Bond Prices and YieldsCHAPTER 10

10-2

Discuss different characteristics (特性 ) in bond contracts (some content was discussed in Ch 2)Review the pricing of bondsAnalyze the price-yield relationship for bondsDefault risk (違約風險 ) of corporate bondsIntroduce the term structure of interest rates※ Bond markets are even more important than stock

markets in terms of size and trading volumeBy the end of 2014, the size of the global bond (debt) market is $149 trillion ($76 trillion)The size of the world stock market is estimated to be $69 trillion by the end of 2014

※ Pricing a bond is simpler than pricing a stock since a bond’s future payment schedule is known today

The Goals of Chapter 10

10-3

10.1 BOND CHARACTERISTICS

10-4

Bond CharacteristicsBond is a security that obligates the issuer (the borrower) to make specified payments to the holder (the lender) over a period of timeBond indenture (債券契約 )– Maturity date (到期日 )– Face or par value (面額 ) (the amount at which the

issuer needs to pay the holder at maturity)– Coupon rate (票面利率 )

A bond’s annual interest payment per dollar of the par valueSemiannual or annual interest paymentsZero-coupon bond (no coupon payments, sold at a discount)

10-5

Treasury BondsMaturities for Treasury bonds (政府公債 )– For T-bills, the maturities are less than one year– T-note maturities range up to 10 years– T-bond maturities range from 10 to 30 years

Both T-bonds and T-notes are issued in denominations of $1,000 (typically) or more– Both with semiannual coupon payments – Bid and ask prices are quoted as a percent of the

par value (see the next slide)For T-bills (國庫券 ) (zero-coupon bonds)– Bid and asked prices are quoted in hundredths,

which are annual discount rates of the face valueT-bills, T-notes, and T-Bonds are traded over the counter, i.e., in a dealer market

10-6

Prices and Yields of U.S. Treasuries

※ The quotation methods for T-bonds, T-notes, and T-bills are already introduced in Ch2

※ The asked yield to maturity (到期收益率 ) as well as the yield to call (贖回收益率 ) will be discussed in detail on Slide 10-30

10-7

Treasury BondsQuoted (clean or flat) price (除息價格 ) vs. Invoice (dirty) price (含息價格 )– If a bond is purchased between coupon payment

dates, the buyer must pay the invoice price, which equals the quoted price plus the accrued interest

The interest is paid in arrear (期滿支付 ) (paid at the end of each period), and part of it should belong to the previous bond holder (i.e., the seller)

– Suppose the semiannual coupon payments is $40, and the bond is quoted as $990. If 30 days have passed since the last coupon payment, and there are 182 days in this semiannual coupon period

The invoice price = $990 + $40 (30/182) = $996.59

10-8

Treasury Bonds– A dirty price is in essence the sum of the present

values of future coupon and principal payments– In fact, dirty prices are calculated first and next the

accrued interest between coupon payment dates is deducted from the dirty price to derive the clean price

– The reason to distinguish clean and dirty pricesClean prices are more stable over time than dirty prices–when clean prices change, it usually reflects an economic reason, for instance, a change in interest rates or in the bond issuer's credit qualityDirty prices, on the other hand, change day to day depending on where the current date is in relation to the coupon payment dates, in addition to any economic reasons

10-9

Corporate BondsListing of corporate bonds (on the next slide)Registered vs. non-registered– Bonds issued in the U.S. today are registered,

meaning that the issuing firm keeps records of the owners (helpful to tax authorities for tax collection)

– Bearer bonds (無記名債券 ) are bonds without any record of ownership. The physical possession of the bond certificate is the only evidence of ownership (common in Eurobond markets)

It is useful for investors who wish to retain anonymity (匿名 )Recovery of the value of a bearer bond in the event of its loss, theft, or destruction is usually impossible

