to c2hapter Financial Market Instruments · Here we examine the securities (instruments) traded in financial markets. We first focus on the instruments traded in the money market

Here we examine the securities (instruments) traded in financial markets. We firstfocus on the instruments traded in the money market and then turn to those tradedin the capital market.

Because of their short terms to maturity, the debt instruments traded in the moneymarket undergo the least price fluctuations and so are the least risky investments. Themoney market has undergone great changes in the past three decades, with theamount of some financial instruments growing at a far more rapid rate than others.

The principal money market instruments are listed in Table 1 along with theamount outstanding at the end of 1970, 1980, 1990, and 2002.

Government of Canada Treasury Bills. These short-term debt instruments of theCanadian government are issued in 1-, 3-, 6-, and 12-month maturities to finance thefederal government. They pay a set amount at maturity and have no interest payments,but they effectively pay interest by initially selling at a discount, that is, at a price lowerthan the set amount paid at maturity. For instance, you might pay $9,000 in May 2004for a one-year treasury bill that can be redeemed in May 2005 for $10,000.

Treasury bills are the most liquid of all the money market instruments, becausethey are the most actively traded. They are also the safest of all money market instru-ments, because there is almost no possibility of default, a situation in which the partyissuing the debt instrument (the federal government, in this case) is unable to makeinterest payments or pay off the amount owed when the instrument matures. The fed-eral government is always able to meet its debt obligations, because it can raise taxesto pay off its debts. Treasury bills are held mainly by banks, although households,corporations, and other financial intermediaries hold small amounts.

Certificates of Deposit A certificate of deposit (CD) is a debt instrument sold by abank to depositors that pays annual interest of a given amount and at maturity paysback the original purchase price. CDs are often negotiable, meaning that they can betraded, and in bearer form (called bearer deposit notes), meaning that the buyer’sname is neither recorded in the issuer’s books nor on the security itself. Thesenegotiable CDs are issued in multiples of $100 000 and with maturities of 30 to 365days, and can be resold in a secondary market, thus offering the purchaser bothyield and liquidity.

Chartered banks also issue non-negotiable CDs. That is, they cannot be sold tosomeone else and cannot be redeemed from the bank before maturity without payinga substantial penalty. Non-negotiable CDs are issued in denominations ranging from

Money MarketInstruments

1

F inanc i a l Marke t I n s t rumen t s

a p p en d i xto c h a p te r

2

$5000 to $100 000 and with maturities of 1 day to 5 years. They are also known asterm deposit receipts or term notes.

CDs are also an extremely important source of funds for trust and mortgage loancompanies. These institutions issue CDs under a variety of names such as, for exam-ple, DRs (Deposit Receipts), GTCs (Guaranteed Trust Certificates), GICs (GuaranteedInvestment Certificates), and GIRs (Guaranteed Investment Receipts).

Commercial Paper. Commercial paper is an unsecured short-term debt instrumentissued in either Canadian dollars or other currencies by large banks and well-knowncorporations, such as General Motors and DaimlerChrysler. Because commercial paperis unsecured, only the largest and most creditworthy corporations issue commercialpaper. The interest rate the corporation is charged reflects the firm’s level of risk. Theinterest rate on commercial paper is low relative to those on other corporate fixed-income securities and slightly higher than rates on government of Canada treasury bills.

Finance and commercial paper are issued in minimum denominations of $50 000and in maturities of 30 to 365 days for finance paper and 1 to 365 days for commercialpaper. Most finance and commercial paper is issued on a discounted basis. Chapter 10discusses why the commercial paper market has had such tremendous growth.

Banker’s Acceptances. These money market instruments are created in the course ofcarrying out international trade and have been in use for hundreds of years. A banker’sacceptance is a bank draft (a promise of payment similar to a cheque) issued by a firm,payable at some future date, and guaranteed for a fee by the bank that stamps it“accepted.” The firm issuing the instrument is required to deposit the required fundsinto its account to cover the draft. If the firm fails to do so, the bank’s guarantee meansthat it is obligated to make good on the draft. The advantage to the firm is that thedraft is more likely to be accepted when purchasing goods abroad, because the for-eign exporter knows that even if the company purchasing the goods goes bankrupt,the bank draft will still be paid off. These “accepted” drafts are often resold in a sec-ondary market at a discount and so are similar in function to treasury bills. Typically,

2 Appendix to Chapter 2

Amount Outstanding (in millions)Type of Instrument 1970 1980 1990 2002

Treasury billsGovernment of Canada 2 762 13 709 113 654 87 604Provincial governments 428 905 12 602 21 022Municipal governments 25 113 514 177

Short-term paperBankers acceptances 291 4 874 46 738 44 882Commercial paper 588 2 555 12 971 21 503

Source: Statistics Canada Cansim II series V37377, V122256, V122257, V122635, and V122652.

Table 1 Principal Money Market Instruments

MONEY RATES

Financial Market Instruments 3

Following the Financial News

The Globe and Mail and the National Post publish daily alisting of interest rates on many different financialinstruments. In The Globe and Mail: Report on Business,this listing can be found in the “Money Rates” coloumn.

The interest rates in the “Money Rates” columnthat are discussed most frequently in the media are as follows:

Bank rate: The interest rate charged by the Bank ofCanada on loans made to members of the CanadianPayments Association.

Target overnight rate: The overnight rate that theBank of Canada is targeting at the midpoint of the oper-ating band for the overnight rate.

Prime rate: The base interest rate on corporate bankloans, an indicator of the cost of business borrowingfrom banks.

Treasury bill rates: The interest rates on governmentof Canada treasury bills, an indicator of general interestrate movements.

Selected U.S. interest rates: Selected U.S. interest ratessuch as the federal funds rate, prime rate, and com-mercial paper rate. These are indicators of generalinterest rate movements in the United States.

Money Market Rates

they are held by many of the same parties that hold treasury bills, and the amountoutstanding has experienced phenomenal growth.

The phenomenal growth in banker’s acceptances in Canada is due to the growthof the Canadian money market and the fact that Canadian chartered banks enjoystronger credit ratings than all but the largest corporations. Moreover, revisions in theBank Act have removed certain restrictions regarding the issuance of banker’s accept-ances and the banks have reduced the stamping fees that they charge for banker’sacceptances—these fees vary from 0.20% to 0.75%.

Repurchase Agreements. Repurchase agreements, or repos, are effectively short-termloans (usually with a maturity of less than two weeks) in which treasury bills serveas collateral, an asset that the lender receives if the borrower does not pay back theloan. Repos are made as follows: a large corporation, such as General Motors, may

ADMINISTERED

Bank of Canada 2.50%Target overnight rate 2.25%Central bank call range 2.00-2.50%Canadian prime 4.00%

MONEY MARKET

(for transactions of$1-million or more)

3-month treasury bills 2.45%6-month treasury bills 2.67%1-year treasury bills 3.37%10-year Canada bonds 5.69%30-year Canada bonds 5.95%1-month banker’s accept. 2.41%2-month banker’s accept. 2.52%3-month banker’s accept. 2.65%Commercial Paper (R-1 low)1-month 2.41%

3-month 2.68%Call money 2.25%Bloomberg News

UNITED STATES

NEW YORK (AP) – Money rates for Thursday asreported by Moneyline Telerate as of 4 p.m.:Prime Rate: 4.75Discount Rate: 1.25Broker call loan rate: 3.50Federal funds market rate: High 1.8125; low 1.8125; last 1.8125Dealers commercial paper: 30-180 days: 1.79-1.95Commercial paper by finance company: 30-270days: 1.90-2.09Bankers acceptances dealer indications: 30 days, 1.84; 60 days, 1.85; 90 days, 1.86; 120 days, 1.91; 150 days, 1.99; 180 days, 2.04Certificates of Deposit Primary: 30 days, 1.48;

90 days, 1.41; 180 days, 1.59Certificates of Deposit by dealer: 30 days, 1.82;60 days, 1.83; 90 days, 1.84; 120 days, 1.89;150 days, 1.97; 180 days, 2.03Eurodollar rates: Overnight, 1.75-1.81; 1 month, 1.75-1.88; 3 months, 1.75-1.88; 6 months, 2.00-2.13; 1 year, 2.56-2.69London Interbank Offered Rate: 3 months, 1.91;6 months, 2.12; 1 year, 2.71Treasury Bill auction results: average discountrate: 3-month as of May. 13: 1.750; 6-month asof May 13: 1.870Treasury Bill annualized rate on weekly averagebasis, yield adjusted for constant maturity, 1-yearas of May. 13: 2.31Treasury Bill market rate, 6 Mos: 1.85-1.84Treasury Note market rate, 10-year: 5.18

Source: The Globe and Mail: Report on Business, May 17, 2002, p. B21. Reprinted with permission.

have some idle funds in its bank account, say $1 million, which it would like tolend for a week. GM uses this excess $1 million to buy treasury bills from a bank,which agrees to repurchase them the next week at a price slightly above GM’s pur-chase price. The effect of this agreement is that GM makes a loan of $1 million tothe bank and holds $1 million of the bank’s treasury bills until the bank repur-chases the bills to pay off the loan. Repurchase agreements are a fairly recent inno-vation in financial markets, having been introduced in 1969. They are now animportant source of bank funds (over $400 billion). The most important lendersin this market are large corporations.

Overnight Funds. These are typically overnight loans by banks to other banks. Theovernight funds designation is somewhat confusing, because these loans are not madeby the federal government or by the Bank of Canada, but rather by banks to otherbanks. One reason why a bank might borrow in the overnight funds market is that itmight find it does not have enough settlement deposits at the Bank of Canada. It canthen borrow these balances from another bank with excess settlement balances.

The overnight market is very sensitive to the credit needs of the deposit-takinginstitutions, so the interest rate on overnight loans, called the overnight funds rate, isa closely watched barometer of the tightness of credit market conditions in the bank-ing system and the stance of monetary policy. When it is high, it indicates that thebanks are strapped for funds, whereas when it is low, banks’ credit needs are low.

Capital market instruments are debt and equity instruments with maturities of greaterthan one year. They have far wider price fluctuations than money market instrumentsand are considered to be fairly risky investments. The principal capital market instru-ments are listed in Table 2, which shows the amount outstanding at the end of 1970,1980, 1990, and 2002.

Capital MarketInstruments


Amount Outstanding (in millions)Type of Instrument 1970 1980 1990 2002Corporate stocks (market value) 24 761 42 541 109 888 268 719Residential mortgages 17 139 90 542 243 737 479 814Corporate bonds 11 307 30 004 72 850 245 238Government of Canada securities (marketable) 9 772 27 862 124 577 281 638Bank commercial loans 11 299 58 751 102 574 122 433Consumer loans 11 439 42 741 97 460 217 881Nonresidential and farm mortgages 3 189 15 129 56 143 52 271

Source: Statistics Canada Cansim II series V122642, V122746, V122640, V37378 V122631, V122707, V122656, V122657, V122658,V122659, V800015, and the authors' calculations.

Table 2 Principal Capital Market Instruments

Stocks. Stocks are equity claims on the net income and assets of a corporation. Theirvalue was $268 billion at the end of 2002. The amount of new stock issues in anygiven year is typically quite small—less than 1% of the total value of shares out-standing. Individuals hold around half of the value of stocks; pension funds, mutualfunds, and insurance companies hold the rest.

Mortgages. Mortgages are loans to households or firms to purchase housing, land, orother real structures, where the structure or land serves as collateral for the loans. Themortgage market is the largest debt market in Canada, with the amount of residentialmortgages (used to purchase residential housing) outstanding more than ninefold theamount of commercial and farm mortgages. Trust and mortgage loan companies andcredit unions and caisses populaires were the primary lenders in the residential mort-gage market until 1967. The revision of the Bank Act in 1967, however, extended theauthority of chartered banks to make conventional residential mortgage loans andchartered banks have entered this market very aggressively in the last two decades. Infact, their market share of residential mortgages has increased from about 50% in1989 to about 65% in 2002.

Banks and life insurance companies make the majority of commercial and farmmortgages. The federal government also plays an active role in the mortgage marketvia the Canada Mortgage and Housing Corporation (CMHC), which provide funds tothe mortgage market by selling bonds and using the proceeds to buy mortgages. Animportant development in the residential mortgage market in recent years is the mort-gage-backed security (see Box 1).


Box 1

Mortgage-Backed SecuritiesA major change in the residential mortgage market inrecent years has been the creation of an active second-ary market for mortgages. Because mortgages have dif-ferent terms and interest rates, they are not sufficientlyliquid to trade as securities on secondary markets. Tostimulate mortgage lending, in late 1986 the govern-ment of Canada introduced the concept of a pass-through mortgage-backed security, patterned after theU.S. Government National Mortgage Association(GNMA, called “Ginnie Mae”). Mortgage-backed secu-rities are not government of Canada securities but theyare guaranteed by the Canada Mortgage and HousingCorporation (CMHC)—a federal government agency.

Under this program, private financial institutionssuch as chartered banks, trust and mortgage loancompanies, and credit unions and caisses populairesgather a group of residential first mortgages withsimilar interest rates and terms to maturity (usuallyfive years) into a bundle (of, say, $1 million). These

mortgages must be individually guaranteed underthe National Housing Act. This bundle is then soldas a security to a third party, usually a large institu-tional investor such as a pension fund. When indi-viduals make their mortgage payments to thefinancial institution, the financial institution passesthe payments through to the owner of the securityby sending a cheque for the total of all payments.Because CMHC guarantees the payments, thesepass-through securities have a very low default riskand are very popular.

Mortgage-backed securities have been so successfulthat they have completely transformed the residentialmortgage market. Throughout the 1970s, over 80% ofresidential mortgages were owned outright by trustand mortgage loan companies, credit unions andcaisses populaires, and chartered banks. Now only afraction is owned outright by these institutions, withthe rest held as mortgage-backed securities.

Corporate Bonds. These are long-term bonds issued by corporations with very strongcredit ratings. The typical corporate bond sends the holder an interest payment twicea year and pays off the face value when the bond matures. Some corporate bonds,called convertible bonds, have the additional feature of allowing the holder to convertthem into a specified number of shares of stock at any time up to the maturity date.This feature makes these convertible bonds more desirable to prospective purchasersthan bonds without it, and allows the corporation to reduce its interest payments,because these bonds can increase in value if the price of the stock appreciates suffi-ciently. Because the outstanding amount of both convertible and nonconvertiblebonds for any given corporation is small, they are not nearly as liquid as other secu-rities such as Government of Canada bonds.

Although the size of the corporate bond market is substantially smaller than thatof the stock market, the volume of new corporate bonds issued each year is substan-tially greater than the volume of new stock issues. Thus the behaviour of the corpo-rate bond market is probably far more important to a firm’s financing decisions thanthe behaviour of the stock market. The principal buyers of corporate bonds are lifeinsurance companies; pension funds and households are other large holders.

Government of Canada Medium- and Long-Term Bonds Medium-term bonds (those withinitial maturities from 3 to 10 years) and long-term bonds (those with initial maturi-ties greater than 10 years) are issued by the federal government to finance its deficit.Because they are the most widely traded bonds in Canada, they are the most liquidsecurity traded in the capital market. They are held by the Bank of Canada, banks,households, and foreign investors.

These debt instruments are issued in either bearer or registered form and indenominations of $1000, $5000, $25 000, $100 000, and $1 million. In the case ofregistered bonds, the name of the owner appears on the bond certificate and is alsorecorded at the Bank of Canada. Some issues have the additional call or redemptionfeature of allowing them to be “called” on specified notice (usually 30 to 60 days).

Canada Savings Bonds These are non-marketable bonds issued by the government ofCanada once a year, generally for about two weeks ending on November 1. CanadaSavings Bonds (CSBs) are floating-rate bonds, available in denominations from $100to $10 000, and offered exclusively to individuals, estates, and specified trusts. Theyare issued as registered bonds and can be purchased from financial institutions orthrough payroll savings plans.

CSBs are different from all other bonds issued by the government of Canada inthat they do not rise or fall in value, like other bonds do. They have the valuableoption of being redeemable at face value plus accrued interest, at any time prior tomaturity, by being presented at any financial institution. In October 1998 the gov-ernment of Canada introduced another type of bonds that are similar to CSBs—theCanada Premium Bonds (CPBs). CPBs offer a slightly higher coupon rate than theCSBs, but can be redeemed only once a year, on the anniversary of the issue date andduring the month after that date.

