23.7Summary and Conclusions

This chapter described several tools that are important forbond portfolio management and corporate financial man-agement of fixed-income securities and interest rate risk.Managers can use these tools to better understand therelation between yield and market price in order to man-age interest rate risk. Among the tools discussed in thechapter were:

•The concept of DV01,in original and term

structure variations, which is a measure of the

slope of the price-yield curve.

•The concept of duration, in MacAuley, modified,

and present value variations, which is a weighted

average of the time at which a set of cash flows is

received.

•The link between duration and DV01and the

formulas needed to use either of them for hedging.

•Immunization and contingent immunization, using

both duration and DV01,which are methods of

converting the cash flows of a bond portfolio into

the cash flows of a zero-coupon bond.

•The link between immunization and hedging and

how to use this link for managing the asset base

and capital structure of financial institutions and

for managing a bond portfolio.

•The concept of convexity, which is similar to the

slope of the price-yield curve, and how it can be

properly used and misused.

The yield curve is not flat and does not shift in a par-allel fashion which complicates the use of these tools. Par-allel shifts lead to arbitrage, thus making it incumbent uponthose who desire a more rigorous analysis of interest ratehedging to develop a sophisticated model of the term struc-ture of interest rates that precludes such arbitrage.Final Remarks.Wall Street firms have developed anumber of sophisticated models that are used to value com-plicated interest rate derivatives, like the “inverse floater”sold to Orange County (see the opening vignette of thischapter). The basic structure of these interest rate deriva-tives valuation models is similar to that of the binomialmodel in Chapter 7. However, here, the risk neutral valua-tion accounts for interest rate factors that evolve along

trees (or grids) with move sizes for these factors deter-mined by the term structure of interest rates and volatility.The level of mathematical sophistication in these models isbeyond the scope of any textbook geared toward thoseseeking a general understanding of corporate finance andfinancial markets. Indeed, the rocket science associatedwith these models has given the rocket scientists—oftenPh.D.’s in mathematics, statistics, economics, physics, andfinance—an unprecedented degree of prestige, compensa-tion, and control over the management and valuation of in-terest rate derivatives.

The level of technical sophistication in corporate fi-nance has also increased considerably over the last twodecades. In some cases, nonfinancial corporations are hir-ing these same rocket scientists to oversee their hedgingoperations. However, this is still the exception. After 23chapters, our message to you is that no matter how sophis-ticated either investment management or corporate man-agement becomes, there are some fundamental principlesin finance that will always be useful.

We believe, for example, that understanding the toolsdescribed in this text is all that is necessary to recognize therisk in Orange County’s investment in inverse floaters andto skirt the disasters from misguided derivatives trades thathave befallen numerous other firms. Aserious student ofthis text, for example, would recognize (perhaps after a bitof thought) that an Orange County inverse floater could betracked by a portfolio with a weight of 3 on a fixed-ratestraight coupon bond and a weight of 2 on par floatingrate bonds. Since the floating rate bond’s duration is zero,this portfolio has three times the duration of a straightcoupon bond (see Result 23.3), and hence, is extremelysensitive to interest rate changes.

In short, one of the basic principles developed in thistextbook—the formation of portfolios to track an invest-ment with risks one is familiar with—can be used to un-derstand the risk of a highly complex security, like an in-verse floater. The lesson here is that one does not needadvanced mathematics or Wall Street’s sophisticated mod-els to master finance. One must, however, go beyond a su-perficial understanding of finance and master its basicprinciples to succeed as a finance professional in the 21stcentury. We wrote this text in the hopes of contributing tosuch an understanding.