Call provisions (贖回權 )– Callable bonds (可贖回債券 ) may be redeemed by

the issuer at a specified call price during the call period

10-10

Corporate Bonds

※ The “MOODY’S/S&P/FITCH” column is the estimation of the default risk (違約風險 ) (or called credit risk (信用風險 )) associated with the bond offered by three major bond rating agencies (the default risk will be discussed in greater detail in Section 10.5) ※ Investment grade bonds (投資等級債券 ) (rated BBB or above) vs. Speculative grade or junk bonds (投機等級或是垃圾債券 ) (rated BB or below)

10-11

Corporate Bonds– Triggered when bond market prices > call price

(usually when the interest rate becomes low)– Refunding (借新償舊 ) strategy when the interest rate

falls: Firms use funds from issuing new bonds (with lower interest costs) to buy back old bonds (with higher interest costs)

– Advantage for firms: callable bonds should offer higher coupons (or be cheaper) than noncallable bonds

Put provision (賣回權 )– The holder of puttable bonds (可賣回債券 ) may

choose to sell the bonds back to the issuer at a pre-specified put price on some date

– This right is often exercised when the coupon rate is much lower than the prevailing interest rate (in this scenario, the bond is worth at a deep discount)

10-12

Corporate Bonds– It is a advantage for bond holders, so puttable bonds

should offer lower coupons (or be more valuable) than nonputtable bonds

Convertible provision (轉換權 ) (convertible bonds, 可轉換債券 )– An option to exchange the bond for a specified

number of shares of common stock of the issuing firm– The market conversion value is the current value of

the shares for which the bond may be exchanged– The conversion premium (轉換溢酬 ) is the excess of

the bond price over its market conversion value– Convertible bondholders could benefit from price

appreciation, so convertible bonds usually offer lower coupon rates than nonconvertible bonds

10-13

Corporate BondsFloating rate bonds (浮動利率型債券 )– The interest payments (coupon rate) are adjusted

according to some measure of market rates, e.g., the prevailing T-bill rate plus 2%

– The 2% is called the yield spread, which reflects the issuer’s credit condition. A lower (worse) credit rating implies a higher yield spread

– The advantage is that the coupon rate can reflect the change of the market situation, and thus the bond price is relatively stable than that of the fixed rate bond

10-14

Other Domestic IssuersState or local governments– Municipal Bonds (interest payment is tax-exempt)

Government agencies– Federal Home Loan Bank Board– Farm Credit Agencies– Ginnie Mae– Fannie Mae– Freddie Mac

※ These bond securities are already introduced in Ch2

10-15

International BondsForeign bonds (外國債券 )– Issued by a borrower from a nation other than the

nation in which the bond is sold, e.g., a German firm sells a dollar-denominated bond in the U.S.

Dollar-denominated foreign bonds are called Yankee bondsYen-denominated foreign bonds are called Samurai bondsPound-denominated foreign bonds are called Bulldog bonds

Eurobonds (歐洲債券 )– Issued in the currency of one nation but sold in other

national markets– Eurodollar bonds are dollar-denominated bonds sold

outside the U.S. (not only in Europe)Euroyen bonds are yen-denominated bonds sold outside JapanEurosterling bonds are pound-denominated bonds sold outside the U.K.

10-16

Innovations in the Bond MarketInverse floaters (反向浮動利率型債券 )– Similar to floating rate bonds except that the coupon

rate on these bond falls when the prevailing interest rate rises

– Benefit (suffer) doubly when rates fall (rise)Asset-backed securities (資產基礎證劵 )– Serviced by the income from a specified group of

assets, like mortgage, auto, or credit card loansA singer can issue ABS to borrow money by linking the coupon rate with the selling amount of his albums

Pay-in-kind bonds (實物支付型債券 )– Issuers of pay-in-kind bonds may choose to pay

interest either in cash or in newly issued bonds if they are short of cash

10-17

Innovations in the Bond MarketCatastrophe bonds (巨災債券 )– Catastrophe: earthquake, hurricane, flood, etc– Holders receive higher coupon rates, but in the

event of a catastrophe, the holders will give up all or part of their principal

– This bond provides a way to transfer the catastrophe risk from insurance companies to the capital market