Provincial and Municipal Government Bonds Provincial and municipal governmentsalso issue bonds to finance expenditures on schools, roads, and other large programs.The securities issued by provincial governments are referred to as provincial bonds orprovincials and those issued by municipal governments as municipal bonds ormunicipals—the securities issued by the federal government are referred to as


Canadas. Provincials and municipals are denominated in either domestic currency orforeign currencies, mostly U.S. dollars, Swiss francs, and Japanese yen. They aremainly held by trusteed pension plans, social security funds (predominantly theCanada Pension Plan), and foreigners.

Government Agency Securities These are long-term bonds issued by variousgovernment agencies such as the Ontario Municipal Improvement Corporation andthe Alberta Municipal Financing Corporation to assist municipalities to finance suchitems as mortgages, farm loans, or power-generating equipment. The provincial gov-ernments guarantee many of these securities. They function much like Canadas,provincials, and municipals and are held by similar parties.

Consumer and Bank Commercial Loans. These are loans to consumers and businessesmade principally by banks, but—in the case of consumer loans—also by finance com-panies. There are often no secondary markets in these loans, which makes them theleast liquid of the capital market instruments listed in Table 2. However, secondarymarkets have been rapidly developing.


In our discussion of interest-rate risk, we saw that when interest rates change, a bondwith a longer term to maturity has a larger change in its price and hence more interest-rate risk than a bond with a shorter term to maturity. Although this is a useful gen-eral fact, in order to measure interest-rate risk, the manager of a financial institutionneeds more precise information on the actual capital gain or loss that occurs when theinterest rate changes by a certain amount. To do this, the manager needs to make useof the concept of duration, the average lifetime of a debt security’s stream of payments.

The fact that two bonds have the same term to maturity does not mean that theyhave the same interest-rate risk. A long-term discount bond with ten years to matu-rity, a so-called zero-coupon bond, makes all of its payments at the end of the ten years,whereas a 10% coupon bond with ten years to maturity makes substantial cash pay-ments before the maturity date. Since the coupon bond makes payments earlier thanthe zero-coupon bond, we might intuitively guess that the coupon bond’s effectivematurity, the term to maturity that accurately measures interest-rate risk, is shorterthan it is for the zero-coupon discount bond.

Indeed, this is exactly what we find in example 1.

EXAMPLE 1: Rate of Capital Gain

Calculate the rate of capital gain or loss on a ten-year zero-coupon bond for which theinterest rate has increased from 10% to 20%. The bond has a face value of $1,000.

SolutionThe rate of capital gain or loss is �49.7%.

g �

where

Pt�1 � price of the bond one year from now � � $193.81

Pt � price of the bond today � � $385.54$1,000

(1 � 0.10 )10

$1,000

(1 � 0.20 )9

Pt�1 � Pt

Pt

Measu r ing In t e re s t -Ra t e R i sk : Dura t i on


4

Thus:

g �

g � �0.497 � �49.7%

But as we have already calculated in Table 2 in Chapter 4, the capital gain on the10% ten-year coupon bond is �40.3%. We see that interest-rate risk for the ten-yearcoupon bond is less than for the ten-year zero-coupon bond, so the effective maturityon the coupon bond (which measures interest-rate risk) is, as expected, shorter thanthe effective maturity on the zero-coupon bond.

To calculate the duration or effective maturity on any debt security, FrederickMacaulay, a researcher at the National Bureau of Economic Research, invented theconcept of duration more than half a century ago. Because a zero-coupon bondmakes no cash payments before the bond matures, it makes sense to define its effec-tive maturity as equal to its actual term to maturity. Macaulay then realized that hecould measure the effective maturity of a coupon bond by recognizing that a couponbond is equivalent to a set of zero-coupon discount bonds. A ten-year 10% couponbond with a face value of $1,000 has cash payments identical to the following set ofzero-coupon bonds: a $100 one-year zero-coupon bond (which pays the equivalentof the $100 coupon payment made by the $1,000 ten-year 10% coupon bond at theend of one year), a $100 two-year zero-coupon bond (which pays the equivalent ofthe $100 coupon payment at the end of two years), … , a $100 ten-year zero-couponbond (which pays the equivalent of the $100 coupon payment at the end of tenyears), and a $1,000 ten-year zero-coupon bond (which pays back the equivalent ofthe coupon bond’s $1,000 face value). This set of coupon bonds is shown in thefollowing time line:

This same set of coupon bonds is listed in column (2) of Table 1, which calculates theduration on the ten-year coupon bond when its interest rate is 10%.

To get the effective maturity of this set of zero-coupon bonds, we would want tosum up the effective maturity of each zero-coupon bond, weighting it by the per-centage of the total value of all the bonds that it represents. In other words, the dura-tion of this set of zero-coupon bonds is the weighted average of the effectivematurities of the individual zero-coupon bonds, with the weights equaling the pro-portion of the total value represented by each zero-coupon bond. We do this in sev-eral steps in Table 1. First we calculate the present value of each of the zero-couponbonds when the interest rate is 10% in column (3). Then in column (4) we divideeach of these present values by $1,000, the total present value of the set of zero-coupon bonds, to get the percentage of the total value of all the bonds that each bondrepresents. Note that the sum of the weights in column (4) must total 100%, as shownat the bottom of the column.

0 1 2 3 4 5 6 7 8 9 10Year When Paid

Amount$100 $100 $100 $100 $100 $100 $100 $100 $100 $100

$1,000

CalculatingDuration

$193.81 � $385.54

$385.54

Measuring Interest-Rate Risk: Duration 9

To get the effective maturity of the set of zero-coupon bonds, we add up theweighted maturities in column (5) and obtain the figure of 6.76 years. This figure forthe effective maturity of the set of zero-coupon bonds is the duration of the 10% ten-year coupon bond because the bond is equivalent to this set of zero-coupon bonds.In short, we see that duration is a weighted average of the maturities of the cashpayments.

The duration calculation done in Table 1 can be written as follows:

(1)

where DUR � durationt � years until cash payment is made

CPt � cash payment (interest plus principal) at time ti � interest raten � years to maturity of the security

This formula is not as intuitive as the calculation done in Table 1, but it does have theadvantage that it can easily be programmed into a calculator or computer, makingduration calculations very easy.

If we calculate the duration for an 11-year 10% coupon bond when the interestrate is again 10%, we find that it equals 7.14 years, which is greater than the 6.76years for the ten-year bond. Thus we have reached the expected conclusion: All elsebeing equal, the longer the term to maturity of a bond, the longer its duration.

DUR � �n

t�1

tCPt

(1 � i )t �n

t�1

CPt

(1 � i )t


(1) (2) (3) (4) (5)Present

Cash Payments Value (PV) Weights Weighted(Zero-Coupon of Cash Payments (% of total Maturity

Bonds) (i � 10%) PV � PV/$1,000) (1 � 4)/100Year ($) ($) (%) (years)

1 100 90.91 9.091 0.090912 100 82.64 8.264 0.165283 100 75.13 7.513 0.225394 100 68.30 6.830 0.273205 100 62.09 6.209 0.310456 100 56.44 5.644 0.338647 100 51.32 5.132 0.359248 100 46.65 4.665 0.373209 100 42.41 4.241 0.38169

10 100 38.55 3.855 0.3855010 1,000 385.54 38.554 3.85500Total 1,000.00 100.000 6.75850

Table 1 Calculating Duration on a $1,000 Ten-Year 10% Coupon Bond When Its Interest Rate Is 10%

You might think that knowing the maturity of a coupon bond is enough to tellyou what its duration is. However, that is not the case. To see this and to give youmore practice in calculating duration, in Table 2 we again calculate the duration forthe ten-year 10% coupon bond, but when the current interest rate is 20%, rather than10% as in Table 1. The calculation in Table 2 reveals that the duration of the couponbond at this higher interest rate has fallen from 6.76 years to 5.72 years. The expla-nation is fairly straightforward. When the interest rate is higher, the cash payments inthe future are discounted more heavily and become less important in present-valueterms relative to the total present value of all the payments. The relative weight forthese cash payments drops as we see in Table 2, and so the effective maturity of thebond falls. We have come to an important conclusion: All else being equal, wheninterest rates rise, the duration of a coupon bond falls.

The duration of a coupon bond is also affected by its coupon rate. For example,consider a ten-year 20% coupon bond when the interest rate is 10%. Using the sameprocedure, we find that its duration at the higher 20% coupon rate is 5.98 years ver-sus 6.76 years when the coupon rate is 10%. The explanation is that a higher couponrate means that a relatively greater amount of the cash payments are made earlier inthe life of the bond, and so the effective maturity of the bond must fall. We have thusestablished a third fact about duration: All else being equal, the higher the couponrate on the bond, the shorter the bond’s duration.


(1) (2) (3) (4) (5)Present

Cash Payments Value (PV) Weights Weighted(Zero-Coupon of Cash Payments (% of total Maturity

Bonds) (i � 20%) PV � PV/$580.76) (1 � 4)/100Year ($) ($) (%) (years)

1 100 83.33 14.348 0.143482 100 69.44 11.957 0.239143 100 57.87 9.965 0.298954 100 48.23 8.305 0.332205 100 40.19 6.920 0.346006 100 33.49 5.767 0.346027 100 27.91 4.806 0.336428 100 23.26 4.005 0.320409 100 19.38 3.337 0.30033

10 100 16.15 2.781 0.2781010 $1,000 161.51 27.808 2.78100Total 580.76 100.000 5.72204

Table 2 Calculating Duration on a $1,000 Ten-Year 10% Coupon Bond When Its Interest Rate Is 20%

Study Guide To make certain that you understand how to calculate duration, practice doing thecalculations in Tables 1 and 2. Try to produce the tables for calculating duration inthe case of an 11-year 10% coupon bond and also for the 10-year 20% coupon bondmentioned in the text when the current interest rate is 10%. Make sure your calcula-tions produce the same results found in this appendix.

One additional fact about duration makes this concept useful when applied to aportfolio of securities. Our examples have shown that duration is equal to theweighted average of the durations of the cash payments (the effective maturities of thecorresponding zero-coupon bonds). So if we calculate the duration for two differentsecurities, it should be easy to see that the duration of a portfolio of the two securi-ties is just the weighted average of the durations of the two securities, with theweights reflecting the proportion of the portfolio invested in each.

EXAMPLE 2: Duration

A manager of a financial institution is holding 25% of a portfolio in a bond with a five-year duration and 75% in a bond with a ten-year duration. What is the duration of theportfolio?

SolutionThe duration of the portfolio is 8.75 years.

(0.25 � 5) � (0.75 � 10) � 1.25 � 7.5 � 8.75 years

We now see that the duration of a portfolio of securities is the weighted averageof the durations of the individual securities, with the weights reflecting the propor-tion of the portfolio invested in each. This fact about duration is often referred to asthe additive property of duration, and it is extremely useful, because it means that theduration of a portfolio of securities is easy to calculate from the durations of the indi-vidual securities.

To summarize, our calculations of duration for coupon bonds have revealedfour facts:

1. The longer the term to maturity of a bond, everything else being equal, thegreater its duration.

2. When interest rates rise, everything else being equal, the duration of a couponbond falls.

3. The higher the coupon rate on the bond, everything else being equal, the shorterthe bond’s duration.

4. Duration is additive: The duration of a portfolio of securities is the weighted aver-age of the durations of the individual securities, with the weights reflecting theproportion of the portfolio invested in each.


Now that we understand how duration is calculated, we want to see how it can beused by the practicing financial institution manager to measure interest-rate risk.Duration is a particularly useful concept, because it provides a good approximation,particularly when interest-rate changes are small, for how much the security pricechanges for a given change in interest rates, as the following formula indicates:

(2)

where %�P � (Pt�1 � Pt)/Pt � percent change in the price of the securityfrom t to t � 1 � rate of capital gain

DUR � durationi � interest rate

EXAMPLE 3: Duration and Interest-Rate Risk

A pension fund manager is holding a ten-year 10% coupon bond in the fund’s portfolioand the interest rate is currently 10%. What loss would the fund be exposed to if theinterest rate rises to 11% tomorrow?

SolutionThe approximate percentage change in the price of the bond is �6.15%.

As the calculation in Table 1 shows, the duration of a ten-year 10% coupon bondis 6.76 years.

whereDUR � duration � 6.76

�i � change in interest rate � 0.11 � 0.10 � 0.01i � current interest rate � 0.10

Thus:

%�P � �6.76 �

%�P � �0.0615 � �6.15%

EXAMPLE 4: Duration and Interest-Rate Risk

Now the pension manager has the option to hold a ten-year coupon bond with a couponrate of 20% instead of 10%. As mentioned earlier, the duration for this 20% couponbond is 5.98 years when the interest rate is 10%. Find the approximate change in thebond price when the interest rate increases from 10% to 11%.

SolutionThis time the approximate change in bond price is �5.4%. This change in bond priceis much smaller than for the higher-duration coupon bond:

%�P � �DUR � �i

1 � i

0.01

1 � 0.10

%�P � �DUR � �i

1 � i

%�P � �DUR � �i

1 � i

Duration andInterest-Rate Risk


whereDUR � duration � 5.98

�i � change in interest rate � 0.11 � 0.10 � 0.01i � current interest rate � 0.10

Thus:

%�P � �5.98 �

%�P � �0.054 � �5.4%

The pension fund manager realizes that the interest-rate risk on the 20% couponbond is less than on the 10% coupon bond, so he switches the fund out of the 10%coupon bond and into the 20% coupon bond.

Examples 3 and 4 have led the pension fund manager to an important conclusionabout the relationship of duration and interest-rate risk: The greater the duration of asecurity, the greater the percentage change in the market value of the security for agiven change in interest rates. Therefore, the greater the duration of a security, thegreater its interest-rate risk.

Other Measures of Duration

Equation (2) can be rearranged, resulting in

%∆P � ��1D

�

UR

i� � ∆i

where the quantity DUR/(1 � i) is referred to as modified duration. The modified dura-tion gives the percentage change in the price of the security for a 1% change in its yieldand is a better measure of the interest-rate risk than Macaulay’s duration, DUR.

Frequently, however, managers of banks and other financial institutions express thesensitivity of fixed income securities to changes in interest rates not in terms of percent-age changes in the price of the security as above, but in terms of dollars gained or lost.In particular, rearranging Equation (14) yields

∆P � �DUR � �1

∆�

i

i� � P (3)

where the quantity

DUR � �1

∆�

i

i� � P

is referred to as dollar duration. Equation (3) gives the change in price in terms ofdollars, ∆P.

0.01

1 � 0.10


EXAMPLE 14: Modified Duration, Dollar Duration, and Interest-RateRisk

A pension fund manager is holding a 10% coupon, $1000 face-value bond with a dura-tion of 2. The interest rate is currently 10%. What is the modified duration? What isthe dollar duration for a 100-basis point change in the interest rate?

Solution

The modified duration is

Modified Duration � �1D

�

UR

i�

where

DUR=duration =2

i=current interest rate =0.10

Thus

Modified Duration � �1 �

20.10� � 1.82

meaning that the percentage change in the price of the security will be 1.82 times thechange in the interest rate.

The dollar duration is

Dollar Duration � DUR � �1

∆�

i

i� � P

where

DUR=duration =2

i=current interest rate =0.10

Di=change in interest rate =0.01

P=price of the security =$1000

Thus

Dollar Duration � 2 � �1 �

0.001.10

� � $1000 � $18.2

Duration analysis applies equally to a portfolio of securites. So by calculating theduration of the fund’s portfolio of securities using the methods outlined here, a pen-sion fund manager can easily ascertain the amount of interest-rate risk the entire fundis exposed to. As we will see in Chapter 9, duration is a highly useful concept for themanagement of interest-rate risk that is widely used by managers of banks and otherfinancial institutions.


16

In Chapter 4, we saw that the return on an asset (such as a bond) measures howmuch we gain from holding that asset. When we make a decision to buy an asset, weare influenced by what we expect the return on that asset to be and its risk. Here weshow how to calculate expected return and risk, which is measured by the standarddeviation.