Grinblatt1711Titman: Financial	VI. Risk Management	23. Interest Rate Risk	© The McGraw1711Hill
Markets and Corporate		Management	Companies, 2002
Strategy, Second Edition

Chapter 23

Interest Rate Risk Management

851

Key Concepts

Result 23.1:If the term structure of interest rates isResult 23.4:Because DV01can be translated into

flat, a bond portfolio with a DV01of zeroduration and vice versa, DV01and

has no sensitivity to interest rateduration are equivalent as tools both for

movements.measuring interest rate risk and forResult 23.2:Let rand rdenote the yield to maturityhedging.

nm

of the same bond computed with yieldsResult 23.5:If the term structure of interest rates is

compounded ntimes a year and mtimes aflat, immunization “guarantees” a fixed

year, respectively. The DV01for a bondvalue for an immunized portfolio at a

(or portfolio) using compounding of mhorizon date. The value obtained is the

times a year is (1 r n) (1 r m)same as the face value of a zero-coupon

nm

times the DV01of the bond using abond with (1) the same market value as

compounding frequency of ntimes a year.the original portfolio and (2) a maturity

Result 23.3:Assuming that the term structure ofdate equal to the horizon date selected as

interest rates is flat, the duration of athe target date to which the duration of

portfolio of bonds is the portfolio-the immunized portfolio is fixed.

weighted average of the durations of the

respective bonds in the portfolio.

Key Terms

contingent immunization	838	MacAuley duration847
convexity819		modified duration831
duration826		present value duration848
DV01820		term structure DV01847
effective duration832		yield beta846
immunization819

Exercises

23.1.	A3-year coupon bond has payments as follows:	23.2.Prove that if the present value of a cash flow is
		represented by

	PV (cash flow) exp(rt) t
Bond Cash Flow at Year	and for all t’s, rshifts up by a constant , the t
123	derivative of the percentage change in the bond’sprice with respect to equals the negative of
$8$8$108	present value duration; that is

1dP

(present value) DUR

This 8 percent coupon bond is currently trading at

Pd

par ($100).

a.What is the annually compounded yield of the23.3.Bond Ahas a DV01of $.10 per $100 face value.

bond?Bond B has a DV01of $.05 per $100 face value.

b.Compute the MacAuley duration and ordinarya.If Daniela buys $1 million (face amount) of

DV01(calculated with respect to the annuallybond A, what should her position (face amount)

compounded bond yield).in bond B be in order to hedge out all interest

c.Using DV01, how much do you expect thisrate risk on the portfolio?

bond’s price to rise if the yield on the bondb.If bond Aand bond B are par bonds (that is,

declines by 10 basis points compoundedthey have prices equal to their face values and

annually?have equal yields to maturity), what dollar

Grinblatt1713Titman: Financial	VI. Risk Management	23. Interest Rate Risk	© The McGraw1713Hill
Markets and Corporate		Management	Companies, 2002
Strategy, Second Edition

852Part VIRisk Management

	amount needs to be spent on bonds Aand B to	b.What is the present value duration of the
	immunize the bond portfolio to a horizon of	straight-coupon bond given the market value of
	seven years if $1 million is spent on the	the bond computed in part a?
	portfolio? (Assume that the DV01s were	c.Assume you hold $1 million face value of the
	calculated with respect to a 1 basis point	straight-coupon bond. How much in market
	decline in each bond’s continuously	value of a 3-year, zero-coupon bond should you
	compoundedyield to maturity.)	hold (in addition to the straight-coupon bond
23.4.	The DV01of a Treasury bond maturing on	position) to have an overall position that is
	November 15, 2021, with an 8 percent coupon	perfectly hedged against a parallel shift in the
	(4percent paid semiannually) and a $100 face	term structure?
	value is $.10. The DV01of a Treasury note	23.6.Compute the duration, DV01,and convexity of a
	maturing on May 15, 2012, with a 7 percent	semiannual straight-coupon bond with three years
	coupon (3.5 percent paid semiannually) and a	to maturity. The bond trades at par with a 6
	$100 face value is $.06.	percent coupon. Assume the term structure of
	a.What is the accrued interest (per $100 face	interest rates is flat.
	value) to be paid on both the bond and the note	23.7.Compute the duration, DV01,and convexity of a
	for a purchase with a settlement date of June	6percent 2-year par bond that pays semiannual
	11, 2002, for each of these fixed-income	coupons. Assume the term structure of interest
	securities? Hint: See Chapter 2.	rates is flat.
	b.If you held a position of $1 million (face value)
		23.8.Discuss how you might use the 6 percent 2-year
	in the Treasury bond, what position should you
		bond in exercise 23.7 to hedge a position in the
	hold in the Treasury note to eliminate all
		3-year bond from exercise 23.6.
	interest rate risk?
		23.9.How would your answer to exercise 23.8 change if
23.5.	A2-year default-free straight-coupon bond has
		the bond in exercise 23.7 were a 10 percent 2-year
	annual coupons of $8 per $100 of face value.
		premium bond with a yield curve still at a flat 6
	Assume that a default-free zero-coupon bond with
		percent?
	one year to maturity sells for $90 per $100 of face
		23.10.Discuss qualitatively how your answer to exercise
	value and that a default-free zero-coupon bond
		23.8 would change if the bond in exercise 23.7
	with two years to maturity sells for $80 per $100
		was a 10 percent 2-year par bond. (This means
	of face value.
		that the term structure of interest rates is not flat.)
	a.What is the no-arbitrage price of the straight-
	coupon 8 percent bond?

