (3), the coefficients multiplied by the exponentials with negative exponentsrepresent the present values of the cash flows. So the equation says that the sumof the present values of the cash flows is zero. This is precisely the definitionof the internal rate of return. (See Exercise 21 for an alternative definition.)

Definition. The internal rate of return (IRR) of an investment is a fixed interestrate that makes the present values of the cash flows sum to zero.

Some observations about this definition are in order. First and foremost,this definition need not be applied only to savings accounts. Any investmenthas the property that money flows in and out at different times—this is all theinformation we require to compute the IRR. Thus, we can compute a singleinterest rate to describe any investment. It would be instructive at this pointto verify that the bond investment described in Section 1.2 has an IRR of r =5.314% by using this interest rate to compute the present values of the cashflows and checking that they do indeed sum to zero. This means that on Febru-ary 22, 1993, we could place $19,631.87 into a savings account with an interestrate of 5.314% (continuously compounded), make withdrawals of $956.25 onthe same schedule as the bond makes interest payments, and have a balance of$17,000.00 remaining on May 15, 1995. From the point of view of performance,we would be ambivalent about choosing between the savings account and thebond. When we have also computed the IRR of your possible investments instocks and the Twentieth Century Growth mutual fund, then we will have abasis for comparing their performances by comparing their IRRs.

It is perhaps best to regard the IRR informally as an answer to the followingquestion about an investment:

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What fixed interest rate must a hypothetical savings account possess in orderthat we could experience the same cash flows from the savings account as thoseactually produced by the investment?

It is this aspect of the definition that allows us to compare disparate investmentsand which makes the IRR such an appealing description of an investment’sperformance.

For all of its simplicity and appeal, there are two drawbacks to the IRR:

• There is no guarantee that there is only one interest rate that satisfies thedefinition. Several authors have addressed this topic [Freund 1995; Paley etal. 1983] in clear and informative articles, so we do not concern ourselveswith it here (but see Exercise 14).

• The equation that results from the definition is very difficult to solve exactly.Being a sum of exponentials, each with a different multiple (usually not aninteger) of the unknown rate, there is no exact method for solving it. Ourmain purpose here is to illustrate some iterative techniques for solving suchdifficult equations, using the computation of the IRR as an example of avery useful and desirable quantity that is best computed via these iterativemethods.

4. Computation of the IRRThe IRR is a favored way to measure the performance of an investment

(a close competitor is the time-weighted rate of return). Therefore, many bookson the analysis of investments contain a definition of the IRR, together with adiscussion of its properties [Fabozzi et al. 1991; Kellison 1970; Khoury and Par-sons 1981]. Many popular computer programs compute the IRR, including theMicrosoft Excel and Lotus 1-2-3 spreadsheets and Intuit’s Quicken personal-finance software. Also, more-specialized programs designed for use by finan-cial professionals, such as Financial Computer Support’s dbCAMS+, includethe IRR in the reports they generate.

While it can be easy to learn about what the IRR is, it can be hard to find adviceon exactly how to compute it. Introductory textbooks on finance sidestep theissue by alluding to “trial and error” techniques [Fabozzi et al. 1991, 76; Khouryand Parsons 1981, 41] or by assuming that the reader is familiar with unnamed“iterative technique[s]” [Kellison 1970, 162]. Software documentation can beequally vague; Quicken also describes its computation as being conducted bytrial and error [Intuit 1995]. An exception is the comprehensive documentationfor dbCAMS+ [Financial Computer Support 1992]; while the documentationdubs the algorithm as trial and error, in reality it is the bisection method, whichwe discuss in Section 5.1.

Given that the IRR is a popular measure of investment performance whichis difficult to compute, how would a person or a computer program convert aseries of cash flows into the IRR?

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For the remainder of the Module, we need some notation. Suppose thatwe have an investment with n cash flows, in amounts c1, c2, . . . , cn that occurat times t1, t2, . . . , tn. Usually, it is convenient to set t1 = 0, but this is notnecessary (see Exercise 15). To compute annual interest rates, we measure timein years. If r is the internal rate of return, we wish to solve the equation

p(r) = c1e−t1r + c2e

−t2r + · · ·+ cne−tnr = 0. (5)

To illustrate: Our previous example has n = 4 with c1 = −600, c2 = −100,c3 = 150, c4 = 657.31, and t1 = 0, t2 = 1, t3 = 2, t4 = 3, which results in an IRRof r = 5.34%. Notice that it is not important which signs we choose to use forthe different directions of the cash flows. We have adopted the point of viewof the investor, so a deposit into the investment is money going away fromthe investor and thus has a negative sign, and therefore a withdrawal from theinvestment is a positive quantity. We could just as easily reverse our viewpointto that of the investment, and switch all of the signs on the cash flows. Since weare solving p(r) = 0, this wholesale change of sign will not affect the solutions(see Exercise 15).

We now detail the application of three iterative methods for computingthe IRR. The first two are very general techniques, and the third is specific tocomputing the IRR of an investment that behaves like a bond.

5. Iterative Solution Methods

5.1 The Bisection MethodThe intermediate value theorem says that if f(x) is a continuous function

on the interval [a, b], and f(a) and f(b) have different signs, then the equationf(x) = 0 has at least one solution x∗ with a ≤ x∗ ≤ b. Since the function ex iscontinuous for all x and the function p(r) is a linear combination of exponentialfunctions, p(r) is a continuous function everywhere. These observations formthe basis of our application of the bisection method to the computation of theIRR.

