課程五：債券市場

本講義僅供上課教學之用。

Bonds are instruments in which issuer promises to repay to lender original principal (the face value or par value) plus interest at an annual coupon rate.

係指政府、企業界或金融機構為籌措資金而發行的可轉讓債權證書，在證書上載明由發行人於到期日按面額償還本金，且除貼現債券外，須定期支付利息。

Bonds 債券

債券依發行單位不同可分類為：1. 中央政府公債：中央政府發行2. 省 ( 市 ) 政府公債： 地方政府發行。3. 金融債券：專業銀行發行。4. 公司債：公開發行公司發行。5. 外國債券：外國金融機構來台發行。 依發行的憑證可區分為：1. 實體債券：發行債券時，以交付實體債票作為債權憑

證者。 2. 登錄公債：中央政府發行公債時，委託清算銀行以電

腦登記相關權利資料，並發給公債存摺，取代交付實體債票，以此方式發行之公債稱之為登錄公債。自 87會計年度起，中央公債均以此方式發行。

Size of Bond Markets

Security Type Market Value ($b, 91)U.S. Treasuries $1,606.8U.S. Agencies $ 256.9State/Local Govt $ 850.0Corporate $ 605.4Mortgages (unsecured) $1,432.3Mortgage-backed $1,043.1Commercial Mortgages $ 733.9Asset-backed $ 73.1TOTAL $6,601.5

債市週報 ( 範例 ) 債券市場方面，由於債券型基金申購回流，利率再度走跌，週五 RP以 2.65%作收，較前週下滑 0.2%。 10天期及 30天期客戶掛牌利率分別介於 2.3％ ~2.4%和 2.5%~2.6%左右。公債買賣斷方面，由於受到經發會和籌碼面利空干擾，不過在短率下跌和投資性買盤進場下，使得殖利率呈開高走低，上週五指標 20年券「央債 90-2」以 4.17％作收，較前週下跌 13.5BP，15年券「央債 90-3」以 4.18％作收，較前週下跌 14BP， 10年券「央債 90-1」殖利率以 3.8%作收，較前週下跌 22.4BP。

Bond Features and Innovations Call provisions 贖回條款 Conversion options 轉換權利 Puttable bonds Floaters 浮動利率債券 International bonds Reverse floaters Indexed bonds

Bond Prices and Yields 債券評價 The price of a bond (B) is determined by

its principal (FV), the coupon payment (C), maturity periods (T) and the underlying yield (y):

T

tTt y

FV

y

CB

1 )1()1(

Bond Pricing Example

Consider an 8% coupon bond with 10 years to maturity. If the discount rate is 10%, what is its price? If the discount rate is 8%? 6%?

What if the bond had 20 years to maturity?

What if the bond had only a 6% coupon?

Bond Price Example

8 % Semiannual Coupon Bond

0.70.80.9

11.11.21.31.4

Years to Maturity

Pri

ce ($

1 P

ar)

10% yield

8% yield

6% yield

Bond Prices Over Time

900

950

1000

1050

1100

Fixed Income Management 固定收益證券管理

As with equity portfolios, we can have active or passive management.Active management either attempts to use interest rate forecasts to predict general market wide movements or analysis of individual securities (or sectors).Passive management takes market prices as fairly set and attempts to maintain risk-return balance. We especially want to insulate the portfolio from interest rate risk.

Interest Rate Sensitivity 敏感性分析 Bond prices and yields are inversely

related. An increase in YTM results in a smaller price

decline than the price gain associated with a similar decrease in YTM (convexity).

Long-term bonds have higher sensitivities than short-term bonds.

Interest rate risk increases at a decreasing rate with maturity.

Low coupon bonds have higher sensitivities than high coupon bonds.

Interest Rate Sensitivity

Coupon T=1 T=10 T=20

8 % -0.92% -6.42% -9.13%

Zero -0.92% -8.80% -16.84%

Consider two bonds when YTM changes from 8% to 9%. What is the effect on the prices of the bonds?