References and Additional Readings

Fabozzi, Frank. Bond Markets, Analysis, and Strategies.Yields, and Stock Prices in the U.S. since 1856.New

3d ed. Englewood Cliffs, NJ: Prentice-Hall, 1996.York: National Bureau of Economic Research, 1938.Kopprasch, Robert. “Understanding Duration andRedington, F. M. “Review of the Principle of Life-Office

Volatility.” New York: Salomon Brothers, 1985;Valuations.” Journal of the Institute of Actuaries78

reprinted in The Handbook of Fixed-Income(1952), pp. 286–340.

Securities,Frank Fabozzi and Irving Pollack, eds.Tuckman, Bruce. Fixed Income Securities: Tools for

Burr Ridge, IL: Irwin Professional Publishing, 1996.Today’s Markets.New York: John Wiley, 1995.

MacAuley, Frederick. Some Theoretical Problems

Suggested by the Movement of Interest Rates, Bond

Grinblatt1715Titman: Financial	VI. Risk Management	23. Interest Rate Risk	© The McGraw1715Hill
Markets and Corporate		Management	Companies, 2002
Strategy, Second Edition

Chapter 23Interest Rate Risk Management	853
PRACTICALINSIGHTSFORPARTVI

Allocating Capital forReal Investment

•Managers are likely to make better capital allocation

decisions if they hedge their risks because it forces

them to use forward and futures prices in their

calculations of NPVrather than potentially ad hoc

estimates of expected value. (Section 21.2)

•Firms sometimes build plants in foreign countries in

which they sell their products as a substitute for

hedging when it is difficult to effectively hedge real

exchange rate risk. (Section 21.8)

Financing the Firm

•The Modigliani-Miller Theorem is very general and

implies that, in the absence of market frictions, firms

are indifferent about the currencies of their debt

obligations and the maturity structure of their debt as

well as their debt-equity ratio. (Section 21.1)

•Firms that use derivatives to reduce their probability of

financial distress can create value by increasing their

debt ratio without increasing their probability of

financial distress. (Section 21.2)

•Firms that have high leverage ratios, but also potentially

have high financial distress costs, are more likely to use

derivatives to hedge. (Section 21.2)

•In general, hedging will benefit debt holders at the

expense of equity holders. (Section 21.6)

•Firms can sometimes reduce their funding costs by

combining debt instruments with swaps and other

derivatives. (Sections 21.7, 22.6)

Knowing Whetherand How to Hedge Risk•Firms can benefit from hedging even when shareholders

are not averse to risk. (Section 21.2)

•Firms that are uncertain about whether or not they will

have positive tax liabilities have an incentive to hedge.