We begin by choosing two values for a and b, one absurdly small and theother absurdly large. For investments with typical returns around 5% or 10%,we might choose a = −20% and b = 40%. We hope that one of these valuesmakes p(r) positive and the other makes p(r) negative. If this is the case, thenwe know that the equation p(r) = 0 has a solution somewhere in [−20%, 40%] =[−0.2, 0.4]. Define r1 = (a+b)/2, the midpoint of the interval [a, b], and computep(r1). If it happens that p(r1) = 0, then we have been extremely lucky andhave found a solution. Otherwise, p(r1) is positive or negative and one of theintervals [a, r1] or [r1, b] has endpoints that cause p(r) to have opposite signs.Concentrating now on this interval, we can again conclude that the equationp(r) = 0 has a solution within this new, smaller interval. We then bisect this

10


interval at its midpoint, r2. Upon examining the value of p(r2) we can again cutthe interval in half and determine which subinterval must contain a solution.

We can continue this process indefinitely, and we may never find the exactvalue of the solution. However, when we do decide to stop, we can statewith certainty that the solution lies in some interval. For every iteration thatwe perform, the size of this bounding interval decreases by half, eventuallycreating an interval that lies within whatever error tolerance we desire. Theprocess of halving the width of the interval allows us to converge to a desiredlevel of accuracy quickly.

While some textbooks state that the IRR can be found by trial and error, usingthe bisection method should be viewed as only a slight upgrade to “efficienttrial and error.” However, its advantage is that the continuity of p(r) and theintermediate value theorem guarantee that we will find a solution to whateverlevel of accuracy we desire in a fairly direct manner.

We illustrate the computation of the IRR via the bisection method by de-termining the performance of your investment in Ina’s three favorite stocksusing the data given in Section 1.1. We begin with the interval [−20%, 40%]and iterate until we achieve two decimal places of accuracy (when expressedas a percentage) (see Table 7).

Table 7.

Example of the bisection method for IRR.

Iteration a m =(a+b)

2b p(a) p(m) p(b)

1 −0.200000 0.100000 0.400000 16558.84 −894.93 −9914.252 −0.200000 −0.050000 0.100000 16558.84 6401.88 −894.933 −0.050000 0.025000 0.100000 6401.88 2452.79 −894.934 0.025000 0.062500 0.100000 2452.79 709.87 −894.935 0.062500 0.081250 0.100000 709.87 −109.08 −894.936 0.062500 0.071875 0.081250 709.87 296.16 −109.087 0.071875 0.076562 0.081250 296.16 92.49 −109.088 0.076562 0.078906 0.081250 92.49 −8.55 −109.089 0.076562 0.077734 0.078906 92.49 41.90 −8.55

10 0.077734 0.078320 0.078906 41.90 16.65 −8.5511 0.078320 0.078613 0.078906 16.65 4.04 −8.5512 0.078613 0.078759 0.078906 4.04 −2.25 −8.5513 0.078613 0.078686 0.078759 4.04 0.89 −2.2514 0.078686 0.078723 0.078759 0.89 −0.68 −2.25

From the last line of the table, we see that the fourteenth iteration tells usthat r must lie in the interval [7.8686%, 7.8723%], which when rounded to twodecimal places yields r = 7.87%. (Notice that the previous iteration would notyield the desired accuracy.) A more precise description of our results is thatr = 7.87045%± .00185%.

We noted earlier that the dbCAMS+ software uses this method. However,one drawback to a computerized implementation of the bisection method isthe need to program a search for the endpoints of the initial interval. It is notan easy task to write a program to “guess” values of a and b that cause p(r) to

11


have differing signs (but see Exercise 22 for a general procedure that applies insome cases).

5.2 Newton’s Method

r x1 x0

y = f(x0) + f’(x0)(x - x0)

(x0, f(x0))

f(x)y

x

Figure 1. Illustration of Newton’s method.

Given a function f(x) and the associated equation f(x) = 0, we can searchfor a solution by using tangent lines to the function as approximations to thefunction itself. This discussion is illustrated in Figure 1. Let r be the exactsolution to f(x) = 0 and let x0 be a first guess at the solution. A tangent lineto the function at x0 passes through the point (x0, f(x0)) and has slope f ′(x0).Thus, the equation of this line is

y = f(x0) + f ′(x0) (x− x0). (6)

Because this line provides a fair approximation of the behavior of f(x) (at leastnear x0), we determine where it crosses the x-axis. This point should be closeto a solution of f(x) = 0. Let x1 denote the x-coordinate of the intersection ofthe tangent line with the x-axis. This means that the point (x1, 0) is on the lineand hence satisfies (6). Making this substitution, we have

0 = f(x0) + f ′(x0) (x1 − x0),

−f(x0) = f ′(x0) (x1 − x0),

−f(x0)/f ′(x0) = x1 − x0,

x1 = x0 − f(x0)

f ′(x0).

With the value of x1 in hand, we can repeat the process by constructing thetangent line to f(x) at x1. Ideally, the intersection of this tangent line with the

12


x-axis will be even closer to the real solution of f(x) = 0. In general, if we havecomputed xm, then we can find xm+1 from

xm+1 = xm − f(xm)

f ′(xm). (7)

Notice that we will have some difficulties if it ever happens that f ′(xm) = 0 (orindeed if f ′(xm) is just very close to zero). Newton’s method usually works welland often gets close to a solution of f(x) = 0 in just a few iterations. However,there are situations where it can go haywire (see Exercise 13). Also, unlikethe bisection method, Newton’s method usually does not give any guaranteesabout how close we are to a solution when we stop the iteration. But we canapply two equivalent subjective criteria to help us decide when to stop applyingNewton’s method. First, we can check to see how “close” f(xm) is to zero, sincethis is the desired property of xm. Or, we can look at how “different” xm+1 isfrom xm: When succesive iterations do not change the value of xm appreciably,we halt.