Expected Return

If a Mobil Oil Corporation bond, for example, has a return of 15% half of the timeand 5% the other half of the time, its expected return (which you can think of as theaverage return) is 10%. More formally, the expected return on an asset is the weightedaverage of all possible returns, where the weights are the probabilities of occurrenceof that return:

Re � p1R1 � p2R2 � . . . � pnRn (1)

where Re � expected returnn � number of possible outcomes (states of nature)

Ri � return in the ith state of naturepi � probability of occurrence of the return Ri

EXAMPLE 1: Expected Return

What is the expected return on the Mobil Oil bond if the return is 12% two-thirds of thetime and 8% one-third of the time?

SolutionThe expected return is 10.68%:

Re � p1R1 � p2R2

where

p1 � probability of occurrence of return 1 � � 0.67

R1 � return in state 1 �12% � 0.12

23

Mode l s o f A s se t P r i c i ng

appendix1to chapter

5

p2 � probability of occurrence of return 2� � .33

R2 � return in state 2 � 8% � 0.08

Thus:

Re � (0.67)(0.12) � (0.33)(0.08) � 0.1068 � 10.68%

The degree of risk or uncertainty of an asset’s returns also affects the demand for theasset. Consider two assets, stock in Fly-by-Night Airlines and stock in Feet-on-the-Ground Bus Company. Suppose that Fly-by-Night stock has a return of 15% half ofthe time and 5% the other half of the time, making its expected return 10%, whilestock in Feet-on-the-Ground has a fixed return of 10%. Fly-by-Night stock has uncer-tainty associated with its returns and so has greater risk than stock in Feet-on-the-Ground, whose return is a sure thing.

To see this more formally, we can use a measure of risk called the standard devi-ation. The standard deviation of returns on an asset is calculated as follows. First cal-culate the expected return, Re; then subtract the expected return from each return toget a deviation; then square each deviation and multiply it by the probability of occur-rence of that outcome; finally, add up all these weighted squared deviations and takethe square root. The formula for the standard deviation, �, is thus:

� � (2)

The higher the standard deviation, �, the greater the risk of an asset.

EXAMPLE 2: Standard Deviation

What is the standard deviation of the returns on the Fly-by-Night Airlines stock and Feet-on-the-Ground Bus Company, with the same return outcomes and probabilitiesdescribed above? Of these two stocks, which is riskier?

SolutionFly-by-Night Airlines has a standard deviation of returns of 5%.

� �

Re � p1R1 � p2R2

where





Re � expected return � (0.50)(0.15) � (0.50)(0.05) � 0.10

12

12

�p1(R1 � R e )2 � p2(R2 � R e )2

�p1(R1 � R e )2 � p2(R2 � R e )2 � . . . � pn(Rn � R e )2

CalculatingStandard Deviationof Returns

13

Models of Asset Pricing 17

Thus:

� �

� � � 0.05 � 5%

Feet-on-the-Ground Bus Company has a standard deviation of returns of 0%.

� �

Re � p1R1

where

p1 � probability of occurrence of return 1 � 1.0


Re � expected return � (1.0)(0.10) � 0.10

Thus:

�

Clearly, Fly-by-Night Airlines is a riskier stock, because its standard deviation ofreturns of 5% is higher than the zero standard deviation of returns for Feet-on-the-Ground Bus Company, which has a certain return.

Benefits of Diversification

Our discussion of the theory of asset demand indicates that most people like to avoidrisk; that is, they are risk-averse. Why, then, do many investors hold many risky assetsrather than just one? Doesn’t holding many risky assets expose the investor to morerisk?

The old warning about not putting all your eggs in one basket holds the key tothe answer: Because holding many risky assets (called diversification) reduces the over-all risk an investor faces, diversification is beneficial. To see why this is so, let’s lookat some specific examples of how an investor fares on his investments when he isholding two risky securities.

Consider two assets: common stock of Frivolous Luxuries, Inc., and commonstock of Bad Times Products, Unlimited. When the economy is strong, which we’llassume is one-half of the time, Frivolous Luxuries has high sales and the return onthe stock is 15%; when the economy is weak, the other half of the time, sales are lowand the return on the stock is 5%. On the other hand, suppose that Bad TimesProducts thrives when the economy is weak, so that its stock has a return of 15%, butit earns less when the economy is strong and has a return on the stock of 5%. Sinceboth these stocks have an expected return of 15% half the time and 5% the other halfof the time, both have an expected return of 10%. However, both stocks carry a fairamount of risk, because there is uncertainty about their actual returns.

Suppose, however, that instead of buying one stock or the other, Irving theInvestor puts half his savings in Frivolous Luxuries stock and the other half in Bad

�0 � 0 � 0%

� � �(1.0 )(0.10 � 0.10 )2

�p1(R1 � R e )2

�(0.50 )(0.0025 ) � (0.50 )(0.0025 ) � �0.0025

�(0.50 )(0.15 � 0.10 )2 � (0.50 )(0.05 � 0.10 )2

18 Appendix 1 to Chapter 5

Times Products stock. When the economy is strong, Frivolous Luxuries stock has areturn of 15%, while Bad Times Products has a return of 5%. The result is that Irvingearns a return of 10% (the average of 5% and 15%) on his holdings of the two stocks.When the economy is weak, Frivolous Luxuries has a return of only 5% and Bad TimesProducts has a return of 15%, so Irving still earns a return of 10% regardless ofwhether the economy is strong or weak. Irving is better off from this strategy of diver-sification because his expected return is 10%, the same as from holding eitherFrivolous Luxuries or Bad Times Products alone, and yet he is not exposed to any risk.

Although the case we have described demonstrates the benefits of diversification,it is somewhat unrealistic. It is quite hard to find two securities with the characteristicthat when the return of one is high, the return of the other is always low.1 In the realworld, we are more likely to find at best returns on securities that are independent ofeach other; that is, when one is high, the other is just as likely to be high as to be low.

Suppose that both securities have an expected return of 10%, with a return of 5%half the time and 15% the other half of the time. Sometimes both securities will earnthe higher return and sometimes both will earn the lower return. In this case if Irvingholds equal amounts of each security, he will on average earn the same return as if hehad just put all his savings into one of these securities. However, because the returnson these two securities are independent, it is just as likely that when one earns thehigh 15% return, the other earns the low 5% return and vice versa, giving Irving areturn of 10% (equal to the expected return). Because Irving is more likely to earnwhat he expected to earn when he holds both securities instead of just one, we cansee that Irving has again reduced his risk through diversification.2

The one case in which Irving will not benefit from diversification occurs when thereturns on the two securities move perfectly together. In this case, when the first secu-rity has a return of 15%, the other also has a return of 15% and holding both securi-ties results in a return of 15%. When the first security has a return of 5%, the otherhas a return of 5% and holding both results in a return of 5%. The result of diversi-fying by holding both securities is a return of 15% half of the time and 5% the otherhalf of the time, which is exactly the same set of returns that are earned by holdingonly one of the securities. Consequently, diversification in this case does not lead toany reduction of risk.

The examples we have just examined illustrate the following important pointsabout diversification:

1. Diversification is almost always beneficial to the risk-averse investor since itreduces risk unless returns on securities move perfectly together (which is anextremely rare occurrence).

2. The less the returns on two securities move together, the more benefit (risk reduc-tion) there is from diversification.


1Such a case is described by saying that the returns on the two securities are perfectly negatively correlated.2 We can also see that diversification in the example above leads to lower risk by examining the standard devi-ation of returns when Irving diversifies and when he doesn’t. The standard deviation of returns if Irving holdsonly one of the two securities is . When Irving holdsequal amounts of each security, there is a probability of 1/4 that he will earn 5% on both (for a total return of5%), a probability of 1/4 that he will earn 15% on both (for a total return of 15%), and a probability of 1/2 thathe will earn 15% on one and 5% on the other (for a total return of 10%). The standard deviation of returns whenIrving diversifies is thus .Since the standard deviation of returns when Irving diversifies is lower than when he holds only one security,we can see that diversification has reduced risk.

�0.25 � (15% � 10% )2 � 0.25 � (5% � 10% )2 � 0.5 � (10% � 10% )2 � 3.5%

�0.5 � (15% � 10% )2 � 0.5 � (5% � 10% )2 � 5%

Diversification and Beta

In the previous section, we demonstrated the benefits of diversification. Here, weexamine diversification and the relationship between risk and returns in more detail.As a result, we obtain an understanding of two basic theories of asset pricing: the cap-ital asset pricing model (CAPM) and arbitrage pricing theory (APT).

We start our analysis by considering a portfolio of n assets whose return is:

Rp � x1R1 � x2R2 � … � xnRn (3)

where Rp � the return on the portfolio of n assetsRi � the return on asset ixi � the proportion of the portfolio held in asset i

The expected return on this portfolio, E(Rp), equals

E(Rp) � E(x1R1) � E(x2R2) � … � E(xnRn)

� x1E(R1) � x2E(R2) � … � xnE(Rn) (4)

An appropriate measure of the risk for this portfolio is the standard deviation of theportfolio’s return (�p) or its squared value, the variance of the portfolio’s return (� p

2),which can be written as:

� p2 � E[Rp � E(Rp)]

2 � E[{x1R1 � … � xnRn} � {x1E(R1) � … � xnE(Rn)}]2

� E[x1{R1 � E(R1)} � … � xn{Rn � E(Rn)}]2

This expression can be rewritten as:

� p2 � E[{x1[R1 � E(R1)] � … � xn[Rn � E(Rn)]} � {Rp � E(Rp)}]

� x1E[{R1 � E(R1)} � {Rp � E(Rp)}] � … � xnE[{Rn � E(Rn)} � {Rp � E(Rp)}]

This gives us the following expression for the variance for the portfolio’s return:

� p2 � x1�1p � x2�2p � xn�np (5)

where

�ip � the covariance of the return on asset i

with the portfolio’s return � E[{Ri � E(Ri)} � {Rp � E(Rp)}]

Equation 5 tells us that the contribution to risk of asset i to the portfolio is xi�ip.By dividing this contribution to risk by the total portfolio risk (� p

2), we have the pro-portionate contribution of asset i to the portfolio risk:

xi�ip/� p2

The ratio �ip /� p2 tells us about the sensitivity of asset i’s return to the portfolio’s return.

The higher the ratio is, the more the value of the asset moves with changes in the


value of the portfolio, and the more asset i contributes to portfolio risk. Our algebraicmanipulations have thus led to the following important conclusion: The marginalcontribution of an asset to the risk of a portfolio depends not on the risk of the assetin isolation, but rather on the sensitivity of that asset’s return to changes in thevalue of the portfolio.

If the total of all risky assets in the market is included in the portfolio, then it iscalled the market portfolio. If we suppose that the portfolio, p, is the market portfolio,m, then the ratio �im/�m

2 is called the asset i’s beta, that is:

�i � �im /�m2 (6)

where

�i � the beta of asset i

An asset’s beta then is a measure of the asset’s marginal contribution to the risk of themarket portfolio. A higher beta means that an asset’s return is more sensitive tochanges in the value of the market portfolio and that the asset contributes more to therisk of the portfolio.

Another way to understand beta is to recognize that the return on asset i can beconsidered as being made up of two components—one that moves with the market’sreturn (Rm) and the other a random factor with an expected value of zero that isunique to the asset (i) and so is uncorrelated with the market return:

Ri � i � �iRm � i (7)

The expected return of asset i can then be written as:

E(Ri) � i � �iE(Rm)

It is easy to show that �i in the above expression is the beta of asset i we defined beforeby calculating the covariance of asset i’s return with the market return using the twoequations above:

�im � E[{Ri � E(Ri)} � {Rm � E(Rm)}] � E[{�i[Rm � E(Rm)] � i}

� {Rm � E(Rm)}]

However, since i is uncorrelated with Rm, E[{i} � {Rm � E(Rm)}] � 0. Therefore,

�im � �i�m2

Dividing through by �m2 gives us the following expression for �i:

�i � �im /�m2

which is the same definition for beta we found in Equation 6.The reason for demonstrating that the �i in Equation 7 is the same as the one we

defined before is that Equation 7 provides better intuition about how an asset’s betameasures its sensitivity to changes in the market return. Equation 7 tells us that when


the beta of an asset is 1.0, it’s return on average increases by 1 percentage point whenthe market return increases by 1 percentage point; when the beta is 2.0, the asset’sreturn increases by 2 percentage points when the market return increases by 1 per-centage point; and when the beta is 0.5, the asset’s return only increases by 0.5 per-centage point on average when the market return increases by 1 percentage point.

Equation 7 also tells us that we can get estimates of beta by comparing the aver-age return on an asset with the average market return. For those of you who know alittle econometrics, this estimate of beta is just an ordinary least squares regression ofthe asset’s return on the market return. Indeed, the formula for the ordinary leastsquares estimate of �i � �im/�m

2 is exactly the same as the definition of �i earlier.

Systematic and Nonsystematic Risk

We can derive another important idea about the riskiness of an asset using Equation7. The variance of asset i’s return can be calculated from Equation 7 as:

� i2 � E[Ri � E(Ri)]

2 � E{�i[Rm � E(Rm)} � i]2

and since i is uncorrelated with market return:

� i2 � � i

2� m2 � �

2

The total variance of the asset’s return can thus be broken up into a component thatis related to market risk, � i

2� m2 , and a component that is unique to the asset, �

2. The� i

2� m2 component related to market risk is referred to as systematic risk and the �

2

component unique to the asset is called nonsystematic risk. We can thus write the totalrisk of an asset as being made up of systematic risk and nonsystematic risk:

Total Asset Risk � Systematic Risk � Nonsystematic Risk (8)

Systematic and nonsystematic risk each have another feature that makes the dis-tinction between these two types of risk important. Systematic risk is the part of anasset’s risk that cannot be eliminated by holding the asset as part of a diversified port-folio, whereas nonsystematic risk is the part of an asset’s risk that can be eliminatedin a diversified portfolio. Understanding these features of systematic and nonsystem-atic risk leads to the following important conclusion: The risk of a well-diversifiedportfolio depends only on the systematic risk of the assets in the portfolio.

We can see that this conclusion is true by considering a portfolio of n assets, eachof which has the same weight on the portfolio of (1/n). Using Equation 7, the returnon this portfolio is:

which can be rewritten as:

Rp � � �Rm � 1�n )�n

i�1

i

Rp � (1�n )�n

i�1

i � (1�n )�n

i�1

�iRm � (1�n )�n

i�1

i


where

� the average of the i’s �

� the average of the �i’s �

If the portfolio is well diversified so that the i’s are uncorrelated with each other, thenusing this fact and the fact that all the i’s are uncorrelated with the market return, thevariance of the portfolio’s return is calculated as:

(average varience of i)

As n gets large the second term, (1/n)(average variance of i), becomes very small, sothat a well-diversified portfolio has a risk of , which is only related to system-atic risk. As the previous conclusion indicated, nonsystematic risk can be eliminatedin a well-diversified portfolio. This reasoning also tells us that the risk of a well-diversifiedportfolio is greater than the risk of the market portfolio if the average beta of the assetsin the portfolio is greater than one; however, the portfolio’s risk is less than the mar-ket portfolio if the average beta of the assets is less than one.

The Capital Asset Pricing Model (CAPM)

We can now use the ideas we developed about systematic and nonsystematic risk andbetas to derive one of the most widely used models of asset pricing—the capital assetpricing model (CAPM) developed by William Sharpe, John Litner, and Jack Treynor.

Each cross in Figure 1 shows the standard deviation and expected return for eachrisky asset. By putting different proportions of these assets into portfolios, we can gen-erate a standard deviation and expected return for each of the portfolios usingEquations 4 and 5. The shaded area in the figure shows these combinations of stan-dard deviation and expected return for these portfolios. Since risk-averse investorsalways prefer to have higher expected return and lower standard deviation of thereturn, the most attractive standard deviation-expected return combinations are theones that lie along the heavy line, which is called the efficient portfolio frontier. Theseare the standard deviation-expected return combinations risk-averse investors wouldalways prefer.