(Section 21.2)

•Firms benefit from hedging if it allows them to avoid

the possibility of costly financial distress. (Section 21.2)•Firms benefit from hedging when external sources of

capital are more expensive than internal sources of

capital. However, when investment opportunities are

positively correlated with the hedgeable risk, firms

should only partially hedge. Options may prove to be a

particularly good hedging vehicle in this case. (Sections

21.2, 22.7)

•Management performance can be evaluated more

accurately if managers are required to hedge extraneous

risks. (Section 21.2)

•In most cases, firms benefit from hedging real rather

than strictly nominal exchange rate changes. (Section

21.8)

•Factor models are perhaps the best way to think about a

firm’s risk exposure. (Sections 21.1, 22.1, 22.8)

•Regression slope coefficients give risk-minimizing

hedge ratios. (Section 22.9)

•Even though futures and forward prices are often

virtually identical, hedge ratios using futures can vastly

differ from hedge ratios using forwards. In particular,

hedges with futures need to be tailed and thus are

generally lower than hedge ratios involving forwards.

(Section 22.3)

•The convenience yield of an asset or commodity affects

the hedge ratio when hedging long-term commitments

with short-term forwards or futures. The larger is the

convenience yield, the more correlated it is with the

price of the commodity, and the more unpredictable it

is, the lower is the risk minimizing hedge ratio.

(Sections 22.4, 22.5)

•Firms that set the ratio of the duration of assets and

liabilities to the inverse of the ratios of their market

values will have equity with a DV01of 0 which means

that their stock price is insensitive to interest rate risk.

(Section 23.3)

•It is possible to lock in (that is, immunize) the future

value of an interest rate sensitive liability or asset at a

horizon date by managing the liability to have a

duration that is matched to the horizon date. (Section

23.4)

•Even though DV01and duration are relatively simple

risk estimation and risk management tools in

comparison with the techniques on Wall Street, they

often provide good approximations of the interest rate

risk of an asset or liability and offer useful insights into

risk and hedging. (Sections 23.1–23.3, 23.7)

Allocating Funds forFinancial Investments

•Bond portfolios that are managed to have DV01s or

durations of zero have no interest rate risk. (Section

23.3)

Grinblatt1717Titman: Financial	VI. Risk Management	23. Interest Rate Risk	© The McGraw1717Hill
Markets and Corporate		Management	Companies, 2002
Strategy, Second Edition

854Part VIRisk Management

EXECUTIVEPERSPECTIVE

David C. Shimko

The failure to account for and control risks has led, fortu-nately and unfortunately, to several risk management andderivatives disasters since the mid-1980s. Unfortunately,shareholders and taxpayers have unwittingly borne thefallout of these debacles. Fortunately, we now have theopportunity to learn from these mistakes.

These are the lessons taught in Part VI of Grinblatt andTitman’s text. Corporate managers who think that riskmanagement is the province of back-office quants¹

shouldread this material, if not for the value in itself, then for theability to use the lessons learned to monitor those who takerisks. We should learn from the past, as Santayana said,“lest we be condemned to repeat it.”

The analysis of Metallgesellschaft AG’s poorly con-ceived hedge, and its implications at the corporate level, isaccurate and easy to follow. Clearly, either hedging wasnot MG’s intent, or hedging was entrusted to managerswith insufficient judgment or experience to hedgeproperly. Alarmingly, the same mistakes are repeated con-stantly in companies around the world. Fortunately formany of these companies, losses have mostly been eithersmall, underscrutinized, or labeled as hedges for account-ing purposes and lost in the paperwork. After studying theMG case, no corporate risk manager should undertake aone-for-one hedge of a longer-dated exposure with ashorter-dated one. Ironically, we know this lesson well inthe interest rate markets. The presentation in Grinblatt andTitman helps us bridge that intuitive gap and apply finan-cial markets knowledge broadly to the practice of corpo-rate finance.