The application of Newton’s method to the computation of the IRR is rela-tively straightforward. Since

p(r) = c1e−t1r + c2e

−t2r + · · ·+ cne−tnr,

we have

p′(r) = −c1t1e−t1r − c2t2e−t2r − · · · − cntne−tnr

and (7) becomes

rm+1 = rm − p(rm)/p′(rm)

= rm +c1e−t1rm + · · ·+ cne

−tnrm

c1t1e−t1rm + · · ·+ cntne−tnrm.

Since we usually have an approximate idea of what kind of return an investmentgenerates, our first guess, r0, should be any reasonable rate of return.

We illustrate the computation of the IRR via Newton’s method by deter-mining the performance of your investment in the Twentieth Century Growthmutual fund using the data given in Section 1.3. There are two peculiar pointsto note about analyzing this investment. In late December of each year, divi-dends were paid and the fund automatically reinvested these amounts for youby purchasing additional shares. This combined event does not constitute acash flow, since no money has returned to the investor and no money has leftthe investor. This contrasts with an investment in stocks, where it is usuallyimpractical or costly to immediately reinvest the relatively small dividends (al-though some companies perform this service for stockholders without charge).Since the reinvestment of dividends is not a cash flow, this investment has onlytwo cash flows—the purchase on February 22, 1993, and the sale on May 15,1995. (See Exercise 5 for an example of an investment in this fund that does

13


have several cash flows.) Since there are only two cash flows, we could computethe IRR exactly without using Newton’s method (see Exercise 8). However, weproceed to illustrate the computation with Newton’s method. Let’s chooser0 = 0% = 0.0 and examine the iterations in Table 8.

Table 8.

Iterations of Newton’s method for IRR.

Iteration (i) ri p(ri) p′(ri) ri+1

0 0.00000000 4428.94000000 −54479.88 0.081294961 0.08129496 378.27929179 −45446.35 0.089618612 0.08961861 3.48934999 −44610.52 0.089696823 0.08969682 0.00030432 −44602.74 0.089696834 0.08969683 0.00000000 −44602.73 0.08969683

We can see that by the fifth iteration, we have achieved what appears to be ahighly accurate result. If we examine eight decimal places in our computations,the value of p(r4) is effectively zero, and thus r5 is not appreciably differentfrom r4. So we would report with confidence that r = 8.969683%.

Under suitable conditions, it can be shown that Newton’s method doublesthe number of decimal places of accuracy with each iteration. An examination ofthe values of ri in Table 8 bears this out. Compare this convergence propertycarefully with the corresponding one for the bisection method, where it is justthe accuracy itself (the difference from the exact value) that doubles with eachiteration.

In practice, a popular method for locating roots of functions is Brent’s method[Brent 1973; Press et al. 1989], which could be viewed as a hybrid of the previoustwo methods. Where Newton’s method uses an approximating linear function(the tangent line), Brent’s method uses an inverse quadratic approximatingfunction. When this quadratic function might behave poorly or erratically, themethod reverts to using bisection.

5.3 Convex MethodWe now describe a method that is detailed in Gerber [1990, 13] that we

refer to here as the convex method, since it relies on a certain function beingconvex. It combines the fast convergence that is characteristic of Newton’smethod with the definite bounding interval that we obtain with the bisectionmethod. However, the drawback to this method is that it can be applied onlyto an investment that behaves like a bond—all of the cash flows, except one,must be of the same sign. So while this method applies to many investments(both bonds and others), it does not apply to all.

Let c0 be a single cash flow out from the investor at time t0 = 0, but expressedas a positive quantity. In the case of a bond, c0 would represent the total purchaseprice. Let c1, c2, . . . , cn be n positive cash flows that occur at times t1, t2, . . . , tnand which represent money returning to the investor.

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In the case of a bond, c1, c2, . . . , cn−1 would be the interest payments, andcn would represent the return of principal at maturity or the sale of the bond.In this case, we can rearrange (5) slightly and describe the IRR as the solutionto

q(r) = c1e−t1r + c2e

−t2r + · · ·+ cne−tnr = c0. (8)

To establish a procedure for computing the IRR in this case, we define twointermediate functions of q(r). The first is

s(r) = ln

[q(r)

c1 + c2 + · · ·+ cn

]= ln[q(r)]− ln[c1 + c2 + · · ·+ cn].

We now examine several properties of s(r). First, the graph of s(r) passesthrough the origin,

s(0) = ln

[q(0)

c1 + c2 + · · ·+ cn

]= ln

[c1 + c2 + · · ·+ cnc1 + c2 + · · ·+ cn

]= ln(1) = 0.

Second, we show that the first derivative of s(r) is always negative:

s′(r) =1

q(r)q′(r)− 0

=q′(r)

q(r)(9)

=−c1t1e−t1r − · · · − cntne−tnrc1e−t1r + · · ·+ cne−tnr

= −c1t1e−t1r + · · ·+ cntne

−tnr

c1e−t1r + · · ·+ cne−tnr.

Since ci > 0 for all i, and ex > 0 for all x, it should be clear that this lastexpression is always negative. Notice that s(r) is negative when r > 0, since itequals zero when r = 0 and is decreasing from there.