The capital asset pricing model assumes that investors can borrow and lend asmuch as they want at a risk-free rate of interest, Rf. By lending at the risk-free rate, theinvestor earns an expected return of Rf and his investment has a zero standard devia-tion because it is risk-free. The standard deviation-expected return combination forthis risk-free investment is marked as point A in Figure 1. Suppose an investor decidesto put half of his total wealth in the risk-free loan and the other half in the portfolio onthe efficient portfolio frontier with a standard deviation-expected return combinationmarked as point M in the figure. Using Equation 4, you should be able to verify thatthe expected return on this new portfolio is halfway between Rf and E(Rm); that is,[Rf � E(Rm)]/2. Similarly, because the covariance between the risk-free return and thereturn on portfolio M must necessarily be zero, since there is no uncertainty about the

�2�2m

�2p � �2�2

m � (1�n )

(1�n )�n

i�1

i�

(1�n )�n

i�1

i


return on the risk-free loan, you should also be able to verify, using Equation 5, thatthe standard deviation of the return on the new portfolio is halfway between zero and�m, that is, (1/2)�m. The standard deviation-expected return combination for this newportfolio is marked as point B in the figure, and as you can see it lies on the linebetween point A and point M. Similarly, if an investor borrows the total amount of herwealth at the risk-free rate Rf and invests the proceeds plus her wealth (that is, twiceher wealth) in portfolio M, then the standard deviation of this new portfolio will betwice the standard deviation of return on portfolio M, 2�m. On the other hand, usingEquation 4, the expected return on this new portfolio is E(Rm) plus E(Rm) � Rf, whichequals 2E(Rm) � Rf. This standard deviation-expected return combination is plottedas point C in the figure.

You should now be able to see that both point B and point C are on the line con-necting point A and point M. Indeed, by choosing different amounts of borrowingand lending, an investor can form a portfolio with a standard deviation-expectedreturn combination that lies anywhere on the line connecting points A and M. Youmay have noticed that point M has been chosen so that the line connecting points Aand M is tangent to the efficient portfolio frontier. The reason for choosing point Min this way is that it leads to standard deviation-expected return combinations alongthe line that are the most desirable for a risk-averse investor. This line can be thoughtof as the opportunity locus, which shows the best combinations of standard deviationsand expected returns available to the investor.

The capital asset pricing model makes another assumption: All investors have thesame assessment of the expected returns and standard deviations of all assets. In thiscase, portfolio M is the same for all investors. Thus when all investors’ holdings ofportfolio M are added together, they must equal all of the risky assets in the market,


F IGURE 1 Risk Expected ReturnTrade-offThe crosses show the combinationof standard deviation and expectedreturn for each risky asset. Theefficient portfolio frontier indicatesthe most preferable standard deviation-expected return combi-nations that can be achieved byputting risky assets into portfolios.By borrowing and lending at therisk-free rate and investing in port-folio M, the investor can obtainstandard deviation-expected returncombinations that lie along theline connecting A, B, M, and C.This line, the opportunity locus,contains the best combinations ofstandard deviations and expectedreturns available to the investor;hence the opportunity locus showsthe trade-off between expectedreturns and risk for the investor.

ExpectedReturn

E(R)

2E(Rm) — Rf

E(Rm)

Rf + E(Rm)2

Rf

EfficientPortfolioFrontier

OpportunityLocus

1/2�m �m 2�m

A

B

M

C

Standard Deviation of Retuns �

+++ +

+ +

+++

++ +

++ +

which is just the market portfolio. The assumption that all investors have the sameassessment of risk and return for all assets thus means that portfolio M is the marketportfolio.Therefore, the Rm and �m in Figure 1 are identical to the market return, Rm,and the standard deviation of this return, �m, referred to earlier in this appendix.

The conclusion that the market portfolio and portfolio M are one and the samemeans that the opportunity locus in Figure 1 can be thought of as showing the trade-off between expected returns and increased risk for the investor. This trade-off isgiven by the slope of the opportunity locus, E(Rm) � Rf, and it tells us that when aninvestor is willing to increase the risk of his portfolio by �m, then he can earn an addi-tional expected return of E(Rm) � Rf. The market price of a unit of market risk, �m,is E(Rm) � Rf. E(Rm) � Rf is therefore referred to as the market price of risk.

We now know that market price of risk is E(Rm) � Rf and we also have learnedthat an asset’s beta tells us about systematic risk, because it is the marginal contribu-tion of that asset to a portfolio’s risk. Therefore the amount an asset’s expected returnexceeds the risk-free rate, E(Ri) � Rf, should equal the market price of risk times themarginal contribution of that asset to portfolio risk, [E(Rm) � Rf]�i. This reasoningyields the CAPM asset pricing relationship:

E(Ri) � Rf � �i[E(Rm) � Rf] (9)

This CAPM asset pricing equation is represented by the upward sloping line in Figure 2,which is called the security market line. It tells us the expected return that the marketsets for a security given its beta. For example, it tells us that if a security has a beta of1.0 so that its marginal contribution to a portfolio’s risk is the same as the marketportfolio, then it should be priced to have the same expected return as the marketportfolio, E(Rm).


F IGURE 2 Security Market LineThe security market line derivedfrom the capital asset pricingmodel describes the relationshipbetween an asset’s beta and itsexpected return.

S

T

ExpectedReturn

E(R)

SecurityMarket

LineE(Rm)

Rf

0.5 1.0 Beta �

To see that securities should be priced so that their expected return-beta combi-nation should lie on the security market line, consider a security like S in Figure 2,which is below the security market line. If an investor makes an investment in whichhalf is put into the market portfolio and half into a risk-free loan, then the beta of thisinvestment will be 0.5, the same as security S. However, this investment will have anexpected return on the security market line, which is greater than that for security S.Hence investors will not want to hold security S and its current price will fall, thusraising its expected return until it equals the amount indicated on the security mar-ket line. On the other hand, suppose there is a security like T which has a beta of 0.5but whose expected return is above the security market line. By including this secu-rity in a well-diversified portfolio with other assets with a beta of 0.5, none of whichcan have an expected return less than that indicated by the security line (as we haveshown), investors can obtain a portfolio with a higher expected return than thatobtained by putting half into a risk-free loan and half into the market portfolio. Thiswould mean that all investors would want to hold more of security T, and so its pricewould rise, thus lowering its expected return until it equaled the amount indicated onthe security market line.

The capital asset pricing model formalizes the following important idea: An assetshould be priced so that is has a higher expected return not when it has a greaterrisk in isolation, but rather when its systematic risk is greater.

Arbitrage Pricing Theory

Although the capital asset pricing model has proved to be very useful in practice,deriving it does require the adoption of some unrealistic assumptions; for example,the assumption that investors can borrow and lend freely at the risk-free rate, or theassumption that all investors have the same assessment of expected returns and stan-dard deviations of returns for all assets. An important alternative to the capital assetpricing model is the arbitrage pricing theory (APT) developed by Stephen Ross ofM.I.T.

In contrast to CAPM, which has only one source of systematic risk, the marketreturn, APT takes the view that there can be several sources of systematic risk in theeconomy that cannot be eliminated through diversification. These sources of risk canbe thought of as factors that may be related to such items as inflation, aggregate out-put, default risk premiums, and/or the term structure of interest rates. The return onan asset i can thus be written as being made up of components that move with thesefactors and a random component that is unique to the asset (i):

Ri � � i1 (factor 1) � � i

2 (factor 2) � … � � ik (factor k) � i (10)

Since there are k factors, this model is called a k-factor model. The � i1 ,…, � i

k describethe sensitivity of the asset i’s return to each of these factors.

Just as in the capital asset pricing model, these systematic sources of risk shouldbe priced. The market price for each factor j can be thought of as E(Rfactor j ) � Rf, andhence the expected return on a security can be written as:

E(Ri) � Rf � � i1 [E(Rfactor 1) � Rf] � … �� i

k [E(Rfactor k) � Rf] (11)


This asset pricing equation indicates that all the securities should have the same mar-ket price for the risk contributed by each factor. If the expected return for a securitywere above the amount indicated by the APT pricing equation, then it would providea higher expected return than a portfolio of other securities with the same averagesensitivity to each factor. Hence investors would want to hold more of this securityand its price would rise until the expected return fell to the value indicated by theAPT pricing equation. On the other hand, if the security’s expected return were lessthan the amount indicated by the APT pricing equation, then no one would want tohold this security, because a higher expected return could be obtained with a portfo-lio of securities with the same average sensitivity to each factor. As a result, the priceof the security would fall until its expected return rose to the value indicated by theAPT equation.

As this brief outline of arbitrage pricing theory indicates, the theory supports abasic conclusion from the capital asset pricing model: An asset should be priced sothat it has a higher expected return not when it has a greater risk in isolation, butrather when its systematic risk is greater. There is still substantial controversy aboutwhether a variant of the capital asset pricing model or the arbitrage pricing theory isa better description of reality. At the present time, both frameworks are consideredvaluable tools for understanding how risk affects the prices of assets.


28

Both models of interest-rate determination in Chapter 4 make use of an asset marketapproach in which supply and demand are always considered in terms of stocks of assets(amounts at a given point in time). The asset market approach is useful in understand-ing not only why interest rates fluctuate but also how any asset’s price is determined.

One asset that has fascinated people for thousands of years is gold. It has been adriving force in history: The conquest of the Americas by Europeans was to a greatextent the result of the quest for gold, to cite just one example. The fascination withgold continues to the present day, and developments in the gold market are followedclosely by financial analysts and the media. This appendix shows how the asset mar-ket approach can be applied to understanding the behavior of commodity markets, inparticular the gold market. (The analysis in this appendix can also be used to under-stand behavior in many other asset markets.)

Supply and Demand in the Gold Market

The analysis of a commodity market, such as the gold market, proceeds in a similarfashion to the analysis of the bond market by examining the supply of and demandfor the commodity. We again use our analysis of the determinants of asset demand toobtain a demand curve for gold, which shows the relationship between the quantityof gold demanded and the price when all other economic variables are held constant.

To derive the relationship between the quantity of gold demanded and its price, weagain recognize that an important determinant of the quantity demanded is itsexpected return:

where Re � expected returnPt � price of gold today

Pet � 1 � expected price of gold next year

ge � expected capital gain

In deriving the demand curve, we hold all other variables constant, particularlythe expected price of gold next year P e

t�1. With a given value of the expected price ofgold next year P e

t�1, a lower price of gold today Pt means that there will be a greater

R e �Pe

t�1 � Pt

Pt

� ge

Demand Curve

App l y ing t he A s se t Marke tApproach to a Commod i t y Marke t : The Ca se o f Go ld

appendix 2to chapter

5

appreciation in the price of gold over the coming year. The result is that a lower priceof gold today implies a higher expected capital gain over the coming year and hencea higher expected return: Re � (P e

t�1� Pt)/Pt. Thus because the price of gold today(which for simplicity we will denote as P) is lower, the expected return on gold ishigher, and the quantity demanded is higher. Consequently, the demand curve Gd

1slopes downward in Figure 1.

To derive the supply curve, expressing the relationship between the quantity suppliedand the price, we again assume that all other economic variables are held constant. Ahigher price of gold will induce producers to mine for extra gold and also possiblyinduce governments to sell some of their gold stocks to the public, thus increasing thequantity supplied. Hence the supply curve Gs

1 in Figure 1 slopes upward. Notice thatthe supply curve in the figure is drawn to be very steep. The reason for this is that theactual amount of gold produced in any year is only a tiny fraction of the outstandingstock of gold that has been accumulated over hundreds of years. Thus the increase inthe quantity of the gold supplied in response to a higher price is only a small fractionof the stock of gold, resulting in a very steep supply curve.

Market equilibrium in the gold market occurs when the quantity of gold demandedequals the quantity of gold supplied:

Gd � Gs

With the initial demand and supply curves of G d1 and G s

1, equilibrium occurs atpoint 1, where these curves intersect at a gold price of P1. At a price above this

MarketEquilibrium

Supply Curve

Applying the Asset Market Approach to a Commodity Market: The Case of Gold 29

F IGURE 1 A Change in theEquilibrium Price of GoldWhen the demand curve shifts right-ward from G1

d to G2d—say, because

expected inflation rises—equilibriummoves from point 1 to point 2, andthe equilibrium price of gold risesfrom P1 to P2.

1

P2

P1

Price of Gold P

Gd1

Gd2

Gs1

Quantity of Gold G

2

equilibrium, the amount of gold supplied exceeds the amount demanded, and thiscondition of excess supply leads to a decline in the gold price until it reaches P1, theequilibrium price. Similarly, if the price is below P1, there is excess demand for gold,which drives the price upward until it settles at the equilibrium price P1.

Changes in the Equilibrium Price of Gold

Changes in the equilibrium price of gold occur when there is a shift in either the sup-ply curve or the demand curve; that is, when the quantity demanded or suppliedchanges at each given price of gold in response to a change in some factor other thantoday’s gold price.

Our analysis of the determinants of asset demand in the chapter provides the factorsthat shift the demand curve for gold: wealth, expected return on gold relative to alter-native assets, riskiness of gold relative to alternative assets, and liquidity of gold rela-tive to alternative assets. The analysis of how changes in each of these factors shift thedemand curve for gold is the same as that found in the chapter.

When wealth rises, at a given price of gold, the quantity demanded increases, andthe demand curve shifts to the right, as in Figure 1. When the expected return on goldrelative to other assets rises—either because speculators think that the future price ofgold will be higher or because the expected return on other assets declines—goldbecomes more desirable; the quantity demanded therefore increases at any given priceof gold, and the demand curve shifts to the right, as in Figure 1. When the relativeriskiness of gold declines, either because gold prices become less volatile or becausereturns on other assets become more volatile, gold becomes more desirable, the quan-tity demanded at every given price rises, and the demand curve again shifts to theright. When the gold market becomes relatively more liquid and gold thereforebecomes more desirable, the quantity demanded at any given price rises, and thedemand curve also shifts to the right, as in Figure 1.

The supply curve for gold shifts when there are changes in technology that make goldmining more efficient or when governments at any given price of gold decide toincrease sales of their holdings of gold. In these cases, the quantity of gold suppliedat any given price increases, and the supply curve shifts to the right.

Study Guide To give yourself practice with supply and demand analysis in the gold market, see ifyou can analyze what happens to the price of gold for the following situations,remembering that all other things are held constant: 1) Interest rates rise, 2) the goldmarket becomes more liquid, 3) the volatility of gold prices increases, 4) the stockmarket is expected to turn bullish in the near future, 5) investors suddenly becomefearful that there will be a collapse in real estate prices, and 6) Russia sells a lot of goldin the open market to raise hard currency to finance expenditures.

Shifts in theSupply Curve forGold

Shift in theDemand Curve for Gold


Applying the Asset Market Approach to a Commodity Market: The Case of Gold 31

Changes in the Equilibrium Price of Gold Due to a Rise inExpected Inflation

Application

To illustrate how changes in the equilibrium price of gold occur when sup-ply and demand curves shift, let’s look at what happens when there is achange in expected inflation.

Suppose that expected inflation is 5% and the initial supply and demandcurves are at G s

1 and Gd1 so that the equilibrium price of gold is at P1 in Figure

1. If expected inflation now rises to 10%, prices of goods and commoditiesnext year will be expected to be higher than they otherwise would have been,and the price of gold next year P e

t�1 will also be expected to be higher thanotherwise. Now at any given price of gold today, gold is expected to have agreater rate of appreciation over the coming year and hence a higher expectedcapital gain and return. The greater expected return means that the quantityof gold demanded increases at any given price, thus shifting the demandcurve from Gd

1 to Gd2. Equilibrium therefore moves from point 1 to point 2,

and the price of gold rises from P1 to P2.By using a supply and demand diagram like that in Figure 1, you should

be able to see that if the expected rate of inflation falls, the price of gold todaywill also fall. We thus reach the following conclusion: The price of goldshould be positively related to the expected inflation rate.

Because the gold market responds immediately to any changes inexpected inflation, it is considered a good barometer of the trend of inflationin the future. Indeed, Alan Greenspan, the chairman of the Board ofGovernors of the Federal Reserve System, at one point advocated using theprice of gold as an indicator of inflationary pressures in the economy. Notsurprisingly, then, the gold market is followed closely by financial analystsand monetary policymakers.