It’s easy to focus on the potential costs of ignoring riskmanagement and thereby miss the hidden opportunitycosts. For example, the value-at-risk (VaR) concept helpsmany managers measure the potential loss associated withan investment activity. Indeed, J.P. Morgan scored a major

coup with the RiskMetrics™product, a relatively simpleanalytic means to determine VaR transparently. Detractorshave rightly argued that the methodology could beimproved, but they are missing the main point. The criticalimportance of VaR is that one can use it to measure riskand, thereby, risk-adjusted performance. VaR is fastbecoming a measure of capital that, for many purposes,works better than traditional capital measures. That is,regardless of the cost of a project, its performance shouldbe measured relative to the capital it actually places at risk.A$100 million mine and a zero-cost swap have the samerisk if both might lose $40 million. Their performanceshould be measured relative to the $40 million value-at-risk, not the cash investment.

Since the mid-1970s, managers have mastered the con-cept of value-added (or EVA, economic value added, orNPV, net present value). In the next 20 years, I hope thesame can be said of VaR and risk management.

The best managed corporations are starting to evaluaterisk and capital management in the same framework. PeterRugg, the CFO of Triton, funded a greater proportion ofhis company’s Colombian oil production with debt than heotherwise would have been able to do because he hadhedged against oil price fluctuations. Essentially, he con-served equity capital by replacing it with contingent capi-tal provided by hedging. (Translation:Swaps markets borethe oil price risk instead of equity holders.) The equity cap-ital was released to seek higher returns elsewhere.

The messages for risk management are clear.Defensively, we need risk management to know our poten-tial losses and to facilitate planning to protect ourselvesagainst those losses. Progressively, we need risk manage-ment to help us make better investment decisions. Everycorporate executive will someday be seen as a riskmanager—only some don’t know this yet! This book willhelp those executives be better risk managers at first, andbetter managers in the end.

¹Quants are employees with considerable quantitative

Dr. Shimko is President of Risk Capital Management Partners, LLC

background or experience, typically hired by a trading floor or

(www.e-rcm.com) specializing in risk management consulting services torisk management department to build mathematical models for

the energy industry. Previously, he headed a consulting group at Bankerspricing and risk assessment.Trust and a risk management research group at JPMorgan.

Grinblatt1719Titman: Financial	VI. Risk Management	23. Interest Rate Risk	© The McGraw1719Hill
Markets and Corporate		Management	Companies, 2002
Strategy, Second Edition

Grinblatt1720Titman: Financial	Back Matter	Appendix A	© The McGraw1720Hill
Markets and Corporate			Companies, 2002
Strategy, Second Edition

APPENDIXA

MATHEMATICALTABLES

ABLEA.1	Future Value of $1 at the End of tPeriods (1 r)t
T

				Interest Rate
Period	1%	2%	3%	4%5%	6%	7%	8%	9%

1	1.0100	1.0200	1.0300	1.0400	1.0500	1.0600	1.0700	1.0800	1.0900
2	1.0201	1.0404	1.0609	1.0816	1.1025	1.1236	1.1449	1.1664	1.1881
3	1.0303	1.0612	1.0927	1.1249	1.1576	1.1910	1.2250	1.2597	1.2950
4	1.0406	1.0824	1.1255	1.1699	1.2155	1.2625	1.3108	1.3605	1.4116
5	1.0510	1.1041	1.1593	1.2167	1.2763	1.3382	1.4026	1.4693	1.5386

6	1.0615	1.1262	1.1941	1.2653	1.3401	1.4185	1.5007	1.5869	1.6671
7	1.0721	1.1487	1.2299	1.3159	1.4071	1.5036	1.6058	1.7138	1.8280
8	1.0829	1.1717	1.2668	1.3686	1.4775	1.5938	1.7182	1.8509	1.9926
9	1.0937	1.1951	1.3048	1.4233	1.5513	1.6895	1.8385	1.9990	2.1719
10	1.1046	1.2190	1.3439	1.4802	1.6289	1.7908	1.9672	2.1589	2.3674