Finally, we show that the second derivative of s(r) is strictly positive. (Forthe reader with a knowledge of probability, an alternative explanation of thisproperty can be found in Exercise 18.) Taking the derivative of s′(r) as writtenin (9), we find

s′′(r) =q(r) q′′(r)− [q′(r)]2

q(r)2

=q′′(r) q(r)2 − [q′(r)]2q(r)

q(r)3. (10)

15


Since q(r) > 0 for any r, we turn our attention to the numerator of this expres-sion:

q′′(r) q(r)2 − [q′(r)]2q(r) = q′′(r) q(r)2 − 2q′(r)2q(r) + q′(r)2q(r)

= q(r)2n∑i=1

t2i cie−tir − 2q(r)q′(r)

n∑i=1

(−ticie−tir)+ q′(r)2

n∑i=1

cie−tir

=

n∑i=1

[q(r)2t2i cie

−tir + 2q(r)q′(r)ticie−tir

+ q′(r)2 cie−tir]

=n∑i=1

[q(r)2 t2i + 2q(r)q′(r) ti + q′(r)2

]cie−tir

=n∑i=1

[q(r) ti + q′(r)]2cie−tir.

Since every term of this sum is a product of nonnegative quantities, we can seethat the sum is nonnegative. Furthermore, only in exceptional circumstances(such as n = 1, or ci = 0 for all i) can the second derivative equal zero (seeExercise 17).

(0,0) r

slope = u(r)

s(r)

(r, s(r))

Figure 2. General shape of s(r).

Now consider another intermediate function, u(r) = s(r)/r. With the aidof Figure 2, we first show that u(r) is an increasing function when r > 0. Thegraph of s(r) passes through the origin and therefore, we can view u(r) as theslope of the secant line joining the points (0, 0) and

(r, s(r)

)on the graph of

16


s(r). Because s(r) is negative when r > 0 (it equals zero when r = 0 andis decreasing from there) and since s(r) has a positive second derivative, theslopes represented by u(r) increase for r > 0.

Our goal was to find an interest rate r that is a solution to (8). Let r∗ be suchan interest rate; that is, let q(r∗) = c0. If we have a reasonable idea of what kindof return our investment has provided, then we can choose numbers a and b sothat 0 < a < r∗ < b. In other words, we can specify an interval (a, b) that weare sure contains the IRR. Now, since u(r) is an increasing function, it followsthat

u(a) < u(r∗) < u(b),

s(a)

a<

s(r∗)

r∗<s(b)

b,

b

s(b)<

r∗

s(r∗)<

a

s(a).

Now, recalling that s(r∗) < 0, we have

s(r∗)

s(a)a < r∗ <

s(r∗)

s(b)b,

ln[q(r∗)/(c1 + c2 + · · ·+ cn)]

ln[q(a)/(c1 + c2 + · · ·+ cn)]a < r∗ <

ln[q(r∗)/(c1 + c2 + · · ·+ cn)]

ln[q(b)/(c1 + c2 + · · ·+ cn)]b,

ln[c0/(c1 + c2 + · · ·+ cn)]

ln[q(a)/(c1 + c2 + · · ·+ cn)]a < r∗ <

ln[c0/(c1 + c2 + · · ·+ cn)]

ln[q(b)/(c1 + c2 + · · ·+ cn)]b.

This gives us a new interval that also contains r∗. But is it a better interval?To answer this question, we wish to show that multiplier of a in the left-handendpoint is greater than one, and that the multiplier of b in the right-handendpoint is less than one.

Examining the left-hand endpoint, we see that since the natural logarithmis an increasing function, all we need to show is that c0 > q(a). However,if a is an interest rate that is smaller than the true IRR (r∗), then the presentvalue of the withdrawals from the investment (q(a)) will be less than the initialdeposit (c0). A similar argument verifies that the right-hand endpoint is alsoan improvement (see Exercise 16).

These observations provide the basis for an iterative technique. We firstchoose 0 < a < b so that we are certain that (a, b) contains the true IRR. Wecompute a new interval from (11) that also contains the true IRR, yet bothendpoints are now closer to the true value than a and b are. Since this newinterval contains the true IRR, we can go ahead and let these new endpointsplay the role of a and b and repeat the process. Just as with the bisection method,when we finish iterating, we have a precise description of our possible error,and it would be reasonable to use the midpoint of our last interval as our best

17


estimate of r∗. Like Newton’s method, in practice we obtain rapid convergence.While this technique has very desirable properties, the price we pay is that wecan apply it only to investments where the cash flows have a certain (but notunusual) description.

We illustrate the computation of the IRR via the convex method by deter-mining the performance of your investment in the U.S. Treasury Note usingthe data given in Section 1.2. We start by choosing a = 0.1% and b = 10%; seeTable 9.

Table 9.

Iterations of the convex method for IRR.

Iteration a b

0 0.001000000000 0.1000000000001 0.052945141402 0.0533301981012 0.053142526332 0.0531440272693 0.053143295661 0.0531433015114 0.053143298660 0.053143298682

Note how accurate the results are after just one iteration (roughly ±.02%)and how quickly the accuracy improves from there. We can state with certaintythat the IRR is in the interval (5.3143298660%, 5.3143298682%), or we can reportthe midpoint of this interval with error as 5.3143298671% ± 0.0000000011%.Comparing the application of four iterations of Newton’s method to computethis same figure, we get similar accuracy—but without any guarantees.

6. ConclusionWith hindsight, which decision in your sophomore year would have been

the most profitable? Clearly, your sister gave the best advice, as the TwentiethCentury Growth mutual fund had the superior performance when measuredwith the internal rate of return. However, performance is not the only measureof an investment to consider.