Indexed bonds (追蹤指數型債券 )– Indexed bonds make payments that are tied to a

general price index or the price of a particular commodity

Mexico had issued bonds with payments depending on the price of oilThe U.S. started issuing TIPS (Treasury Inflation Protected Securities) since 1997

10-18

Innovations in the Bond Market– The par value of the TIPS is adjusted according to

the growth rate of the consumer price index (CPI)– Interest incomes of TIPS thus increase in proportion to the

CPI and bring holders the same level of purchasing power– RORs on TIPS are real risk-free RORs

– A TIPS with three-year maturity, $1,000 par value, and 4% coupon rate paid annually

Interest + Price appreciation 40.80 + 20 42.02 + 30.6Nominal ROR = 6.08% ( 7.12%)Initial price 1000 1020

1 + Nominal ROR 1.0608 1.0712Real ROR = 1 1 4% ( 1 4%)1 + Inflation 1.02 1.03

10-19

10.2 BOND PRICING

10-20

Bond Pricing

PB = theoretical value of the bondCt = interest or coupon payments at period tT = number of periods to maturityr = one-period discount rate (折現率 ) or one-period yield to

maturity (到期收益率 ) or one-period discount yield (折現收益率 )

1

Present value of coupons + Present value of par valuePar value

(1 ) (1 )

B

Ttt T

t

PCr r

※If Ct’s are constant, the above summation term represents a geometric series (等比級數 ), which can be evaluated as follows

1

Par value 1 1 11 Par value(1 ) (1 ) (1 ) (1 )Annuity factor( , ) + Par value PV factor( , )

T

B t T T Tt

CP Cr r r r r

C r T r T

※Annuity factor(r, T) (年金因子 ) is the sum of the PVs of $1 for each of T periods※PV factor(r, T) (現值因子 ) is the PV of $1 at the end of T periods

10-21

Price of 8%, 30-yr. with yield at 10%Coupon = 8% × ½ × 1,000 = 40 (semiannual or per period)

Discount Rate = 10% × ½ = 5% (semiannual or per period)

Maturity = 30 years = 60 periods

Par Value = 1,000

60

601

$40 $1,000(1.05) (1.05)

$40 Annuity factor(5%, 60)+$1,000 PV factor(5%, 60)$810.71 (< par value)

B tt

P

10-22

Yield at 6% or 8%Discount Rate = 3% or 4% (semiannual or per period)

60

601

$40 $1,000(1.03) (1.03)

$40 Annuity factor(3%, 60)+$1,000 PV factor(3%, 60)=$1,276.76 (> par value)

B tt

P

60

601

$40 $1,000(1.04) (1.04)

$40 Annuity factor(4%, 60)+$1,000 PV factor(4%, 60)=$1,000 (= par value)

B tt

P

discount yield coupon rate, par value discount bond ( )discount yield coupon rate, par value discount yield coupon rate, par value premium bo

B

B

B

PPP

折價債券

nd ( )

溢價債券

※

10-23

The Inverse Relationship Between Bond Prices and Yields

※ Bond prices and discount yields (or discount interest rates or required rates of return) have an inverse relationship

※ When yields become high, the value of the bond will be low ※ When yields approach zero, the value of the bond approaches the sum of all promised cash flows

10-24

Price of Perpetual BondPerpetual bond (永續債券 )– A bond with no maturity date, which means it pays a

steady stream of coupon interest forever– Set T to be infinity in the bond pricing formula, which

implies a permanent series of constant cash flows and the par value will never be paid off

– As a result, the price of a perpetual bond equals C / r, where C is the coupon payment per period and r is the one-period discount rate or one-period yield

1

Par value 1 1 11 Par value(1 ) (1 ) (1 ) (1 )1 Par value 0

B tt

CP Cr r r r r

CCr r

10-25

Valuing Bonds using Excel

※ PRICE(settlement date, maturity date, annual coupon rate, yield to maturity, redemption value as percent of par value, number of coupon payments per year, different day count convention): return the quoted (or clean) price