11	1.1157	1.2434	1.3842	1.5395	1.7103	1.8983	2.1049	2.3316	2.5804
12	1.1268	1.2682	1.4258	1.6010	1.7959	2.0122	2.2522	2.5182	2.8127
13	1.1381	1.2936	1.4685	1.6651	1.8856	2.1329	2.4098	2.7196	3.0658
14	1.1495	1.3195	1.5126	1.7317	1.9799	2.2609	2.5785	2.9372	3.3417
15	1.1610	1.3459	1.5580	1.8009	2.0789	2.3966	2.7590	3.1722	3.6425

16	1.1726	1.3728	1.6047	1.8730	2.1829	2.5404	2.9522	3.4259	3.9703
17	1.1843	1.4002	1.6528	1.9479	2.2920	2.6928	3.1588	3.7000	4.3276
18	1.1961	1.4282	1.7024	2.0258	2.4066	2.8543	3.3799	3.9960	4.7171
19	1.2081	1.4568	1.7535	2.1068	2.5270	3.0256	3.6165	4.3157	5.1417
20	1.2202	1.4859	1.8061	2.1911	2.6533	3.2071	3.8697	4.6610	5.6044

21	1.2324	1.5157	1.8603	2.2788	2.7860	3.3996	4.1406	5.0338	6.1088
22	1.2447	1.5460	1.9161	2.3699	2.9253	3.6035	4.4304	5.4365	6.6586
23	1.2572	1.5769	1.9736	2.4647	3.0715	3.8197	4.7405	5.8715	7.2579
24	1.2697	1.6084	2.0328	2.5633	3.2251	4.0489	5.0724	6.3412	7.9111
25	1.2824	1.6406	2.0938	2.6658	3.3864	4.2919	5.4274	6.8485	8.6231

30	1.3478	1.8114	2.4273	3.2434	4.3219	5.7435	7.6123	10.063	13.268
40	1.4889	2.2080	3.2620	4.8010	7.0400	10.286	14.974	21.725	31.409
50	1.6446	2.6916	4.3839	7.1067	11.467	18.420	29.457	46.902	74.358
60	1.8167	3.2810	5.8916	10.520	18.679	32.988	57.946	101.26	176.03

856

Grinblatt1721Titman: Financial	Back Matter	Appendix A	© The McGraw1721Hill
Markets and Corporate			Companies, 2002
Strategy, Second Edition

Mathematical Tables

857

TABLEA.1(concluded)