32

An alternative method for measuring interest-rate risk, called duration gap analysis,examines the sensitivity of the market value of the financial institution’s net worth tochanges in interest rates. Duration analysis is based on Macaulay’s concept of duration,which measures the average lifetime of a security’s stream of payments (described inthe appendix to Chapter 4). Recall that duration is a useful concept, because it pro-vides a good approximation, particularly when interest-rate changes are small, of thesensitivity of a security’s market value to a change in its interest rate using the fol-lowing formula:

(1)

where%�P � (Pt � 1 � Pt)/Pt � percent change in market value of the securityDUR � duration

i � interest rate

After having determined the duration of all assets and liabilities on the bank’s bal-ance sheet, the bank manager could use this formula to calculate how the marketvalue of each asset and liability changes when there is a change in interest rates andthen calculate the effect on net worth. There is, however, an easier way to go aboutdoing this, derived from the basic fact about duration we learned in the appendix toChapter 4: Duration is additive; that is, the duration of a portfolio of securities is theweighted average of the durations of the individual securities, with the weights reflect-ing the proportion of the portfolio invested in each. What this means is that the bankmanager can figure out the effect that interest-rate changes will have on the marketvalue of net worth by calculating the average duration for assets and for liabilities andthen using those figures to estimate the effects of interest-rate changes.

To see how a bank manager would do this, let’s return to the balance sheet of theFirst Bank. The bank manager has already used the procedures outlined in the appen-dix to Chapter 4 to calculate the duration of each asset and liability, as listed in Table1. For each asset, the manager then calculates the weighted duration by multiplyingthe duration times the amount of the asset divided by total assets, which in this caseis $100 million. For example, in the case of securities with maturities less than oneyear, the manager multiplies the 0.4 year of duration times $5 million divided by$100 million to get a weighted duration of 0.02. (Note that physical assets have nocash payments, so they have a duration of zero years.) Doing this for all the assets and

%�P � �DUR � �i

1 � i

Dura t i on Gap Ana l y s i s

appendix 1t o c h a p te r

9

adding them up, the bank manager gets a figure for the average duration of the assetsof 2.70 years.

The manager follows a similar procedure for the liabilities, noting that total lia-bilities excluding capital are $95 million. For example, the weighted duration forcheckable deposits is determined by multiplying the 2.0-year duration by $15 milliondivided by $95 million to get 0.32. Adding up these weighted durations, the managerobtains an average duration of liabilities of 1.03 years.

Duration Gap Analysis 33

WeightedAmount Duration Duration

($ millions) (years) (years)

AssetsReserves and cash items 5 0.0 0.00Securities

Less than 1 year 5 0.4 0.021 to 2 years 5 1.6 0.08Greater than 2 years 10 7.0 0.70

Residential mortgagesVariable-rate 10 0.5 0.05Fixed-rate (30-year) 10 6.0 0.60

Commercial loansLess than 1 year 15 0.7 0.111 to 2 years 10 1.4 0.14Greater than 2 years 25 4.0 1.00

Physical capital 5 0.0 0.00Average duration 2.70

LiabilitiesCheckable deposits 15 2.0 0.32Money market deposit accounts 5 0.1 0.01Savings deposits 15 1.0 0.16CDs

Variable-rate 10 0.5 0.05Less than 1 year 15 0.2 0.031 to 2 years 5 1.2 0.06Greater than 2 years 5 2.7 0.14

Overnight funds 5 0.0 0.00Borrowings


Average duration 1.03

Table 1 Duration of the First Bank’s Assets and Liabilities

EXAMPLE 1: Duration Gap Analysis

The bank manager wants to know what happens when interest rates rise from 10% to11%. The total asset value is $100 million, and the total liability value is $95 million.Use Equation 1 to calculate the change in the market value of the assets and liabilities.

SolutionWith a total asset value of $100 million, the market value of assets falls by $2.5 mil-lion ($100 million � 0.025 � $2.5 million):

%� P � �DUR �

where

DUR � duration � 2.70�i � change in interest rate � 0.11 � 0.10 � 0.01

i � interest rate � 0.10

Thus:

%� P � �2.70 � � �0.025 � �2.5%

With total liabilities of $95 million, the market value of liabilities falls by $0.9 million($95 million � 0.009 � �$0.9 million):

%� P � �DUR �

where

DUR � duration � 1.03�i � change in interest rate � 0.11 � 0.10 � 0.01


Thus:

%� P � �1.03 � � �0.009 � �0.9%

The result is that the net worth of the bank would decline by $1.6 million (�$2.5million � (�$0.9 million) � �$2.5 million � $0.9 million � �$1.6 million).

The bank manager could have gotten to the answer even more quickly by calcu-lating what is called a duration gap, which is defined as follows:

DURgap � DURa � (2)

where DURa � average duration of assetsDURl � average duration of liabilities

L � market value of liabilitiesA � market value of assets

�L

A� DURl�

0.01

1 � 0.10

�i

1 � i

0.01

1 � 0.10

�i

1 � i



Based on the information provided in Example 1, use Equation 2 to determine the dura-tion gap for First Bank.

SolutionThe duration gap for First Bank is 1.72 years:

DURgap � DURa �

where

DURa � average duration of assets � 2.70L � market value of liabilities � 95A � market value of assets � 100

DURl � average duration of liabilities � 1.03

Thus:

DURgap � 2.70 � � 1.72 years

To estimate what will happen if interest rates change, the bank manager uses theDURgap calculation in Equation 1 to obtain the change in the market value of networth as a percentage of total assets. In other words, the change in the market valueof net worth as a percentage of assets is calculated as:

� �DURgap � (3)


What is the change in the market value of net worth as a percentage of assets if inter-est rates rise from 10% to 11%? (Use Equation 3.)

SolutionA rise in interest rates from 10% to 11% would lead to a change in the market valueof net worth as a percentage of assets of �1.6%:

� �DURgap �

where

DURgap � duration gap � 1.72�i � change in interest rate � 0.11 � 0.10 � 0.01


Thus:

� �1.72 � � �0.016 � �1.6%0.01

1 � 0.10

�NW

A

�i

1 � i

�NW

A

�i

1 � i

�NW

A

� 95

100� 1.03�

�L

A� DURl�


With assets totaling $100 million, Example 3 indicates a fall in the market valueof net worth of $1.6 million, which is the same figure that we found in Example 1.

As our examples make clear, both income gap analysis and duration gap analysisindicate that the First Bank will suffer from a rise in interest rates. Indeed, in thisexample, we have seen that a rise in interest rates from 10% to 11% will cause themarket value of net worth to fall by $1.6 million, which is one-third the initial amountof bank capital. Thus the bank manager realizes that the bank faces substantial inter-est-rate risk because a rise in interest rates could cause it to lose a lot of its capital.Clearly, income gap analysis and duration gap analysis are useful tools for telling afinancial institution manager the institution’s degree of exposure to interest-rate risk.

Study Guide To make sure that you understand income gap and duration gap analysis, you shouldbe able to verify that if interest rates fall from 10% to 5%, the First Bank will find itsincome increasing and the market value of its net worth rising.

So far we have focused on an example involving a banking institution that has bor-rowed short and lent long so that when interest rates rise, both income and the networth of the institution fall. It is important to recognize that income and duration gapanalysis applies equally to other financial institutions. Furthermore, it is important foryou to see that some financial institutions have income and duration gaps that areopposite in sign to those of banks, so that when interest rates rise, both income andnet worth rise rather than fall. To get a more complete picture of income and dura-tion gap analysis, let us look at a nonbank financial institution, the Friendly FinanceCompany, which specializes in making consumer loans.

The Friendly Finance Company has the following balance sheet:

The manager of the Friendly Finance Company calculates the rate-sensitive assetsto be equal to the $5 million of securities with maturities less than one year plus the$50 million of consumer loans with maturities of less than one year, for a total of $55

Example of aNonbankingFinancialInstitution


Friendly Finance Company

Assets Liabilities

Cash and deposits $3 million Commercial paper $40 millionSecurities Bank loans

Less than 1 year $5 million Less than 1 year $3 million1 to 2 years $1 million 1 to 2 years $2 millionGreater than 2 years $1 million Greater than 2 years $5 million

Consumer loans Long-term bonds andLess than 1 year $50 million other long-term debt $40 million1 to 2 years $20 million Capital $10 millionGreater than 2 years $15 million

Physical capital $5 millionTotal $100 million Total $100 million

million of rate-sensitive assets. The manager then calculates the rate-sensitive liabili-ties to be equal to the $40 million of commercial paper, all of which has a maturity ofless than one year, plus the $3 million of bank loans maturing in less than a year, fora total of $43 million. The calculation of the income gap is then:

GAP � RSA � RSL � $55 million � $43 million � $12 million

To calculate the effect on income if interest rates rise by 1%, the manager multipliesthe GAP of $12 million times the change in the interest rate to get the following:

� I � GAP � �i � $12 million � 1% � $120,000

Thus the manager finds that the finance company’s income will rise by $120,000when interest rates rise by 1%. The reason that the company has benefited from theinterest-rate rise, in contrast to the First Bank, whose profits suffer from the rise ininterest rates, is that the Friendly Finance Company has a positive income gapbecause it has more rate-sensitive assets than liabilities.

Like the bank manager, the manager of the Friendly Finance Company is also inter-ested in what happens to the market value of the net worth of the company when inter-est rates rise by 1%. So the manager calculates the weighted duration of each item inthe balance sheet, adds them up as in Table 2, and obtains a duration for the assets of1.16 years and for the liabilities, 2.77 years. The duration gap is then calculated to be:

Since the Friendly Finance Company has a negative duration gap, the manager real-izes that a rise in interest rates by 1 percentage point from 10% to 11% will increasethe market value of net worth of the firm. The manager checks this by calculating thechange in the market value of net worth as a percentage of assets:

With assets of $100 million, this calculation indicates that net worth will rise in mar-ket value by $1.2 million.

Even though the income and duration gap analysis indicates that the FriendlyFinance Company gains from a rise in interest rates, the manager realizes that if inter-est rates go in the other direction, the company will suffer a fall in income and mar-ket value of net worth. Thus the finance company manager, like the bank manager,realizes that the institution is subject to substantial interest-rate risk.

Although you might think that income and duration gap analysis is complicatedenough, further complications make a financial institution manager’s job even harder.

One assumption that we have been using in our discussion of income and dura-tion gap analysis is that when the level of interest rates changes, interest rates on allmaturities change by exactly the same amount. That is the same as saying that we con-ducted our analysis under the assumption that the slope of the yield curve remainsunchanged. Indeed, the situation is even worse for duration gap analysis, because theduration gap is calculated assuming that interest rates for all maturities are the same—in other words, the yield curve is assumed to be flat. As our discussion of the term

Some Problemswith Income andDuration GapAnalysis

�NW � �DURgap ��i

1 � i� �(�1.33 ) �

0.01

1 � 0.10� 0.012 � 1.2%

DURgap � DURa � �L

A� DURl� � 1.16 � � 90

100� 2.77� � �1.33 years


structure of interest rates in Chapter 6 indicated, however, the yield curve is not flat,and the slope of the yield curve fluctuates and has a tendency to change when thelevel of the interest rate changes. Thus to get a truly accurate assessment of interest-rate risk, a financial institution manager has to assess what might happen to the slopeof the yield curve when the level of the interest rate changes and then take this infor-mation into account when assessing interest-rate risk. In addition, duration gap analy-sis is based on the approximation in Equation 1 and thus only works well for smallchanges in interest rates.

A problem with income gap analysis is that, as we have seen, the financial insti-tution manager must make estimates of the proportion of supposedly fixed-rate assetsand liabilities that may be rate-sensitive. This involves estimates of the likelihood ofprepayment of loans or customer shifts out of deposits when interest rates change.Such guesses are not easy to make, and as a result, the financial institution manager’sestimates of income gaps may not be very accurate. A similar problem occurs incalculating durations of assets and liabilities, because many of the cash payments areuncertain. Thus the estimate of the duration gap might not be accurate either.


WeightedAmount Duration Duration

($ millions) (years) (years)

AssetsCash and deposits 3 0.0 0.00Securities


Consumer loansLess than 1 year 50 0.5 0.251 to 2 years 20 1.5 0.30Greater than 2 years 15 3.0 0.45

Physical capital 5 0.0 0.00Average duration 1.16

LiabilitiesCommercial paper 40 0.2 0.09Bank loans


Long-term bonds and otherlong-term debt 40 5.5 2.44

Average duration 2.77

Table 2 Duration of the Friendly Finance Company’s Assets and Liabilities

Do these problems mean that managers of banks and other financial institutionsshould give up on gap analysis as a tool for measuring interest-rate risk? Financialinstitutions do use more sophisticated approaches to measuring interest-rate risk,such as scenario analysis and value-at-risk analysis, which make greater use of com-puters to more accurately measure changes in prices of assets when interest rateschange. Income and duration gap analyses, however, still provide simple frameworksto help financial institution managers to get a first assessment of interest-rate risk, andthey are thus useful tools in the financial institution manager’s toolkit.


Strategies for Managing Interest-Rate RiskApplication

Once financial institution managers have done the duration and income gapanalysis for their institutions, they must decide which alternative strategies topursue. If the manager of the First Bank firmly believes that interest rates willfall in the future, he or she may be willing to take no action knowing that thebank has more rate-sensitive liabilities than rate-sensitive assets, and so willbenefit from the expected interest-rate decline. However, the bank manageralso realizes that the First Bank is subject to substantial interest-rate risk,because there is always a possibility that interest rates will rise rather than fall,and as we have seen, this outcome could bankrupt the bank. The managermight try to shorten the duration of the bank’s assets to increase their rate sen-sitivity either by purchasing assets of shorter maturity or by converting fixed-rate loans into adjustable-rate loans. Alternatively, the bank manager couldlengthen the duration of the liabilities. With these adjustments to the bank’sassets and liabilities, the bank would be less affected by interest-rate swings.

For example, the bank manager might decide to eliminate the incomegap by increasing the amount of rate-sensitive assets to $49.5 million toequal the $49.5 million of rate-sensitive liabilities. Or the manager couldreduce rate-sensitive liabilities to $32 million so that they equal rate-sensitiveassets. In either case, the income gap would now be zero, so a change ininterest rates would have no effect on bank profits in the coming year.

Alternatively, the bank manager might decide to immunize the marketvalue of the bank’s net worth completely from interest-rate risk by adjustingassets and liabilities so that the duration gap is equal to zero. To do this, themanager can set DURgap equal to zero in Equation 2 and solve for DURa:

DURa � � DURl � � 1.03 � 0.98

These calculations reveal that the manager should reduce the average dura-tion of the bank’s assets to 0.98 year. To check that the duration gap is setequal to zero, the calculation is:

DURgap � 0.98 � � 0� 95

100 � 1.03�

95

100

L

A


In this case, as in Equation 3, the market value of net worth would remainunchanged when interest rates change. Alternatively, the bank manager couldcalculate the value of the duration of the liabilities that would produce aduration gap of zero. To do this would involve setting DURgap equal to zeroin Equation 2 and solving for DURl:

DURl � DURa � � 2.70 � � 2.84

This calculation reveals that the interest-rate risk could also be eliminated byincreasing the average duration of the bank’s liabilities to 2.84 years. The man-ager again checks that the duration gap is set equal to zero by calculating:

DURgap � 2.70 � � 0

Study Guide To see if you understand how a financial institution manager can protectincome and net worth from interest-rate risk, first calculate how the FriendlyFinance Company might change the amount of its rate-sensitive assets or itsrate-sensitive liabilities to eliminate the income gap. You should find that theincome gap can be eliminated either by reducing the amount of rate-sensitiveassets to $43 million or by raising the amount of rate-sensitive liabilities to$55 million. Also do the calculations to determine what modifications to theduration of the assets or liabilities would immunize the market value ofFriendly Finance’s net worth from interest-rate risk. You should find thatinterest-rate risk would be eliminated if the duration of the assets were set to2.49 years or if the duration of the liabilities were set to 1.29 years.