※ COUPDAYBS(settlement date, maturity date, number of coupon payments per year, different day count convention): return the number of days from the beginning of the coupon period to the settlement date

※ COUPDAYS(settlement date, maturity date, number of coupon payments per year, different day count convention): return the number of days in the coupon period that contains the settlement date

※ Different day count convention (計日慣例 ): 0 is 30/360 in USA (for corporate bonds); 1 is actual/actual (for T-bonds); 2 is actual/360; 3 is actual/365; 4 is 30/360 in Europe)

10-26

10.3 BOND YIELDS

10-27

Yield to MaturityYield to maturity (YTM) (到期收益率 ) is the discount rate that makes the present value of a bond’s payments equal to its market price– 8% coupon rate, 30-year bond sold at $1,276.76:

– YTM can be interpreted as the average rate of return if the bond is purchased today and held until maturity

– Note that the financial press reports yield on an annualized basis, and annualizes the bond’s semiannual yield using the simple interest approach, resulting an annual percentage rate (APR)

– Yields annualized using the simple interest approach are also called bond equivalent yields (債券等值收益率 )

60

601

$40 $1,000$1,276.76(1 ) (1 )

3% per period or 6% annually

tt r r

r

10-28

Use Excel to calculate YTM

※ YIELD(settlement date, maturity date, annual coupon rate, bond (clean) price, redemption value as percent of par value, number of coupon payments per year, different day count convention)

※ The accurate formula in the above example should be “=YIELD(B3, B4, B5, B6, B7, B8, 1)”

10-29

Alternative Measures of YieldCurrent Yield (當期收益率 )– The ratio of the bond’s annual coupon payment

over the bond price, e.g., $80/$1,276.76 = 6.27% in the example on Slide 10-27

– For premium bonds, coupon rate > current yield > YTM

coupon rate > current yield: because the coupon payments are divided by the par value for the coupon rate and by the market value for the current yieldcurrent yield > YTM: because the YTM reflects the “build-in” capital loss for premium bonds that the bond price will eventually fall to $1,000 at maturity, but the current yield does not account for this capital loss (see Slide 10-35)

– For discount bonds, coupon rate < current yield < YTM

10-30

Alternative Measures of YieldYield to Call (YTC, 贖回收益率 )– Similar to YTM except that call price replaces the

par value and call date replaces the maturity date

※ Holders of premium bonds are often more interested in the bond’s YTC rather than YTM, because the bond price is high and it is with higher probability to be retired on the call date

10-31

Price-yield relationship for Callable and Straight Bonds

※ At high rates, the probability of calling back is negligible because the present value of the bond is less than the call price, which is $1,100

※ At low rates, the present value of the bond exceeds the call price, so the bond is possible to be called back

※ If we assume the firm will redeem the bond as soon as it can do so, the bond price will never exceed the call price

10-32

Improper Implication of YTM on the Reinvestment Rate (再投資收益率 )

2

2

For case A: $1,000(1 RCR) $1,210 RCR 10%(With a reinvestment rate equal to the 10%, the realized compound return (RCR) equals YTM)

For case B: $1,000(1 RCR) $1,208 RCR 9.9

1%

※ The above example highlights the problem with the conventional YTM, in which all coupons are assumed to be reinvested at the bond’s yield to maturity

※ Realized compound return (實現複合報酬率 ) (RCR): compound rate of return on a bond with all coupons reinvested until maturity, i.e., bond price×(1+RCR)n = sum of all future values of coupon and principal payments after reinvestment

2

2

The bond price today is $1,000, implying a YTM of 10%,$100 $1,100i.e., $1,0001.1 (1.1)

$1,000(1.1) $100(1.1) $1,100=$1,210

10-33

Alternative Measures of YieldHorizon analysis (水平分析 ) to calculate the RCR– The forecast of total payoff depends on the

forecasted YTM of the bond when you sell it and the rate at which you are able to reinvest coupon income