				Interest Rate
10%	12%	14%	15%	16%18%20%	24%	28%	32%	36%

1.10001.12001.14001.15001.16001.18001.20001.24001.28001.32001.3600

1.21001.25441.29961.32251.34561.39241.44001.53761.63841.74241.8496

1.33101.40491.48151.52091.56091.64301.72801.90662.09722.30002.5155

1.46411.57351.68901.74901.81061.93882.07362.36422.68443.03603.4210

1.61051.76231.92542.01142.10032.28782.48832.93163.43604.00754.6526

1.77161.97382.19502.31312.43642.69962.98603.63524.39805.28996.3275

1.94872.21072.50232.66002.82623.18553.58324.50775.62956.98268.6054

2.14362.47602.85263.05903.27843.75894.29985.58957.20589.217011.703

2.35792.77313.25193.51793.80304.43555.15986.93109.223412.16615.917

2.59373.10583.70724.04564.41145.23386.19178.594411.80616.06021.647

2.85313.47854.22624.65245.11736.17597.430110.65715.11221.19929.439

3.13843.89604.81795.35035.93607.28768.916113.21519.34327.98340.037

3.45234.36355.49246.15286.88588.599410.69916.38624.75936.93754.451

3.79754.88716.26137.07577.987510.14712.83920.31931.69148.75774.053

4.17725.47367.13798.13719.265511.97415.40725.19640.56564.359100.712

4.59506.13048.13729.357610.74814.12918.48831.24351.92384.954136.97

5.05456.86609.276510.76112.46816.67222.18638.74166.461112.14186.28

5.55997.690010.57512.37514.46319.67326.62348.03985.071148.02253.34

6.11598.612812.05614.23216.77723.21431.94859.568108.89195.39344.54

6.72759.646313.74316.36719.46127.39338.33873.864139.38257.92468.57

7.400210.80415.66818.82222.57432.32446.00591.592178.41340.45637.26

8.140312.10017.86121.64526.18638.14255.206113.57228.36449.39866.67

8.954313.55220.36224.89130.37645.00866.247140.83292.30593.201178.7

9.849715.17923.21228.62535.23653.10979.497174.63374.14783.021603.0

10.83517.00026.46232.91940.87462.66995.396216.54478.901033.62180.1

17.449	29.960	50.950	66.212	85.850	143.37	237.38	634.821645.54142.110143.
45.259	93.051	188.88	267.86	378.72	750.38	1469.8	5455.919427.66521.*
117.39	289.00	700.23	1083.7	1670.7	3927.4	9100.4	46890.***
304.48	897.60	2595.9	4384.0	7370.2	20555.	56348.	****

*The factor is greater than 99,999.

Grinblatt1723Titman: Financial	Back Matter	Appendix A	© The McGraw1723Hill
Markets and Corporate			Companies, 2002
Strategy, Second Edition

858	Appendix A

TABLEA.2Present Value of $1 to be Received aftertPeriods 1/(1 r)^t

				Interest Rate
Period	1%	2%	3%	4%5%	6%	7%	8%	9%

1	0.9901	0.9804	0.9709	0.9615	0.9524	0.9434	0.9346	0.9259	0.9174
2	0.9803	0.9612	0.9426	0.9246	0.9070	0.8900	0.8734	0.8573	0.8417
3	0.9706	0.9423	0.9151	0.8890	0.8638	0.8396	0.8163	0.7938	0.7722
4	0.9610	0.9238	0.8885	0.8548	0.8227	0.7921	0.7629	0.7350	0.7084
5	0.9515	0.9057	0.8626	0.8219	0.7835	0.7473	0.7130	0.6806	0.6499

6	0.9420	0.8880	0.8375	0.7903	0.7462	0.7050	0.6663	0.6302	0.5963
7	0.9327	0.8706	0.8131	0.7599	0.7107	0.6651	0.6227	0.5835	0.5470
8	0.9235	0.8535	0.7894	0.7307	0.6768	0.6274	0.5820	0.5403	0.5019
9	0.9143	0.8368	0.7664	0.7026	0.6446	0.5919	0.5439	0.5002	0.4604
10	0.9053	0.8203	0.7441	0.6756	0.6139	0.5584	0.5083	0.4632	0.4224

11	0.8963	0.8043	0.7224	0.6496	0.5847	0.5268	0.4751	0.4289	0.3875
12	0.8874	0.7885	0.7014	0.6246	0.5568	0.4970	0.4440	0.3971	0.3555
13	0.8787	0.7730	0.6810	0.6006	0.5303	0.4688	0.4150	0.3677	0.3262
14	0.8700	0.7579	0.6611	0.5775	0.5051	0.4423	0.3878	0.3405	0.2992
15	0.8613	0.7430	0.6419	0.5553	0.4810	0.4173	0.3624	0.3152	0.2745

16	0.8528	0.7284	0.6232	0.5339	0.4581	0.3936	0.3387	0.2919	0.2519
17	0.8444	0.7142	0.6050	0.5134	0.4363	0.3714	0.3166	0.2703	0.2311
18	0.8360	0.7002	0.5874	0.4936	0.4155	0.3503	0.2959	0.2502	0.2120
19	0.8277	0.6864	0.5703	0.4746	0.3957	0.3305	0.2765	0.2317	0.1945
20	0.8195	0.6730	0.5537	0.4564	0.3769	0.3118	0.2584	0.2145	0.1784