Suppose that your financial situation as a sophomore was somewhat tenu-ous, and you were not sure that you would be able to afford tuition for yoursenior year. Thus, you thought there was a real possibility that you would needto cash out your investment in the summer preceding your senior year. Whatreturn would you earn in this case? On July 1, 1994, a share of this fund wasworth $21.38, so a sale would net $20,983.25 and the IRR resulting from a saleon this date is 3.55%. Clearly, the difference in selling a year later is substan-tial. This illustrates one way of defining the risk of an investment. If the valuefluctuates widely, and there is a chance that you may need to spend the moneyquickly during a time of low value, then you may not realize the large gainsthat could be obtained by waiting and cashing out during a period of higher

18


value. Notions from statistics, such as the standard deviation, can be used toquantify how variable an investment is and in turn quantify the risk.

Another type of risk is the possibility that a bond may default by not mak-ing its interest payments as scheduled or not returning all of the principal atmaturity. You might be skeptical of making a loan to Sizzler International—acompany that would appear to be in financial trouble if its stock price is a truereflection of its prospects for future profitability. However, a U.S. Treasury Noteis considered the safest investment possible, since there is virtually no way thatthe U.S. government would default on one of its bonds (temporarily ignoringthe U.S. budget crisis of 1995–1996!). Notice also that the Treasury Note is theone investment where you could have calculated the IRR as a sophomore, priorto making your decision, since the purchase price, interest payments, and facevalue were all known at that time. So while the U.S. Treasury Note had thelowest overall IRR of your relative’s three suggestions, you might have decidedthat your grandfather’s advice was sound, and that a sure 5.31% was a goodinvestment. Then you could lie awake at night worrying about your course-work and your graduate school applications rather than the fluctuating valueof your investments. The precise mathematical analysis of an investment’s riskcan take several different forms and is another subject of interest in its ownright.

For an investor evaluating a proposed investment, the tradeoff between riskand performance is the fundamental issue—greater performance is usuallyaccompanied by greater risk. So, an important part of investment analysis(and portfolio monitoring) is measuring the performance of an investment. Inconjuction with a measure of risk, one can make informed decisions about thesuitability of an investment. We have attempted to show here that the IRRis a natural measure of performance, although its computation would appeardaunting. However, it can be computed quickly and accurately (especially withthe aid of computers) using iterative methods that are grounded in several ofthe basic concepts of differential calculus.

7. ExercisesSeveral of the following exercises ask for the IRR of investments that are

variations of those described in Sections 1.1–1.3, so you will need to get someof the needed information from there. For each investment that asks you tocompute the IRR, you could use any or all of the following methods:(a) the bisection method;(b) Newton’s method;(c) the convex method, if applicable;(d) a spreadsheet program;(e) verification that the solution given is correct.You may find a computer program (possibly in a language like Mathematica orMaple) or a programmable calculator convenient for doing the computations.

19


1. Suppose that you found it necessary to sell your investment in the TwentiethCentury Growth mutual fund on July 1, 1994, when a share was worth $21.38.Verify the statement from Section 6 that the IRR for this shorter investmentis 3.55%.

2. Suppose that you found it necessary to sell your investment in J.C. Penney,Disney, and Sizzler on July 1, 1994. The price per share for each on thatdate was: J.C. Penney, $53.50; Disney, $42.00; Sizzler, $5.875. Find the IRRfor the combined investment in these three stocks. (Since you owned thestocks at the end of the second quarter on June 30, 1994, you will receive thedividend that is paid a few weeks later. At the discount brokerage houseFreeman-Welwood, commissions on stock purchases and sales are computedas follows: if the value of the stock is less than $5,000, then the commissionis $30 + 0.008 × (value); for stock valued between $5,000 and $15,000, thecommission is $30 + 0.006× (value).)

3. Suppose that you found it necessary to sell your investment in the U.S.Treasury Note on July 1, 1994. On this date, each $100 of face value could besold for $10430

32 . Find the IRR that results from selling this bond prior to itsmaturity.

4. Because income tax laws treat interest and capital gains differently, whena bond is traded, the purchaser usually pays the seller for any interest thatthe bond has earned but that has not been received by the seller (accruedinterest). For example, if a bond is being sold a month after making its lastinterest payment, the buyer pays the seller the purchase price together witha month’s worth of interest. Compute the IRR of the bond in Section 1.2when the accrued interest due the seller is added to the purchase price ofthe bond.

5. A popular way to invest in mutual funds is to make periodic additionalinvestments of a fixed dollar amount. This way, you buy few shares whenthe price is high and more shares when the price is low. This is knownas dollar-cost averaging. Let’s suppose that you invest your $20,000 in theTwentieth Century Growth mutual fund on February 22, 1993, as before.However, four times a year, you invest an additional $1,000 that you haveearned working in the school cafeteria. Table 10 gives the dates and shareprices for these additional investments. What is the IRR for this investmentif you sell all of your shares on May 15, 1995, as before? (Do not forget torecalculate the reinvested dividends based on the new, larger share balances.)

6. Stock in Boeing (a manufacturer of airplanes) did very well during the timethat you invested your inheritance. Using the following figures, computethe IRR for an investment in Boeing. The price per share on February 22,1993, was $33.625; and on May 15, 1995, it was $54.975. Dividends per sharewere $0.25 each quarter, paid on the 10th of June, September, December, andMarch for the preceding quarter.

20


Table 10.