One problem with eliminating a financial institution’s interest-rate riskby altering the balance sheet is that doing so might be very costly in theshort run. The financial institution may be locked into assets and liabilitiesof particular durations because of its field of expertise. Fortunately, recentlydeveloped financial instruments, such as financial futures, options, andinterest-rate swaps, help financial institutions manage their interest-rate riskwithout requiring them to rearrange their balance sheets. We discuss theseinstruments and how they can be used to manage interest-rate risk inChapter 13.

� 95

100 � 2.84�

100

95

A

L

To understand how well a bank is doing, we need to start by looking at a bank’sincome statement, the description of the sources of income and expenses that affectthe bank’s profitability.

The end-of-year 2001 income statement for the Big Six (Bank of Montreal, CIBC,National Bank, Royal Bank of Canada, Scotiabank, and TD Canada Trust) plus theLaurentian Bank of Canada and the Canadian Western Bank appears in Table 1.

Operating Income. Operating income is the income that comes from a bank’s ongoingoperations. Most of a bank’s operating income is generated by interest on its assets,particularly loans. As we see in Table 1, in 2001 interest income represented 72% ofcommercial banks’ operating income. Interest income fluctuates with the level ofinterest rates, and so its percentage of operating income is highest when interest ratesare at peak levels. That is exactly what happened in 1981, when interest rates roseabove 20% and interest income rose to 90% of total bank operating income.

Noninterest income, which made up 28% of operating income in 2001, is gener-ated partly by service charges on deposit accounts, but the bulk of it comes from theoff-balance-sheet activities, which generate fees or trading profits for the bank. Theimportance of these off-balance-sheet activities to bank profits has been growing inrecent years. Whereas in 1980 other noninterest income from off-balance-sheet activ-ities represented only 5% of operating income, it reached 20% in 2001.

Operating Expenses. Operating expenses are the expenses incurred in conducting thebank’s ongoing operations. An important component of a bank’s operating expensesis the interest payments that it must make on its liabilities, particularly on its deposits.Just as interest income varies with the level of interest rates, so do interest expenses.Interest expenses as a percentage of total operating expenses reached a peak in 1981,when interest rates were at their highest, and fell in recent years as interest ratesmoved lower. Noninterest expenses include the costs of running a banking business:salaries for tellers and officers, rent on bank buildings, purchases of equipment suchas desks and vaults, and servicing costs of equipment such as computers.

The final item listed under operating expenses is provisions for credit losses.When a bank has a bad debt or anticipates that a loan might become a bad debt inthe future, it can write up the loss as a current expense in its income statement underthe “provision for credit losses” heading. Provisions for loan losses are directly relatedto loan loss reserves. When a bank wants to increase its loan loss reserves account by,say, $1 million, it does this by adding $1 million to its provisions for loan losses. Loan

Bank’s IncomeStatement

41

Measu r ing Bank Pe r fo rmance

appendix 2to c h a p te r

9

loss reserves rise when this is done because by increasing expenses when losses havenot yet occurred, earnings are being set aside to deal with the losses in the future.

Provisions for loan losses have been a major element in fluctuating bank profitsin recent years. The 1980s brought the third-world debt crisis; a sharp decline inenergy prices in 1986, which caused substantial losses on loans to energy producers;and a collapse in the real estate market. As a result, provisions for loan losses wereparticularly high in the late 1980s. Since then, losses on loans have begun to subside,and in 2001, provisions for loan losses dropped to only 5.9% of operating expenses.

Income. Subtracting the $99 492 million in operating expenses from the $113 138million of operating income in 2001 yields net operating income of $14 066 million.Net operating income is closely watched by bank managers, bank shareholders, andbank regulators because it indicates how well the bank is doing on an ongoing basis.


Share ofOperating

Amount Income or($ millions) Expenses (%)

Operating IncomeInterest income 81 473 72.01

Interest on loans 60 856 53.79Interest on securities 16 898 14.94Deposits with other banks 3 719 3.29

Noninterest income 31 665 27.99Total operating income 113 138 100.00

Operating ExpensesInterest expenses 53 451 53.72

Interest on deposits 41 994 42.21Bank debentures 1 804 1.81Other liabilities 9 653 9.70

Noninterest expenses 40 168 40.37Salaries and employee benefits 21 931 22.04Premises and equipment 8 712 8.76Other 9 525 9.57

Provisions for credit losses 5 873 5.90Total operating expenses 99 492 100.00

Net Operating Income 13 646Provisions for income taxes –3 842

Net Income 9 804

Source: www.fdic.gov/banks/statistical/statistics/0106/cbr

Table 1 Income Statement for All Federally Insured Commercial Banks, 2002

One item, net extraordinary items, which are events or transactions that are bothunusual and infrequent, is added or deducted to the net operating income figure toget the figure for net income before taxes. Net income before taxes is more commonlyreferred to as profits before taxes. Subtracting the $3 842 million of provisions forincome taxes then results in $9 804 million of net income. Net income, more com-monly referred to as profits after taxes, is the figure that tells us most directly how wellthe bank is doing because it is the amount that the bank has available to keep asretained earnings or to pay out to stockholders as dividends.

Although net income gives us an idea of how well a bank is doing, it suffers from onemajor drawback: It does not adjust for the bank’s size, thus making it hard to com-pare how well one bank is doing relative to another. A basic measure of bank prof-itability that corrects for the size of the bank is the return on assets (ROA), mentionedearlier in the chapter, which divides the net income of the bank by the amount of itsassets. ROA is a useful measure of how well a bank manager is doing on the jobbecause it indicates how well a bank’s assets are being used to generate profits. At theend of 2001, the assets of the Big Eight banks amounted to $1 485.5 billion, so usingthe $9.8 billion net income figure from Table 1 gives us a return on assets of:

ROA � � � 0.0066 � 0.66%

net income\assets

Although ROA provides useful information about bank profitability, we havealready seen that it is not what the bank’s owners (equity holders) care about most.They are more concerned about how much the bank is earning on their equity invest-ment, an amount that is measured by the return on equity (ROE), the net income perdollar of equity capital. At the end of 2001 equity capital for all the Big 8 banks was$70.6 billion, so the ROE was therefore:

ROA � � � 0.1383 � 13.88%

Another commonly watched measure of bank performance is called the net inter-est margin (NIM), the difference between interest income and interest expenses as apercentage of total assets:

As we have seen earlier in the chapter, one of a bank’s primary intermediationfunctions is to issue liabilities and use the proceeds to purchase income-earning assets.If a bank manager has done a good job of asset and liability management such that thebank earns substantial income on its assets and has low costs on its liabilities, profitswill be high. How well a bank manages its assets and liabilities is affected by the spreadbetween the interest earned on the bank’s assets and the interest costs on its liabilities.This spread is exactly what the net interest margin measures. If the bank is able to raisefunds with liabilities that have low interest costs and is able to acquire assets with highinterest income, the net interest margin will be high, and the bank is likely to be highlyprofitable. If the interest cost of its liabilities rises relative to the interest earned on itsassets, the net interest margin will fall, and bank profitability will suffer.

NIM �interest income � interest expenses

assets

9.8__70.6

net income___assets

9.8__1485.5

net income___assets

Measures of BankPerformance

Measuring Bank Performance 43

Table 2 provides measures of return on assets (ROA), return on equity (ROE), and thenet interest margin (NIM) for the Big 6 plus the Laurentian Bank of Canada and theCanadian Western Bank from 1991 to 2001. Because the relationship between bankequity capital and total assets for those eight domestic banks remained fairly stable inthe 1990s, both the ROA and ROE measures of bank performance move closelytogether and indicate that the early 1990s, there was an increase in bank profitability.The rightmost column, net interest margin, indicates that the spread between interestincome and interest expenses declined throughout the 1990s.

The explanation of the weak performance of the eight domestic banks in the early1990s is that they had made many risky loans in the late 1980s that turned sour. Theresulting huge increase in loan loss provisions in that period directly decreased netincome and hence caused the fall in ROA and ROE. (Why bank profitability deterio-rated and the consequences for the economy are discussed in Chapters 9 and 11.)

Beginning in 1994, bank performance improved substantially. The return onequity rose to nearly 12% in 1994 and remained above 13% in the 1995–2001period. Similarly, the return on assets rose from the 0.5% level in the 1991–1993period to around the 0.66% level in 1994–2001.

Recent Trends inBank PerformanceMeasures


Return on Return on Net InterestYear Assets (ROA) (%) Equity (ROE) (%) Margin (NIM)(%)

1991 0.68 13.08 2.861992 0.32 5.92 2.791993 0.47 8.72 2.651994 0.59 11.62 2.531995 0.67 13.2 2.341996 0.71 14.93 2.071997 0.71 16.37 1.921998 0.57 13.39 1.751999 0.71 15.69 1.812000 0.71 15.25 1.732001 0.66 13.89 1.80

Source: www2.fdic.gov/qbp

Table 2 Measures of Bank Performance, 1980–2002

The Bank o f Canada ’ s Ba l anceShee t and the Mone ta r y Ba se


15Just as any other bank has a balance sheet that lists its assets and liabilities, so doesthe Bank of Canada. We examine each of its categories of assets and liabilities becausechanges in them are an important way the Bank affects the reserve position of banksand manipulates interest rates and the money supply.

Assets Amount Percent Liabilities Amount Percent

Government of Canada Bank of Canada notes outstanding 41 146.7 93.60securities

Treasury bills 13 113.1 29.83Securities maturing Depositswithin three years 8 571.3 19.50 Government of Canada 534.6 1.22

Securities not maturing Chartered banks 1 065.5 2.42within three years 18 648.7 42.42 Other members of the CPA 125.8 0.29

Other deposits 415.0 0.94

Other investments 2.6 0.01 Securities sold under repurchase agreements

Advances to members of the CPA 534.9 1.22 Other liabilities 671.2 1.53

Foreign currency assetsU.S. dollars 674.2 1.53Other currencies 4.4 0.01

Securities purchased under resale agreements 1 904.8 4.33

Other assets 504.8 1.15

Total 43 958.8 100.00 Total 43 958.8 100.00

Source: Bank of Canada Annual Report 2002

Table 1 Balance Sheet of the Bank of Canada ($ millions, end of 2002)

1. Government of Canada Securities. These are the Bank’s holdings of securities. Thetotal amount of securities is controlled by open market operations (the Bank’s purchaseand sale of these securities). As shown in Table 1, “Government of Canada Securities” isby far the largest category of assets accounting for over 90% of the balance sheet.

2. Other investments. This category mostly includes investments held under short-term foreign currency swap arrangements with the Exchange Fund Account (EFA)of the Department of Finance. In particular, as part of its cash-management operationswithin the Canadian banking system, the Bank of Canada temporarily acquires foreigncurrency investments from the EFA at the prevailing exchange rate with a commitmentto reverse the transaction at the same exchange rate at a future date. Moreover, theBank of Canada is a participant in two foreign currency swap facilities with foreign cen-tral banks—the U.S. Federal Reserve (in the amount of U.S. $2 billion) and the Bancode Mexico (in the amount of Can $1 billion).

3. Advances. These are loans the Bank of Canada makes to members of theCanadian Payments Association. The Bank of Canada charges the bank rate onadvances under the Large Value Transfer System (LVTS). The Canadian paymentssystems are discussed in detail in Chapter 17.

4. Foreign currency assets. These include deposits denominated in foreigncurrencies, which the Bank keeps with domestic and foreign banks and with othercentral banks. Although these assets are part of Canada’s foreign exchange reserves,they must not be confused with the foreign currency assets held by the ExchangeFund Account of the government of Canada.

5. Securities purchased under resale agreements. These are Special Purchase andResale Agreements (SPRAs) with primary dealers, a subgroup of government secu-rities distributors, in which the Bank of Canada purchases government of Canada secu-rities with an agreement to sell them back the next business day at a predeterminedprice. The balance sheet entry “Securities purchased under resale agreements” repre-sents the value receivable by the Bank of Canada upon resale of the securities. As youwill see in Chapter 17, the Bank enters into SPRAs at the target rate for the overnightinterest rate if overnight funds in the money market are traded above the target rate.

These first five assets are important because they earn interest. Because the lia-bilities of the Bank generally pay low interest rates, the Bank makes millions of dol-lars every year—its assets earn income, and its liabilities cost little. Although it returnsmost of its earnings to the federal government, the Bank does spend some of it on“worthy causes,” such as supporting economic research.

6. All other assets. These include securities denominated in foreign currencies aswell as physical goods such as computers, office equipment, and buildings owned bythe Bank of Canada.

1. Bank of Canada notes outstanding. The Bank of Canada issues notes (those blue,purple, green, red, and brown pieces of paper in your wallet that say “Bank of Canada

Liabilities

Assets


note” at the top). The Bank of Canada notes outstanding is the amount of these notesthat is in the hands of the public and the depository institutions. Coins issued by theCanadian Mint are not a liability of the Bank of Canada. The coins and Bank ofCanada notes that we use in Canada are collectively known as currency.

2. Reserves. All the direct clearers have an account at the Bank of Canada in whichthey hold settlement deposits. Reserves consist of settlement balances at the Bank ofCanada plus currency that is physically held by banks (called vault cash because it isheld in bank vaults, cash tills, and automated banking machines).

3. Government of Canada deposits. The government keeps deposits at the Bank ofCanada, against which it writes all its cheques. The government also maintainsdeposit accounts with direct clearers. The maintenance of these government accountswith the Bank of Canada and the direct clearers gives the Bank of Canada an addi-tional instrument of monetary control, called government deposit transfers.

4. Securities sold under repurchase agreements. These are Sale and RepurchaseAgreements (SRAs) with primary dealers, in which the Bank of Canada sells gov-ernment of Canada securities (Treasury bills and bonds) with an agreement to buythem back the next business day at a predetermined price. The balance sheet entry“Securities sold under repurchase agreements” represents the value payable by theBank of Canada upon repurchase of the securities. As you will see in Chapter 17, theBank of Canada enters into SRAs at the target rate for the overnight interest rate ifovernight funds in the money market are traded below the target rate.

5. All other liabilities. This item includes the deposits with the Bank of Canadaowned by foreign governments, foreign central banks, and international agencies(such as the World Bank and the United Nations). It also includes all the remainingBank of Canada liabilities not included elsewhere on the balance sheet.

The first and third liabilities on the balance sheet, Bank of Canada notes outstandingand bank settlement balances, are often referred to as the monetary liabilities of theBank of Canada. When we add to these liabilities the amount of coins in the hands ofthe public and depository institutions, we get a construct called the monetary base.The monetary base is an important part of the money supply, because increases in itwill lead to a multiple increase in the money supply (everything else being constant).This is why the monetary base is also called high-powered money. The monetary baseMB is expressed as:

MB � (Bank of Canada notes outstanding) � (Settlement balances) �(Coins outstanding)

� C � R

where C denotes currency in circulation (coins and Bank of Canada notes held bythe public) and R denotes bank reserves (vault cash plus settlement balances).

The items on the right-hand side of this equation indicate how the base is usedand are called the uses of the base. Unfortunately, this equation does not tell us thefactors that determine the base (the sources of the base), but the Bank of Canada balancesheet in Table 1 comes to the rescue because it has the property that the total assets on

Monetary Base

The Bank of Canada’s Balance Sheet and the Monetary Base 47

the left-hand side must equal the total liabilities on the right-hand side. Because the“Bank of Canada notes outstanding” and “settlement balances” items in the uses of thebase are Bank of Canada liabilities, the “assets equals liabilities” property of the Bankof Canada sheet enables us to solve for these items in terms of the Bank of Canada bal-ance sheet items that are included in the sources of the base: Specifically, Bank ofCanada notes outstanding and bank settlement balances equal the sum of all the Bankof Canada assets minus all the other Bank of Canada liabilities:

( )� ( ) � ( ) � Advances

� Foreign assets � SPRAs � Government deposits � SRAs � Other assets (net)

The two balance sheet items related to other assets and other liabilities have beencollected into one term called Other assets (net), defined as “All other assets” minus“All other liabilities.” This is a technical item, affecting the monetary base, but is notan instrument of monetary control. Substituting all the right-hand-side items in theequation for “Bank of Canada notes outstanding + Settlement balances” in the uses ofthe base equation, we obtain the following expression describing the sources of themonetary base:

MB � Securities and investments � Advances � Foreign assets � SPRAs� Other assets (net) � Coins outstanding � Government deposits � SRAs (1)

Accounting logic has led us to a useful equation that clearly identifies the eightfactors affecting the monetary base listed in Table 2. As Equation 1 and Table 2 depict,increases in the first six factors increase the monetary base, and increases in the lasttwo factors reduce the monetary base.