21	0.8114	0.6598	0.5375	0.4388	0.3589	0.2942	0.2415	0.1987	0.1637
22	0.8034	0.6468	0.5219	0.4220	0.3418	0.2775	0.2257	0.1839	0.1502
23	0.7954	0.6342	0.5067	0.4057	0.3256	0.2618	0.2109	0.1703	0.1378
24	0.7876	0.6217	0.4919	0.3901	0.3101	0.2470	0.1971	0.1577	0.1264
25	0.7798	0.6095	0.4776	0.3751	0.2953	0.2330	0.1842	0.1460	0.1160

30	0.7419	0.5521	0.4120	0.3083	0.2314	0.1741	0.1314	0.0994	0.0754
40	0.6717	0.4529	0.3066	0.2083	0.1420	0.0972	0.0668	0.0460	0.0318
50	0.6080	0.3715	0.2281	0.1407	0.0872	0.0543	0.0339	0.0213	0.0134

Grinblatt1724Titman: Financial	Back Matter	Appendix A	© The McGraw1724Hill
Markets and Corporate			Companies, 2002
Strategy, Second Edition

Mathematical Tables

859

TABLEA.2(concluded)

				Interest Rate
10%	12%	14%	15%	16%18%20%	24%	28%	32%	36%

0.90910.89290.87720.86960.86210.84750.83330.80650.78130.75760.7353

0.82640.79720.76950.75610.74320.71820.69440.65040.61040.57390.5407

0.75130.71180.67500.65750.64070.60860.57870.52450.47680.43480.3975

0.68300.63550.59210.57180.55230.51580.48230.42300.37250.32940.2923

0.62090.56740.51940.49720.47610.43710.40190.34110.29100.24950.2149

0.56450.50660.45560.42320.41040.37040.33490.27510.22740.18900.1580

0.51320.45230.39960.37590.35380.31390.27910.22180.17760.14320.1162

0.46650.40390.35060.32690.30500.26600.23260.17890.13880.10850.0854

0.42410.36060.30750.28430.26300.22550.19380.14430.10840.08220.0628

0.38550.32200.26970.24720.22670.19110.16150.11640.08470.06230.0462

0.35050.28750.23660.21490.19540.16190.13460.09380.06620.04720.0340

0.31860.25670.20760.18690.16850.13720.11220.07570.05170.03570.0250

0.28970.22920.18210.16250.14520.11630.09350.06100.04040.02710.0184

0.26330.20460.15970.14130.12520.09850.07790.04920.03160.02050.0135

0.23940.18270.14010.12290.10790.08350.06490.03970.02470.01550.0099

0.21760.16310.12290.10690.09300.07080.05410.03200.01930.01180.0073

0.19780.14560.10780.09290.08020.06000.04510.02580.01500.00890.0054

0.17990.13000.09460.08080.06910.05080.03760.02080.01180.00680.0039

0.16350.11610.08290.07030.05960.04310.03130.01680.00920.00510.0029

0.14860.10370.07280.06110.05140.03650.02610.01350.00720.00390.0021

0.13510.09260.06380.05310.04430.03090.02170.01090.00560.00290.0016

0.12280.08260.05600.04620.03820.02620.01810.00880.00440.00220.0012

0.11170.07380.04910.04020.03290.02220.01510.00710.00340.00170.0008

0.10150.06590.04310.03490.02840.01880.01260.00570.00270.00130.0006

0.09230.05880.03780.03040.02450.01600.01050.00460.00210.00100.0005

0.0573	0.0334	0.0196	0.0151	0.0116	0.0070	0.0042	0.00160.00060.00020.0001
0.0221	0.0107	0.0053	0.0037	0.0026	0.0013	0.0007	0.00020.0001**
0.0085	0.0035	0.0014	0.0009	0.0006	0.0003	0.0001	****

*The factor is zero to four decimal places.

Grinblatt1725Titman: Financial	Back Matter	Appendix A	© The McGraw1725Hill
Markets and Corporate			Companies, 2002
Strategy, Second Edition

860	Appendix A

TABLEA.3

Present Value of an Annuity of $1 perPeriod fortPeriods [1 1/(1 r)t ]/r