Twentieth Century Growth Fund, per-share price

1993 1994 1995

January 1 $22.15 $18.60April 1 $23.59 21.96 19.98July 1 23.62 21.38October 1 25.37 22.04

7. Compute the IRR for each of the individual stocks (J.C. Penney, Disney,Sizzler) in your stock portfolio. Then calculate the simple average of thesethree returns. Is the average return equal to the one obtained in Section 5.1?Is this result what you would expect? Why or why not?

8. Your investment in the Twentieth Century Growth mutual fund had onlytwo cash flows—the purchase and the sale. In this situation we can computethe IRR exactly. Do so.

9. Many investments are made specifically for the cash flows they generate. Forexample, a retired person might purchase a bond so that the periodic interestpayments could be spent on daily living expenses. The IRR is an appropriatemeasure of the return on such investments, since the withdrawals are spentsoon after they are received. However, in saving for your graduate schooltuition, you would most likely reinvest any withdrawals until after you finishyour undergraduate degree. This is one advantage of mutual funds: Theyallow for convenient and automatic reinvestment of any dividend receivedthrough the purchase of fractional shares. Reinvesting stock dividends orbond interest payments can be difficult since these investments cannot bemade with small amounts.

Suppose then that you have purchased the U.S. Treasury Note as de-scribed above, only now you deposit your interest payments into a savingsaccount that pays continuously compounded interest at a 5% annual rate.What is the IRR for this new investment (the combination of the bond andthe savings account)?

10. Upon completing your Ph.D. in graduate school, suppose that you obtain aposition as a university professor. Many universities provide their facultywith retirement plans run by Teachers Insurance and Annuity Association(TIAA). You have the choice of having your retirement monies invested ina TIAA Annuity, which is a fund that behaves somewhat like a bond, or theCollege Retirement Equities Fund (CREF), which is a fund that is similar toa mutual fund that invests only in stocks. For the ten-year period endingon June 30, 1995, TIAA had an annual return of 8.98%, while CREF had anannual return of 13.30%. Assuming that

• your investment compounds continuously,

21


• the historical returns continue, and

• you are 35 years away from retirement,

compare how your first year’s retirement contribution grows depending onif you elect to have it invested in TIAA versus CREF.

11. Suppose that when you have finished your career as a university professor atthe age of 65, you decide to use a portion of your retirement savings to pur-chase an annuity. An annuity is a contract with a life insurance company—inthis case, if you give the insurance company $500,000, then they promise togive you $51,930.95 yearly for the rest of your life. You may die young, andthis would be a poor investment, or you could live long, and this would bea wise investment. From your point of view, this may seem like a gamble;from the insurance company’s point of view, however, they know that ifthey sell many such annuities, everything will average out slightly in theirfavor. Suppose the insurance company believes that the average 65-year-oldwill live only long enough to collect 12 yearly payments. What interest rateis the insurance company using when they determine the price and annualpayment for this annuity? Suppose that you die at 69 and collect only 4payments. What is the resulting IRR? And if you live to be 95, what IRRwould result from receiving 30 payments?

12. Often the IRR of a bond held until it matures (known as the yield to maturity)is approximated by the following procedure [Shao 1980, 478]. The averageinvestment P is the average of the purchase price and the face value. Theamortized premiumA is the purchase price minus the face value, divided bythe number of years to maturity. (If this quantity is negative, then it shouldbe referred to as a discount rather than as a premium.) Annual interest, I ,is the total amount of interest paid by the bond in a year. Then the yield tomaturity can be approximated by (I − A)/P . Compute this approximationfor the bond purchase described in Section 1.2. How does this compare withthe values computed in Section 5.3 and Exercise 4? Does this approximationmethod seem plausible? (Note that this approximation could be used tocompute a starting value when using Newton’s method to compute the IRRof a bond.)

13. Consider an investment with cash flows described by t1 = 0, t2 = 1, t3 = 2and c1 = 8721.39, c2 = −18666.52, c3 = 10000. Describe the behavior ofNewton’s method with an initial guess of r0 = 0.05 = 5%.

14. After seeing the success of your sister’s recommendation of the TwentiethCentury Growth Fund, you decide to take her up on an offer to becomeher business partner. You agree to invest $10,000 of the money earmarkedfor your graduate school tuition in her espresso coffee stand. Business isvery good at first; at the end of one year, your sister gives you back $7,000in profits. However, during the second year, several new well-publicizedmedical studies indicate links between drinking coffee and heart disease.

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Sales plummet and your sister must declare bankruptcy. As her businesspartner, you must put up another $1,200 at the end of the second year to coveryour share of the debts. Compute the IRR for this “investment” by usingNewton’s method—first with r0 = −90%, and again with r0 = −120%. Canyou explain the two different values? (Hint: Studying Exercise 19 might behelpful.) Which value is “correct”?

15. Suppose that we have an investment with cash flows c1, c2, . . . , cn at timest1, t2, . . . , tn. Prove that the following transformations of the investmentyield the same IRR:

a) Replace c1, c2, . . . , cn by c∗1, c∗2, . . . , c

∗n, where c∗i = ci/c1. Thus, we can as-

sume that c1 = 1.

b) Replace t1, t2, . . . , tn by t∗1, t∗2, . . . , t

∗n, where t∗i = ti−t1. Thus, we can assume

that t1 = 0.

16. Explain why the multiplier of b in the convex method is less than one. Thatis, given the assumption that the interval (a, b) contains the true IRR, explainwhy

ln[c0/(c1 + c2 + · · ·+ cn)]

ln[q(b)/(c1 + c2 + · · ·+ cn)]< 1.

17. Study the derivation of the convex method in the case where n = 1. Whatchanges must be made? In particular, show that s′′(r) = 0. Can we stillapply this method in this case?