10-34

10.4 BOND PRICES OVER TIME

10-35

Premium and Discount BondsPremium bond: coupon rate > discount rate– “Built-in” capital losses (內置資本損失 ): bond price

will decline to par over its life periodDiscount bond: coupon rate < discount rate– “Built-in” capital gains (內置資本利得 ): bond price

will increase to par over its life period ※Because fewer of these above-market (below-market)

payments remains for premium (discount) bonds

10-36

Premium and Discount BondsExpected holding-period return:– A 3-year bond, 7% (9%) coupon rate with annual

payments, and the discount yield in the market is 8%

– The expected HPR for discount and premium bonds

”※ Built in” capital gains (losses) of a discount (premium) bond compensate (offset) its low (high) coupon rates

※Both premium and discount bonds offer the same expected HPR, which equals the yield to maturity

※The reason is that a bond must offer a yield competitive with other bonds with the same level of credit risk

At =0: $70 ($90) Annuity factor(8%, 3) + $1,000 PV factor(8%, 3) = $974.23 ($1,025.77)t

At =1: $70 ($90) Annuity factor(8%, 2) + $1,000 PV factor(8%, 2) = $982.17 ($1,017.83)t

($70 $982.17 $974.23) / $974.23 8%($90 $1,017.83 $1,025.77) / $1,025.77 8%

10-37

OID Bonds and Treasury STRIPSOriginal issue discount bonds (OID bonds, 折價發行債券 ) are those bonds sold at a discount at issuance– The zero-coupon bond is the most extreme OID bond

T-bills are the most important short-term (shorter than 1 year) zero-coupon bondsLonger-term zero-coupon bonds are commonly created from coupon-bearing notes and bonds

– A broker can ask the U.S. Department of Treasury to break down a 10-year coupon bond into its 20 semiannual coupon payment and the final principal repayment, each of which can be viewed as a zero-coupon bond

– The Treasury program for coupon stripping is called STRIPS (Separate Trading of Registered Interest and Principal of Securities, 本息分離債券 ), and these zero-coupon securities are called Treasury strips

10-38

Tax issues about the OID bondsThe tax authorities recognize that the “built-in” price appreciation on OID bonds represents an implicit interest payment to the holder of the bond– The forecasted price appreciation, treated as interest

income, is subject to personal or corporate taxes– Gains or losses deviated from the forecasted price

appreciation are treated as capital gains or losses

10-39

Tax issues about the OID bonds

– Internal Revenue Service (IRS, 國家稅務局 ) is the United States federal government agency that collects taxes and enforces the tax laws

– The above taxation rule is not only for zero-coupon bonds but also for all OID bonds

– IRS always applies the “constant yield method” to calculate the price appreciation schedule for OID bonds

10-40

10.5 DEFAULT RISK AND BOND PRICING

10-41

Default Risk and RatingsDefault risk (DR, 違約風險 ): If issuers go bankruptcy, bondholders will not receive all the payments they have been promised– Due to the DR, the fix-income securities are not so

safe as this name suggests– Credit rating agencies (信用風險評等機構 )

Moody’s Investor Service, Standard & Poor’s, Fitch– Common factors used by rating agencies

Coverage ratio (earnings to interest costs) ↑ DR ↓Leverage ratio (debt-to-equity ratio) ↑ DR ↑Liquidity ratio (current ratio: current assets/current liabilities or quick ratio: (current assets – inventories)/current liabilities) ↑ DR ↓Profitability ratios (ROA and ROE) ↑ DR ↓Cash flow-to-debt ratio (total cash flow to outstanding debt) ↑ DR ↓

10-42

Default Risk and RatingsNotation Systems

Moody's S&P and FitchAaa AAAAa1 AA+Aa2 AAAa3 AA-A1 A+A2 AA3 A-

Baa1 BBB+Baa2 BBBBaa3 BBB-Ba1 BB+Ba2 BBBa3 BB-B1 B+B2 BB3 B-Caa CCCCaa CCC CD D

10-43

Financial Ratios and Default Risk

Default Risk and Ratings

10-44

Protection Against DefaultSinking funds (償債準備基金條款 )– The payment of the par value at maturity generates

the cash pressure for firms– Issuing firms may agree to establish a sinking fund to

spread the payment burden over several yearsThe fund can be used to repurchase a fraction of outstanding bonds in the open market each yearFirms may repurchase a fraction of outstanding bonds, at the lower of the market price and the sinking fund call price (Hurts bond holders: firms choose to buy back discount bonds at their market price and premium bonds at the sinking fund call price)