Securitiesand investments

Settlementbalances

Bank of Canadanotes outstanding


Table 2 Factors Affecting the Monetary BaseS U M M A R Y

Change in Change in Factor Factor Monetary Base

Factors That Increase the Monetary Base

1. Securities and investments ↑ ↑2. Advances to members of the CPA ↑ ↑3. Foreign currency assets ↑ ↑4. Securities purchased under resale

agreements (SPRAs) ↑ ↑5. Other assets (net) ↑ ↑6. Currency outstanding ↑ ↑

Factors That Decrease the Monetary Base

7. Government deposits with the Bank of Canada ↑ ↓

8. Securities sold under repurchase agreements (SRAs) ↑ ↓

The derivation of a money multiplier for the M2+ definition of money requires onlyslight modifications to the analysis in the chapter. The definition of M2+ is

M2+ � C � D � T � MMF

where C � currency in circulationD � all chequable depositsT � all time and savings deposits

MMF � money market mutual funds

We again assume that all desired quantities of these variables rise proportionallywith chequable deposits so that the equilibrium ratios

c � currency ratio, C/Dt � time deposit ratio, T/Df � money market fund ratio, MMF/D

set by depositors and the desired reserve ratio r set by banks are treated as con-stants. Replacing C by c � D, T by t � D, and MMF by f � D in the definition ofM2+ just given, we get

M2� � (c � D) � D � (t � D) � (f � D)� (1 � c � t � f) � D

Substituting in the expression for D from equation 2 in the chapter,1 we have

M2� � � MB

To see what this formula implies about the M2+ money multiplier, we continuewith the same numerical example in the chapter, with the additional information

1 � c � t � fc � r

1From the derivation here it is clear that the quantity of chequable deposits D is unaffected by thedepositor ratios t and f even though time deposits and money market mutual funds are included inM2+. This is just a consequence of the fact that the desired reserve ratios on time deposits and moneymarket mutual funds are zero (because they are not payable on demand), so T and MMF do not appearin any of the equations in the derivation of D in the chapter.

Appendix to Chapter 16

The M2+ Money Multiplier


that T = $320 billion and MMF = $80 billion so that t = 2 and f = 0.5. The resultingvalue of the multiplier for M2+ is

m � � � 12.5

An important feature of the M2+ multiplier is that it is substantially above theM1+ multiplier of 4.2 that we found in the chapter. The crucial concept in under-standing this difference is that a lower desired reserve ratio for time deposits ormoney market mutual funds means that they undergo more multiple expansionbecause fewer reserves are needed to support the same amount of them. Timedeposits and MMFs have a lower desired reserve ratio than chequable deposits—zero—and they will therefore have more multiple expansion than chequabledeposits will. Thus the overall multiple expansion for the sum of these depositswill be greater than for chequable deposits alone, and so the M2+ money multi-plier will be greater than the M1+ money multiplier.

FA C T O R S T H AT D E T E R M I N E T H E M 2+ M O N E Y M U LT I P L I E R

The economic reasoning analyzing the effect of changes in the desired reserveratio and the currency ratio on the M2+ money multiplier is identical to that usedfor the M1+ multiplier in the chapter. An increase in the desired reserve ratio r willdecrease the amount of multiple deposit expansion, thus lowering the M2+ moneymultiplier. An increase in c means that depositors have shifted out of chequabledeposits into currency, and since currency has no multiple deposit expansion, theoverall level of multiple deposit expansion for M2+ must also fall, lowering theM2+ multiplier.

We thus have the same results we found for the M1+ multiplier: the M2+money multiplier and M2+ money supply are negatively related to thedesired reserve ratio r and the currency ratio c.

An increase in either t or f leads to an increase in the M2+ multiplier because thedesired reserve ratios on time deposits and money market mutual funds are zeroand hence are lower than the desired reserve ratio on chequable deposits.

Both time deposits and money market mutual funds undergo more multi-ple expansion than chequable deposits. Thus a shift out of chequable depositsinto time deposits or money market mutual funds, increasing t or f, implies thatthe overall level of multiple expansion will increase, raising the M2+ moneymultiplier.

A decline in t or f will result in less overall multiple expansion, and the M2+money multiplier will decrease, leading to the following conclusion: the M2+money multiplier and M2+ money supply are positively related to both thetime deposit ratio t and the money market fund ratio f.

The response of the M2+ money supply to all the depositor and desired reserveratios is summarized in Table 16A-1.

Response toChanges in tand f

Changes in cand r

3.750.3

1 � 0.25 � 2 � 0.50.25 � 0.05

The M2+ Money Multiplier 51

TA B L E 1 6 A - 1 Response of the M2+ Money Supply toChanges in MBn, A, r, c, t, and f

Variable Change in M2+ MoneyVariable Supply Response Reason

MBn

↑ ↑ More MB to support C and D

A ↑ ↑ More MB to support C and D

r ↑ ↓ Less multiple deposit expansion

c ↑ ↓ Less overall deposit expansion

t ↑ ↑ More multiple deposit expansion

f ↑ ↑ More multiple deposit expansion

Note: Only increases (↑) in the variables are shown; the effects of decreases in the variables on the money supplywould be the opposite of those indicated in the “Response” column.

A Ma thema t i c a l T rea tmen t o f t he Baumo l -Tob in and Tob in Mean -Va r i ance Mode l s


22The derivation of a money multiplier for the M2 definition of money requires onlyslight modifications to the analysis in the chapter. The definition of M2 is:

M2 � C � D � T � MMF

where C � currency in circulationD � checkable depositsT � time and savings deposits

MMF � money market mutual funds

We again assume that all desired quantities of these variables rise proportionallywith checkable deposits so that the equilibrium ratios

c � currency ratio, C/Dt � time deposit ratio, T/Df � money market fund ratio, MMF/D

set by depositors and the desired reserve ratio r set by banks are treated as constants.Replacing C by c � D, T by t � D, and MMF by f� D in the definition of M2+ justgiven, we get

M2� � (c � D) � D � (t � D) (f � D)

� (1 � c � t � f) � D

Substituting in the expression for D from Equation 2 in the chapter,1 we have

To see what this formula implies about the M2+ money multiplier, we continuewith the same numerical example in the chapter, with the additional informationthat T � $320 billion and MMF � $80 billion so that t � 2 and mm � 0.5. Theresulting value of the multiplier for M2+ is:

m � � � 12.53.75_0.3

1 + 0.25 + 0.5___0.25 + 0.05

M2 �1 � c � t � mm

r � e � c � MB

53


An important feature of the M2+ multiplier is that it is substantially above theM1+ multiplier of 4.2 that we found in the chapter. The crucial concept in under-standing this difference is that a lower required reserve ratio for time deposits ormoney market mutual fund means that they undergo more multiple expansionbecause fewer reserves are needed to support the same amount of them. Timedeposits and MMFs have a lower required reserve ratio than checkable deposits—zero—and they will therefore have more multiple expansion than checkable depositswill. Thus the overall multiple expansion for the sum of these deposits will be greaterthan for checkable deposits alone, and so the M2+ money multiplier will be greaterthan the M1+ money multiplier.

Factors That Determine the M2 Money Multiplier

The economic reasoning analyzing the effect of changes in the desired reserve ratioand the currency ratio on the M2+ money multiplier is identical to that used for theM1+ multiplier in the chapter. An increase in the desired reserve ratio r will decreasethe amount of multiple deposit expansion, thus lowering the M2+ money multiplier.An increase in c means that depositors have shifted out of checkable deposits into cur-rency, and since currency has no multiple deposit expansion, the overall level of mul-tiple deposit expansion for M2+ must also fall, lowering the M2+ multiplier.

We thus have the same results we found for the M1 multiplier: The M2+ moneymultiplier and M2+ money supply are negatively related to the required reserveratio r, the currency ratio c, and the excess reserves ratio e.

An increase in either t or f leads to an increase in the M2+ multiplier, because thedesired reserve ratios on time deposits and money market mutual fund shares are zeroand hence are lower than the desired reserve ratio on checkable deposits.

Response toChanges in t and f

Changes in c and r

1To minimize costs, the second derivative must be greater than zero. We find that it is, because:

2An alternative way to get Equation 1 is to have the individual maximize profits, which equal the interest onbonds minus the brokerage costs. The average holding of bonds over a period is just:

Thus profits are:

Then:

This equation yields the same square root rule as Equation 1.

d PROFITS

dC�

� i

2�

bT0

C2 � 0

PROFITS � �i

2(T0 � C ) �

bT0

C

T0

2�

C

2

d2COSTS

dC2 ��2

C3 (�bT0 ) �2bT0

C3 � 0

A Mathematical Treatment of the Baumol-Tobin and Tobin Mean-Variance Models 55

Both time deposits and money market mutual funds undergo more multipleexpansion than checkable deposits. Thus a shift out of checkable deposits into timedeposits or money market mutual funds, increasing t or f, implies that the overall levelof multiple expansion will increase, raising the M2+ money multiplier.

A decline in t or f will result in less overall multiple expansion, and the M2+money multiplier will decrease, leading to the following conclusion: The M2+ moneymultiplier and M2+ money supply are positively related to both the time depositratio t and the money market fund ratio f.

The response of the M2+ money supply to all the depositor and required reserveratios is summarized in Table 16A-1.

Baumol-Tobin Model of Transactions Demand for Money

The basic idea behind the Baumol-Tobin model was laid out in the chapter. Here weexplore the mathematics that underlie the model. The assumptions of the model areas follows:

1. An individual receives income of T0 at the beginning of every period. 2. An individual spends this income at a constant rate, so at the end of the period,

all income T0 has been spent. 3. There are only two assets—cash and bonds. Cash earns a nominal return of zero,

and bonds earn an interest rate i.

F IGURE 1 Indifference Curves ina Mean-Variace ModelThe indifference curves areupward-sloping, and higher indif-ference curves indicate that utilityis higher. In other words, U3 � U2 � U1.

Expected Return �

HigherUtility

Standard Deviation of Returns �

U3U2

U1

3 This assumption is not critical to the results. If E(g) ≠ 0, it can be added to the interest term i, and the analy-sis proceeds as indicated.

4. Every time an individual buys or sells bonds to raise cash, a fixed brokerage feeof b is incurred.

Let us denote the amount of cash that the individual raises for each purchase orsale of bonds as C, and n � the number of times the individual conducts a transac-tion in bonds. As we saw in Figure 3 in the chapter, where T0 � 1,000, C � 500, andn � 2:

Because the brokerage cost of each bond transaction is b, the total brokerage costs fora period are:

Not only are there brokerage costs, but there is also an opportunity cost to holdingcash rather than bonds. This opportunity cost is the bond interest rate i times aver-age cash balances held during the period, which, from the discussion in the chapter,we know is equal to C/2. The opportunity cost is then:

Combining these two costs, we have the total costs for an individual equal to:

The individual wants to minimize costs by choosing the appropriate level of C.This is accomplished by taking the derivative of costs with respect to C and setting itto zero.1 That is:

Solving for C yields the optimal level of C:

Because money demand Md is the average desired holding of cash balances C/2,

(1)

This is the famous square root rule.2 It has these implications for the demand formoney:

1. The transactions demand for money is negatively related to the interest rate i. 2. The transactions demand for money is positively related to income, but there are

economies of scale in money holdings—that is, the demand for money rises lessthan proportionally with income. For example, if T0 quadruples in Equation 1,

Md �1

2�2bT0

i� �bT0

2i

C � �2bT0

i

d COSTS

dC�

� bT0

C2 �i

2� 0

COSTS �bT0

C�

iC

2

iC

2

nb �bT0

C

n �T0

C


A Mathematical Treatment of The Baumol-Tobin and Tobin Mean-Variance Models 57

the demand for money only doubles. 3. A lowering of the brokerage costs due to technological improvements would

decrease the demand for money. 4. There is no money illusion in the demand for money. If the price level doubles,

T0 and b will double. Equation 1 then indicates that M will double as well. Thusthe demand for real money balances remains unchanged, which makes sensebecause neither the interest rate nor real income has changed.

F IGURE 2 Optimal Choice of theFraction of the Portfolio in BondsThe highest indifference curve isreached at a point B, the tangencyof the indifference curve with theopportunity locus. This pointdetermines the optimal risk �*,and using Equation 2 in the bot-tom half of the figure, we solve forthe optimal fraction of the portfo-lio in bonds A*.

�

�

�*

A*

A

B

Slope = i/�g

Slope = 1/�g

Eq. 3OpportunityLocus

Eq. 2

4The indifference curves have been drawn so that the usual result is obtained that as i goes up, A* goes up aswell. However, there is a subtle issue of income versus substitution effects. If, as people get wealthier, they arewilling to bear less risk, and if this income effect is larger than the substitution effect, then it is possible to getthe opposite result that as i increases, A* declines. This set of conditions is unlikely, which is why the figure isdrawn so that the usual result is obtained. For a discussion of income versus substitution effects, see DavidLaidler, The Demand for Money: Theories and Evidence, 4th ed. (New York: HarperCollins, 1993).


F IGURE 3 Optimal Choice of theFraction of the Portfolio in Bonds asthe Interest Rate RisesThe interest rate on bonds risesfrom i1 to i2, rotating the opportu-nity locus upward. The highestindifference curve is now at pointC, where it is tangent to the newopportunity locus. The optimallevel of risk rises from �1

* to �2*,

and then Equation 2, in the bot-tom haf of the figure, shows thatthe optimal fraction of the portfo-lio in bonds rises from A1

* to A2*.

�

�

�*

A*

A

B

C

Slope = i2/�g

Slope = 1/�g

Slope = i1/�g

1 �*2

1

A*2

Tobin Mean-Variance Model

Tobin’s mean-variance analysis of money demand is just an application of the basicideas in the theory of portfolio choice. Tobin assumes that the utility that peoplederive from their assets is positively related to the expected return on their portfolioof assets and is negatively related to the riskiness of this portfolio as represented bythe variance (or standard deviation) of its returns. This framework implies that anindividual has indifference curves that can be drawn as in Figure 1. Notice that theseindifference curves slope upward because an individual is willing to accept more riskif offered a higher expected return. In addition, as we go to higher indifference curves,utility is higher, because for the same level of risk, the expected return is higher.

Tobin looks at the choice of holding money, which earns a certain zero return, orbonds, whose return can be stated as:

RB � i � g

where i � interest rate on the bondg � capital gain

Emp i r i c a l E v i dence on the Demand fo r Money


22

1James Tobin, “Liquidity Preference and Monetary Policy,” Review of Economics and Statistics 29 (1947): 124–131.2A problem with Tobin’s procedure is that idle balances are not really distinguishable from transactions balances.As the Baumol-Tobin model of transactions demand for money makes clear, transactions balances will be relatedto both income and interest rates, just like idle balances.3See David E. W. Laidler, The Demand for Money: Theories and Evidence, 4th ed. (New York: HarperCollins, 1993).Only one major study has found that the demand for money is insensitive to interest rates: Milton Friedman,“The Demand for Money: Some Theoretical and Empirical Results,” Journal of Political Economy 67 (1959):327–351. He concluded that the demand for money is not sensitive to interest-rate movements, but as later workby David Laidler (using the same data as Friedman) demonstrated, Friedman used a faulty statistical procedurethat biased his results: David E. W. Laidler, “The Rate of Interest and the Demand for Money: Some EmpiricalEvidence,” Journal of Political Economy 74 (1966): 545–555. When Laidler employed the correct statistical pro-cedure, he found the usual result that the demand for money is sensitive to interest rates. In later work, Friedmanhas also concluded that the demand for money is sensitive to interest rates.4Kevin Clinton, “The Demand for Money in Canada: 1955–1970: Some Single Equation Estimates and StabilityTests,” Canadian Journal of Economics 6 (1973): 53–61; Norman Cameron, “The Stability of Canadian Demandfor Money Functions,” Canadian Economics 12 (1979): 258–281; Stephen Poloz, “Simultaneity and the Demandfor Money in Canada,” Canadian Journal of Economics 13 (1980): 407–420.