18. (This question requires some advanced concepts from mathematical proba-bility.) Using the notation developed in the derivation of the convex method,consider the discrete random variable T whose probability distribution isgiven by

P (T = ti) =cie−tir

q(r), 1 ≤ i ≤ n.

Prove that the variance of T equals the second derivative of s(r), from whichit follows easily that s′′(r) ≥ 0.

19. In Section 3, we gave an example of an investment in a savings account andasked for the interest rate (the IRR, really) in effect for that account. This ledto the equation

−600 + (−100)e−r + 150e−2r + 657.31e−3r = 0.

If we make the substitution x = e−r, we obtain the cubic polynomial equa-tion

−600− 100x+ 150x2 + 657.31x3 = 0.

Solve this polynomial equation for x (possibly with the aid of a calculator orcomputer), and find the corresponding values of r for each solution.

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20. (Writing Exercise) Rather than follow your relative’s advice concerningyour inheritance, you decide instead to buy a used Mercedes to drive for afew years, and then you will sell it to raise money for your graduate schooltuition. Having read one too many automobile advertisements, you reasonaway your relative’s advice by telling yourself that the purchase of such afine automobile is a good “investment.” Compute an IRR for this investmentand explain your rationale for the cash flows you use. Is this a “good”investment? Suppose that you are content with your Mercedes purchase atthe end of your senior year, even though the IRR might be way below thatof the other three investments. What dollar value are you implicitly placingon the joy and convenience of driving a fine automobile?

You may want to make use of the following facts. The price of a used1989 Mercedes 190E was about $19,100 in February 1993. It could be soldfor about $15,500 in May 1995. Registration and license costs about $200 peryear, insurance is about $350 per year, and gasoline sells for about $1.30 pergallon. Repairing a Mercedes has a reputation for being expensive. Yourtransportation needs are currently met by a public bus pass costing $35 amonth.

21. (Writing Exercise) If we decide to view interest as being paid once a year(rather than continuously), then the present value of P dollars, t years fromnow, at an interest rate of r isM = P (1+r)−t. Use this definition to create analternate version of the definition of the IRR. (This is the definition that mosttextbooks on finance prefer to use.) An interest rate re obtained from thisdefinition would be known as an effective interest rate. It can be convertedinto what is known as a nominal interest rate, rn, according to re = ern − 1.For more on this topic, see [Lindstrom 1988].

Derive the form of Newton’s method in this case and apply it to yourinvestment in the U.S. Treasury Note. Convert the effective rate you findinto a nominal rate. How does this nominal rate compare with the onecomputed in Section 5.3 using the definition given in the article? Discussyour observations and any explanations you might be able to offer.

22. (Writing Exercise) One drawback to using the bisection method is the ne-cessity of “guessing” initial values of a and b so that p(a) and p(b) differin sign. While experience with a particular investment may make this eas-ier, having an automated procedure is useful for computer implementa-tions. Consider here the case of an investment where p(0) is positive andlimr→∞ p(r) is negative (an interest-bearing bond purchased at face valueand held to maturity would be an example). We then use the substitutionv = 1/(1 + r). Since there is a root r > 0, we know that 0 < v < 1.Transform p(r) into a function of v so that the individual terms look likecie−tir = cie

(v−1)ti/v . Then we can safely apply the bisection method top(v) by choosing a = 0 and b = 1. A final estimate of a root v can then betransformed back into a value of r.

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Write a complete description of how this transformed problem is derivedand used, being careful about how the function p(v) is evaluated at v = 0.As an example, recompute the IRR of your investment in the U.S. TreasuryNote.

23. (Project) The secant method is yet another iterative method for finding solu-tions to equations. It is similar to the bisection method, but has some of theflavor of Newton’s method. In this project you are given a rough descrip-tion of the central ideas behind this method. Then you can derive a morecareful and precise description of the relevant assumptions and conclusions.With this formulation in hand, apply this method to the three alternative in-vestments of your inheritance and compare the performance of this newmethod.

Suppose that we want to solve f(x) = 0 and we know that f(a) andf(b) have different signs. Applying the intermediate value theorem (whathypothesis do we need here?), we know that there is a solution somewhere inthe interval [a, b]. Construct the line joining the points (a, f(a)) and (b, f(b)),and find its x-intercept, x0. Now, consider the intervals [a, x0] and [x0, b] andconcentrate on the one where f(x) has different signs when evaluated at theendpoints. Repeat the process to find a new x-intercept, x1. By continuingthis process, we find smaller and smaller intervals that we know contain asolution.

Notice that this method is very similar to the bisection method; but in-stead of splitting the interval exactly in half, we are using a line derivedfrom the function f(x) (much as in Newton’s method) to determine how tosplit the interval into two subintervals. The rate at which the secant methodconverges to a root is discussed in Maron and Lopez [1993] and Vianello andZanovello [1992].

8. Solutions to the Exercises

2. 7.077%.

3. 4.41%.

4. 4.03%. This figure appears in the price quotation in the February 22, 1993,issue of Barron’s and is labeled as the “yield to maturity (YTM).”

5. The final share balance of 1,639.727 is worth $34,663.83 on May 15, 1995.This results in an IRR of 9.36%.

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6. Having not been given a specific dollar amount to invest, we can track theperformance of any number of shares (one would be simplest), and notconsider the effect of commissions. Exercise 15 gives a justification for this.We obtain an IRR of 24.32%. Notice that the convex method applies to thissort of analysis of a stock’s performance.