– Different from the call provisionFor sinking funds, a fraction of outstanding bonds can be repurchased regardless they are at premiums or discountsUsually the sinking fund call price is set at par, whereas the call prices are above the par value in general

10-45

Protection Against Default– Serial bond issue: firms sell bonds with sequential

maturity dates such that the principal repayment burden is spread over time just like the sinking fund scheme

Dividend restrictions (股利限制條款 )– One commonly used dividend payment restriction is

that the dividend distribution is allowed only when the firm makes profits (after the payments of interest expenses) and the maximum distributed dividend cannot exceed the profits

– It protects bondholders because it forces the firm to retain assets rather than pay them out to stockholders

10-46

Protection Against DefaultSubordination of future debt (後續借款次順位求償條款 )– Bondholders do not like the issuing firm to raise

additional debts because more outstanding debts implies higher default probability

– Subordination clauses (次級條款 ) require additional debts to be subordinated in priority to existing debt (senior bonds) (高級或是優先債券 )

Later bonds are called junior bonds (低級或是非優先債券 )– Subordination is sometimes called a “me-first rule,”

meaning the senior (earlier) bondholders are to be paid first in the event of bankruptcy

10-47

Protection Against DefaultCollateral (擔保條款 )– Collateral is a specific asset pledged (抵押 ) against

possible default on a bondIf the issuer defaults, this specific asset is sold and the sale proceeds are used exclusively to liquidate (償付 ) the bond

– A bond backed by collateral is called a secured bond (有擔保債券 ), whereas the unsecured bond (無擔保債券 ), also called debenture (信用債券 ), by contrast do not provide specific collateral

If the collateral is a property, the bond is called the mortgage bond (不動產抵押債券 )If the collateral is other securities held by the firm, the bond is a collateral trust bond (擔保信託債券 )In the case of equipments, the bond is known as an equipment obligation bond (設備責任債券 )

10-48

YTM and Default RiskPromised YTM vs. Expected YTM– The promised (or stated) YTM is calculated under the

assumption that firms meet the obligations of bonds– The expected YTM is based on expected cash flows,

which may be far less than the promised cash flowsTo estimate expected cash flows, both the default probability (or the default time point) and the recovery rate (回復率 ) in the event of default should be considered

10-49

Yield Spreads (Default Premium) on Long-Term Bonds

※ The default premium (違約溢酬 ) is defined as the yield spread (收益率利差 ) between the promised YTM of a corporate bond and that of a comparable Treasury bond

※ The above figure suggests that when a bond comes more likely subject to default risk, its default premium is higher to reflect the higher tendency to default

※ The default premium of bonds increases during the crises, especially for junk bonds because in the crises, investors lose confidence of the ability of the issuers of junk bonds to fulfill the obligation

Subprime Crisis and Financial Tsunami

10-50

Credit Default Swaps (CDS)Credit Default Swaps (信用違約交換 ) (CDS) is an insurance policy on the default risk of a corporate bond or loan– Invented by JPMorgan in 1997, CDSs were designed to

shift the default risk from the protection buyer (保護買方 ) to the protection seller (保護賣方 ) (see the next slide)

– CDSs are traded in OTC markets– Price of CDS (or called CDS spread (信用違約交換利差 )): If a BBB-rated bond + a CDS a AAA-rated

(assumed to be the rating of the protection seller) bond, the fair price of this CDS ought to approximately the yield spread between AAA-rated and BBB-rated bonds

– Thus, during the subprime crisis and financial tsunami, the yield spreads of corporate bonds as well as the CDS spreads skyrocketed

10-51

Credit Default Swaps (CDS)