59


Tobin also assumes that the expected capital gain is zero3 and its variance is �g2. That is,

E(g) � 0 and so E(RB) � i � 0 � i

Var(g) � E[g � E(g)]2 � E(g2) � �g2

where E � expectation of the variable inside the parenthesesVar � variance of the variable inside the parentheses

If A is the fraction of the portfolio put into bonds (0 ≤ A ≤ 1) and 1 � A is thefraction of the portfolio held as money, the return R on the portfolio can be writ-ten as:

R � ARB � (1 � A)(0) � ARB � A(i � g)

Then the mean and variance of the return on the portfolio, denoted respectively as �and �2, can be calculated as follows:

� � E(R) � E(ARB) � AE(RB) � Ai

�2 � E(R � �)2 � E[A(i � g) � Ai]2 � E(Ag)2 � A2E(g2) � A2�g2

Taking the square root of both sides of the equation directly above and solving for Ayields:

(2)

Substituting for A in the equation � � Ai using the preceding equation gives us:

(3)

Equation 3 is known as the opportunity locus because it tells us the combinationsof � and � that are feasible for the individual. This equation is written in a form inwhich the � variable corresponds to the Y axis and the � variable to the X axis. Theopportunity locus is a straight line going through the origin with a slope of i/�g. It isdrawn in the top half of Figure 2 along with the indifference curves from Figure 1.

The highest indifference curve is reached at point B, the tangency of the indiffer-ence curve and the opportunity locus. This point determines the optimal level of risk�* in the figure. As Equation 2 indicates, the optimal level of A, A*, is:

A ��

�g

� �i

�g

�

A �1

�g

�

5David E. W. Laidler, “Some Evidence on the Demand for Money,” Journal of Political Economy 74 (1966): 55–68;Allan H. Meltzer, “The Demand for Money: The Evidence from the Time Series,” Journal of Political Economy 71(1963): 219–246; Karl Brunner and Allan H. Meltzer, “Predicting Velocity: Implications for Theory and Policy,”Journal of Finance 18 (1963): 319–354.6Interest sensitivity is measured by the interest elasticity of money demand, which is defined as the percentagechange in the demand for money divided by the percentage change in the interest rate.

Empirical Evidence on the Demand for Money 61

This equation is solved in the bottom half of Figure 2. Equation 2 for A is a straightline through the origin with a slope of 1/� g. Given �*, the value of A read off this lineis the optimal value A*. Notice that the bottom part of the figure is drawn so that aswe move down, A is increasing.

Now let’s ask ourselves what happens when the interest rate increases from i1 toi2. This situation is shown in Figure 3. Because � g is unchanged, the Equation 2 linein the bottom half of the figure does not change. However, the slope of the opportu-nity locus does increase as i increases. Thus the opportunity locus rotates up and wemove to point C at the tangency of the new opportunity locus and the indifferencecurve. As you can see, the optimal level of risk increases from �*

1 and �*2 the optimal

fraction of the portfolio in bonds rises from A*1 to A*

2. The result is that as the interestrate on bonds rises, the demand for money falls; that is, 1 � A, the fraction of theportfolio held as money, declines.4

Tobin’s model then yields the same result as Keynes’s analysis of the speculativedemand for money: It is negatively related to the level of interest rates. This model,however, makes two important points that Keynes’s model does not:

1. Individuals diversify their portfolios and hold money and bonds at the same time. 2. Even if the expected return on bonds is greater than the expected return on

money, individuals will still hold money as a store of wealth because its return ismore certain.

Here we examine the empirical evidence on the two primary issues that distinguishthe different theories of money demand and affect their conclusions about whetherthe quantity of money is the primary determinant of aggregate spending: Is thedemand for money sensitive to changes in interest rates, and is the demand for moneyfunction stable over time?

James Tobin conducted one of the earliest studies on the link between interest ratesand money demand using U.S. data.1 Tobin separated out transactions balances fromother money balances, which he called “idle balances,” assuming that transactionsbalances were proportional to income only, and idle balances were related to interestrates only. He then looked at whether his measure of idle balances was inverselyrelated to interest rates in the period 1922–1941 by plotting the average level of idlebalances each year against the average interest rate on commercial paper that year.When he found a clear-cut inverse relationship between interest rates and idle bal-ances, Tobin concluded that the demand for money is sensitive to interest rates.2

Additional empirical evidence on the demand for money strongly confirmsTobin’s finding.3 Also, studies of the demand for money in Canada, using post-wardata, by Kevin Clinton, Norman Cameron, and Stephen Poloz found that the demandfor money is sensitive to interest rates.4 Does this sensitivity ever become so high thatwe approach the case of the liquidity trap in which monetary policy is ineffective? Theanswer is almost certainly no. Keynes suggested in The General Theory that a liquid-

Interest Ratesand MoneyDemand

7Stephen M. Goldfeld, “The Demand for Money Revisited,” Brookings Papers on Economic Activity 3 (1973): 577–638.8See, for example, William R. White, “The Demand for Money in Canada and the Control of Monetary Aggregates:Evidence from the Monthly Data.” Bank of Canada Staff Research Study 12, Ottawa: Bank of Canada, 1976.9Stephen M. Goldfeld, “The Case of the Missing Money,” Brookings Papers on Economic Activity 3 (1976): 683–730.10Charles Freedman, “Financial Innovation in Canada: Causes and Consequences,” American Economic Review,Papers and Proceedings, 73 (May 1983): 101–106; Ed Fine, “Institutional Developments Affecting MonetaryAggregates,” in Monetary Seminar 90 (Ottawa: Bank of Canada, 1990): pp. 555–563.11Francesco Caramaza, “The Demand for M2 and M2+ in Canada,” Bank of Canada Review, December 1989: 3-19.


ity trap might occur when interest rates are extremely low. (However, he did state thathe had never yet seen an occurrence of a liquidity trap.)

Typical of the evidence demonstrating that the liquidity trap has never occurredis that of David Laidler, Karl Brunner, and Allan Meltzer, who looked at whether theinterest sensitivity of money demand increased in periods when interest rates werevery low.5 Laidler and Meltzer looked at this question by seeing whether the interestsensitivity of money demand differed across periods, especially in periods such as the1930s when interest rates were particularly low.6 They found that there was no ten-dency for interest sensitivity to increase as interest rates fell—in fact, interest sensi-tivity did not change from period to period. Brunner and Meltzer explored thisquestion by recognizing that higher interest sensitivity in the 1930s as a result of a liq-uidity trap implies that a money demand function estimated for this period shouldnot predict well in more normal periods. What Brunner and Meltzer found was thata money demand function, estimated mostly with data from the 1930s, accuratelypredicted the demand for money in the 1950s. This result provided little evidence infavour of the existence of a liquidity trap during the Great Depression period.

The evidence on the interest sensitivity of the demand for money found by dif-ferent researchers is remarkably consistent. Neither extreme case is supported by thedata: The demand for money is sensitive to interest rates, but there is little evidencethat a liquidity trap has ever existed.

If the money demand function, like Equation 4 or 6 in Chapter 22, is unstable andundergoes substantial unpredictable shifts, as Keynes thought, then velocity is unpre-dictable, and the quantity of money may not be tightly linked to aggregate spending,as it is in the modern quantity theory. The stability of the money demand function isalso crucial to whether the central bank should target interest rates or the money sup-ply (see Chapter 24). Thus it is important to look at the question of whether themoney demand function is stable, because it has important implications for howmonetary policy should be conducted.

As our discussion of the Brunner and Meltzer article indicates, evidence on thestability of the demand for money function is related to the evidence on the existenceof a liquidity trap. Brunner and Meltzer’s finding that a money demand function esti-mated using data mostly from the 1930s predicted the demand for money well in thepostwar period not only suggests that a liquidity trap did not exist in the 1930s, butalso indicates that the money demand function has been stable over long periods of

Stability of MoneyDemand

12David Longworth and Joseph Attah-Mensah, “The Canadian Experience with Weighted Monetary Aggregates,”Bank of Canada Working Paper 95-10.13John P. Cockerline and John Murray, “A Comparison of Alternative Methods of Monetary Aggregation: SomePreliminary Evidence,” Bank of Canada Technical Report 28, Ottawa: Bank of Canada, 1981.14Francesco Caramaza, Doug Hostland, and Kim McPhail, “Studies on the Demand for M2 and M2+ in Canada,”in Monetary Seminar 90 (Ottawa: Bank of Canada, 1990): pp. 1–114.15Steve Ambler and Alain Paquet, “Cointegration and the Demand for M2 and M2+ in Canada,” in MonetarySeminar 90 (Ottawa: Bank of Canada, 1990): pp. 125–168.16This research is discussed in John P. Judd and John L. Scadding, “The Search for a Stable Money DemandFunction, “Journal of Economic Literature 20 (1982): 993–1023.17Apostolos Serletis, The Demand for Money: Theoretical and Empirical Approaches, Kluwer Academic (2001), isthe state-of-the-art regarding recent theoretical and empirical approaches to the demand for money.18Thomas F. Cooley and Stephen F. Le Roy, “Identification and Estimation of Money Demand,” AmericanEconomic Review 71 (1981): 825–844, is especially critical of the empirical research on the demand for money.

time. The evidence that the interest sensitivity of the demand for money did notchange from period to period also suggests that the money demand function is stable,since a changing interest sensitivity would mean that the demand for money functionestimated in one period would not be used to predict that of another period.

By the early 1970s, the evidence using data from the postwar period stronglysupported the stability of the money demand function when M1 was used as thedefinition of the money supply. For example, a well-known U.S. study by StephenGoldfeld published in 1973 found not only that the interest sensitivity of M1 moneydemand did not undergo changes in the postwar period, but also that the M1 moneydemand function predicted extremely well throughout the postwar period.7 Similarly,studies of the demand for money in Canada concluded that narrow money demandfunctions were quuite stable.8 As a result of this evidence, the M1 money demandfunction became the conventional money demand function used by economists. Infact, this evidence provided the foundation for the Bank of Canada’s experiment withtargeting the growth rate of M1 and for its strategy of gradualism from 1975 to 1982.

The Case of the Missing Money. The stability of the demand for money, then, was awell-established fact when, starting in 1974, conventional M1 money demand func-tions in the United States and Canada began to severely overpredict the demand formoney. Stephen Goldfeld labeled this phenomenon of instability in the demand formoney function “the case of the missing money.”9 It presented a serious challenge tothe usefulness of the money demand function as a tool for understanding how mone-tary policy affects aggregate economic activity. In addition, it had important implica-tions for how monetary policy should be conducted. As a result, the instability of theM1 money demand function stimulated an intense search for a solution to the mysteryof the missing money so that a stable money demand function could be resurrected.

The search for a stable money demand function took three directions. The firstdirection focused on whether an incorrect definition of money could be the reason whythe demand for money function had become so unstable. As Charles Freedman and EdFine argue, competition between banks and near-banks, technological innovation, andhigh interest rates caused the payments mechanism and cash management techniquesto undergo rapid changes after the beginning of money targeting in 1975.10 This hasled some researchers to suspect that the rapid pace of financial innovation has meantthat the conventional definitions of the money supply no longer apply. They searched

A l geb ra o f t he I S LM Mode l


24

63


for a stable money demand function by actually looking directly for the missingmoney; that is, they looked for financial instruments that have been incorrectly left outof the definition of money used in the money demand function.

Daily interest savings accounts, introduced in 1979, and daily interest chequingaccounts, introduced in 1981, are one example. These accounts provided chequingprivileges and paid daily interest (computed on the daily closing balance), therebyoffering the small saver the opportunity to earn near-market interest rates. As a result,people found these accounts attractive and were encouraged to substitute them fordemand deposits (part of M1). These accounts, however, were included in the M2definition of the money supply and hence the demand for M1 decreased and that forM2 increased. Recent evidence using later data has cast some doubt on whetherincluding daily interest saving and chequing accounts, and other highly liquid assets,in measures of the money supply produces money demand functions that are stable.11

The second direction of search for a stable money demand function was to useweighted monetary aggregates (discussed in Chapter 3). However, the results of esti-mating money demand functions using weighted monetary aggregates do not supportthe existence of a stable money demand function. For example, David Longworth andJoseph Atta-Mensah of the Bank of Canada compared the empirical performance ofweighted monetary aggregates with the corresponding simple-sum aggregates andfound that the theoretically superior weighted aggregates do not produce a stablemoney demand function.12 This is also consistent with earlier results by JohnCockerline and John Murray, also of the Bank of Canada.13

The third direction of search for a stable money demand function was to reeval-uate the conventional money demand specifications, be looking for new variables toinclude in the money demand function that will make it stable. Francesco Caramaza,Doub Hostland, and Kim McPhail, for example, found that the earning-price ratio hasa significant negative effect on the demand for broad money.14 Other researchers, suchas Steve Ambler and Alain Paquet, added the real stock of Canada Savings Bonds(CSB) as well as dummy variables (to capture seasonal factors and postal strikes).15

These attempts to produce a stable money demand function have been criticizedon the grounds that the theoretical justification for including them in the moneydemand function is weak. Also, later research questions whether these alterations tothe money demand function will lead to continuing stability in the future.16

Conclusion. The main conclusion from the research on the money demand functionseems to be that the most likely cause of its instability is the rapid pace of financialinnovation occurring after 1973, which has changed what items can be counted asmoney. The evidence is still somewhat tentative, however, and a truly stable and sat-isfactory money demand function has not yet been found. And so the search for a sta-ble money demand function goes on.17

The recent instability of the money demand function calls into question whetherour theories and empirical analyses are adequate.18 It also has important implicationsfor the way monetary policy should be conducted because it casts doubt on the use-fulness of the money demand function as a tool to provide guidance to policymakers.In particular, because the money demand function has become unstable, velocity isnow harder to predict, and as discussed in Chapter 21, setting rigid money supplytargets in order to control aggregate spending in the economy may not be an effectiveway to conduct monetary policy.

Algebra of the ISLM Model 65

The use of algebra to analyze the ISLM model allows us to extend the multiplier analy-sis in Chapter 23 and to obtain many of the results of Chapters 23 and 24 veryquickly.

Basic Closed-Economy ISLM Model

The goods market can be described by the following equations:

Consumption function: C � � mpc (Y � T) (1)Investment function: I � � di (2)Taxes: T � (3)Government spending: G � (4)Goods market equilibrium condition: Y � Yad � C � I � G (5)

The money market is described by these equations:

Money demand function: Md � d � eY � fi (6)Money supply: Ms � (7)Money market equilibrium condition: Md � Ms (8)

The uppercase terms are the variables of the model; , , and , are the values of thepolicy variables that are set exogenously (outside the model); and , , and d areautonomous components of consumer expenditure, investment spending, and moneydemand that are also determined exogenously (outside the model). Except for theinterest rate i, the lowercase terms are the parameters, the givens of the model, andall are assumed to be positive. The definitions of these variables and parameters areas follows:

C � consumer spendingI � investment spending

G � � government spendingY � outputT � � taxes

Md � money demandMs � � money supply

i � interest rate� autonomous consumer spending

d � interest sensitivity of investment spending� autonomous investment spending related to business confidence

d � autonomous money demande � income sensitivity of money demandf � interest sensitivity of money demand

mpc � marginal propensity to consume

Substituting for C, I, and G in the goods market equilibrium condition and then solv-ing for Y, we obtain the IS curve:

(9)Y �1

1 � mpc(C � I � mpc T � G � di )

IS and LM Curves

MI

C

M

T

G

MICMTG

MM

GTIC

Solving for i from Equations 6, 7, and 8, we obtain the LM curve:

(10)

The solution to the model occurs at the intersection of the IS and LM curves, whichinvolves solving for Y and i simultaneously, using Equations 9 and 10, as follows:

(11)

(12)i �1

f(1 � mpc ) � d3e(C � I � mpc T � G ) � Md(1 � mpc ) � M(1 � mpc ) 4

Y �1

1 � mpc � de�f�C � I � mpc T � G �dMd

f�

dM

f �

Solution of theModel

i �Md � M � eY

f


Documents

to c2hapter Financial Market Instruments · Here we examine the securities (instruments) traded in financial markets. We first focus on the instruments traded in the money market