7. J.C. Penney, 13.41%; Disney, 11.75%; Sizzler,−24.82%. The average is 0.11%,significantly different from the IRR for the entire portfolio. Since we did notinvest our $20,000 equally into the three companies, we would not expectthe average of their returns to equal the IRR. This illustrates another niceproperty of the IRR: its ability to account for investments of varying sizesin a portfolio.

8. r =ln[

24,428.9420,000.00

]814365

= .089696831 = 8.9696831%

9. On May 15, 1995, the balance in the savings account will be $5,029.53. Sincethe interest payments were all reinvested, the combined investment hasonly two cash flows: a deposit of $19,581.87 and a withdrawal of $22,029.53(the $17,000 face value plus the savings account balance). The IRR can thenbe computed simply as in Exercise 8 to be r = 5.28127.

This example points out one criticism of the IRR. The IRR effectivelypresumes that withdrawals can be reinvested at the same rate as the over-all IRR. Here the consideration of reinvesting the withdrawals at 5% haspulled the IRR down by 0.033%, from 5.314% (as computed earlier withoutreinvestment) to 5.281%.

10. At retirement, an investment in CREF would grow to be about four anda half times as large as the investment in TIAA. The exact multiple ise35(0.1330−0.0898) = e1.512 = 4.5358.

11. 12 payments, r = .0350 = 3.50%. 4 payments, r = −0.3253 = −32.53%. 30payments, r = .0930 = 9.30%.

12. P = $18, 290.935, A = $1, 157.72, I = $1, 912.50, so the approximate yieldis 0.0413 = 4.13%.

13. If r0 = 0.05, then r1 = 0.09. Then r2 = 0.05 and r3 = 0.09 again, and theprocess repeats. Note that this investment has no IRR, since there is no realsolution to p(r) = 0. Viewed as a quadratic in e−r, p(r) can be seen to havea negative discriminant.

14. The two values for the IRR are r = ln(0.3) = −120.40% and r = ln(0.4) =91.63%. These exact values can be found from the roots of a quadraticequation.

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19. The only real root of the cubic polynomial is x = 0.947992 (the other tworoots are complex numbers, x = −0.588098 ± 0.785513 i). Solving e−r =0.947992 yields r = 0.05341 = 5.341%.

ReferencesBrent, Richard P. 1973. Algorithms for Minimization without Derivatives. Engle-

wood Cliffs, NJ: Prentice-Hall. Chapters 3, 4.

Fabozzi, Frank J., T. Dessa Fabozzi, and Irving M. Pollack. 1991. The Handbookof Fixed Income Securities. Burr Ridge, IL: Business One Irwin.

Financial Computer Support, Inc. 1992. dbCMS+ 3.x Manual. Oakland, Mary-land.

Freund, Dwight D. 1995. A genuine application of synthetic division, Descartes’rule of signs and all that stuff. College Mathematics Journal 26 (2): 106–110.

Gerber, Hans U. 1990. Life Insurance Mathematics. New York: Springer-Verlag.

Intuit, Inc. 1995. Average Annual Return (IRR). Technical Support FaxbackDocument #6027.

Kellison, Stephen G. 1970. The Theory of Interest. Homewood, IL: Richard D.Irwin, Inc.

Khoury, Sarkis J., and Torrence D. Parsons. 1981. Mathematical Methods inFinance and Economics. New York: North Holland.

Lindstrom, Peter A. 1988. Nominal vs. Effective Rates of Interest. UMAP Mod-ules in Undergraduate Mathematics and Its Applications: Module 474.Reprinted in UMAP Modules: Tools for Teaching 1987, edited by Paul J. Camp-bell, 21–54. Arlington, MA: COMAP.

Maron, Melvin J., and Robert J. Lopez. 1993. The secant method and the goldenmean. The American Mathematical Monthly 100 (7): 676–678.

Paley, Hiram, Peter F. Colwell, and Roger E. Cannaday. 1984. Internal Rates ofReturn. UMAP Modules in Undergraduate Mathematics and Its Applica-tions: Module 640. Reprinted in UMAP Modules: Tools for Teaching 1983,493–548. Lexington, MA: COMAP.

Press, William H., Brian P. Flannery, Saul A. Teukolsky, and William T. Vetter-ling. 1989. Numerical Recipes in Pascal. New York: Cambridge UniversityPress. Chapter 9.

Shao, Stephen P. 1980. Mathematics for Management and Finance. 4th ed. Cincin-nati: South-Western Publishing Co.

Vianello, M., and R. Zanovello. 1992. On the superlinear convergence of thesecant method. American Mathematical Monthly 99 (8): 758-761.

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AcknowledgmentsHelpful conversations and suggestions were received from colleagues M.

Allison Beezer, Alva Butcher, and Bryan Smith and students Tony Austin, GavinConant, Eric Grouse, Eric Munger, Lissa Petersen, and Nathalie Williams. Thereferees also made several useful suggestions. The assistance of each of theseindividuals is gratefully acknowledged.

About the AuthorRobert Beezer received a B.S. in mathematics from Santa Clara University

in 1978 and a Ph.D. in mathematics from the University of Illinois at Urbana–Champaign in 1984. Since that time he has been on the faculty of the Universityof Puget Sound, with the exception of a visiting appointment at the Universityof the West Indies, St. Augustine, Trinidad in 1991. His research interestsrevolve around algebraic graph theory and regular graphs, and other topics incombinatorics. When he is not teaching calculus or chasing after his two veryyoung sons, he enjoys swimming, bicycling, volleyball, hiking, computers, and,of course, tracking the performance of his meager investments.

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UMAP Module 750