※ When the default event occurs, the protection seller should compensate the protection buyer any losses in the default event

※ For the protection buyer, CDS provides insurance against the possibility that a borrower might not pay

※ For the protection seller, CDS provides a way to earn profits by bearing default risk without ever holding the credit instrument itself

10-52

Credit Default Swaps (CDS)CDSs can be used to speculate (投機 ) on financial health of firms– CDS buyers need not physically hold the underlying bond

or loan, i.e., participants simply have a viewpoint but do not need to have any actual credit exposure

– The total size of outstanding CDS contracts reaches a peak of $63 trillion before the credit crisis (US GDP is about $14 trillion per year)

Possible reform on CDSs to limit risk: Trade with collateral requirements on exchanges, that also can increase transparency of positions of investors

10-53

10.6 THE YIELD CURVE

10-54

Term Structure of Interest RatesTerm structure of interest rates (利率的期間結構 ): Relationship between yields to maturity and terms to maturityYield curve (收益率曲線 )–a graph of the yields (YTMs) on bonds w.r.t. different years to maturity– Flat, rising, inverted, and hump-shaped yield curves

(see figures on the next slide)– The rising yield curve is also called the normal yield

curve because it is most commonly observed: Long-term yield > short-term yield(Long-term yield – short-term yield) is also known as term spread (期間利差 )

The explanations for the term structure– The expectations theory (期望理論 )– The liquidity preference theory (流動性偏好理論 )

10-55

Treasury Yield Curves

※ For rising yield curves, they are upward sloping, i.e., bonds with shorter maturities generally offer lower YTM than longer-maturity bonds

※ On the contrary, the inverted yield curve is downward sloping (usually interpreted as a signal of a coming recession)

※ The hump-shaped (駝峰型 ) yield curve is first rising and then falling along the maturity dimension

10-56

The Expectations TheoryExpectations Theory (期望理論 )– Upward (downward) sloping yield curves indicate

that the market is expecting higher (lower) future short term rates (see Slide 10-56)

– The expectations theory asserts that yields to maturity are determined solely by expectations about future short-term interest rates

– We cannot directly observe the expectation of future short-term interest rates, but we can observe yields of different maturities today and then infer the market’s expectation of future short-term rates, which is also called the (implied) forward rate ((隱含 ) 遠期利率 ) (see Slide 10-57)

10-57

Returns to Two 2-year Investment Strategies

※ r1 and r2 are short-term interest rates for the first and the second year

※ y1 and y2 are yields with maturities of one year and two years, respectively

※ By definition, r1 equals the YTM for the 1-year investment horizon, y1

※ The above example demonstrates numerically that if the market expects higher future short-term rates, i.e., E(r2) > r1, the yield curve is with upward sloping, i.e., y2 > y1

10-58

Forward Rates Implied in the Yield Curve

)1403.1()11.1()12.1()1()1()1(

23

1

1=

–= +++ –

fyy nnnnn

For example, using a 2-yr and 3-yr yields

Longer term yield, y3 = 12%

Shorter term yield, y2 = 11%

Forward rate f3, a one-year rate after two years = 14.03%

– The formula to calculate the one-year (implied) forward rate after (n–1) years

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The Liquidity Preference TheoryLiquidity Preference Theory (流動性偏好理論 )– Lenders prefer greater liquidity provided by short-

term bonds even if they offer relatively lower expected return (So, lenders will demand the liquidity premium (流動性溢酬 ) and thus the higher yield to hold longer-term bonds)

– For borrowers, they prefer to issuing long-term bonds so as to eliminate the re-borrowing interest rate risk and thus are willing to pay higher yields on these bonds

Synthesis of the expectations and liquidity preference theories could explain different shapes of the term structures of interest rates, especially for hump-shaped yield curves

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Illustrative Yield Curves ※ The expected rising interest

rates, plus a liquidity premium, makes the yield curve more steeply upward-sloping

※ The expected falling interest rates makes the yield curve slope downward

※ Together with the liquidity preference theory, the net effect of these two opposite forces could form a “hump-shaped” curve