Interest Rate Sensitivity

Long maturity bonds are more sensitive to yield changes than short maturity bonds.

Size of coupon payments also important. How can we determine the relative

importance of these components? We need some measure which accounts

for the fact that we get money at different times.

We need a measure of the effective maturity of the bond.

Duration 存續期間 Duration recognizes that each of the

individual payments (whether coupon or interest) is important and attempts give each payment its relative importance.

Duration measures the effective maturity of a bond by taking a weighted average of the time until each payment is made.

The weights are the proportion of the bond price that the present value of each payment represents.

Duration

B

yP

TyC

t

DT

T

tt

)1()1(1

w

CFy

t

tt

( )1

B o n d P riceD t w

t

T

t

1

Duration Example Consider a 9% annual coupon bond with 5

years to maturity and current yield of 8%. What is its duration (in years)?

Duration and Interest Rate Risk Duration is useful because it generalizes the idea

that longer bonds are more sensitive to rate changes than shorter bonds. Duration gives us a quantitative measure of interest rate risk.

Duration makes a linear approximation to the bond price’s sensitivity to yield changes

Current yields change on our 5 year bond from 8% to 8.25%, what is the approximate bond price change?

B

BD

y

y

1

Duration Rules The duration of a zero coupon bond equals

its time to maturity. Duration is higher when the coupon rate is

lower. Duration generally increases with time to

maturity. Duration is higher when YTM is lower. Duration of a perpetuity is (1+y)/y.

Duration Example

Time 6% 8% 10% 12%

1 1.000 1.000 1.000 1.000

5 4.439 4.312 4.204 4.11010 7.615 7.247 6.996 6.744

20 11.231 10.604 10.182 9.880 13.500 13.500 13.500 13.500

Assume annual coupon payments

Net Worth Immunization 免疫策略 Many institutions try to insulate the value

of their portfolios from interest rate risk. Banks/Thrifts engage in “gap

management” to hedge low-duration liabilities (deposits) against high-duration assets (loans) to insulate net worth against interest rates:

D V D Vassets assets liabs liabs 0

Immunization Example You will be paying $10,000 per year in

tuition and room and board at the end of the next two years. Bonds currently yield 8%. What is the present value and duration

of your obligation? What maturity zero coupon bonds would

immunize your obligation? Suppose you buy the above zero. What

happens to your net position if rates immediately increase to 9%? What if they immediately fall to 7%?

Suppose you could only buy 1 year and 2 year zeros, could you construct a portfolio to immunize your obligation?

Convexity 債券凸性

We have seen that using duration to predict price changes in bonds for a given change in interest rates leads to a linear relationship

However, we know that the true relationship between changes in rates and changes in prices is non-linear. It is curved. We call the curvature of the price-yield relationship the convexity of the bond.

B

BD

y

y

1

Duration and Interest Rate Risk

Bond Price vs Yield

55

75

95

115

135

155

175

6 7 8 9 10 11 12 13 14Yield(%)

Bo

nd

Pri

ce (

% o

f P

ar)

Actual

Dur at io n Es t .

Interest Rate Swaps

Swaps have become a major fixed-income tool to manage interest rate risk (over $5 trillion).

Usually two parties with different exposure to interest rate risk will engage in swap:

S&L with short-term floating (variable) rate deposits and long-term fixed-rate mortgages (assets).

Corporation with long-term non-callable fixed rate bonds and short-term floating (variable) rate assets.

If rates rise, S&L loses; if rates fall, corporation loses. How do they both win?

Interest Rate Swaps

“Plain vanilla swap” works if S&L agrees to make fixed rate payments to corporation on some notional amount and fixed rate AND corporation pays at agreed-upon short-term rate (ex. LIBOR) on notional amount to S&L.

Swaps provide a means to restructure balance sheets and manage risk cheaply.

Reinvestment Risk 再投資風險Problem: A bank makes a $100,000 installment loan at annual interest rate of 10%. The contract specifies: 1) interest to be compounded annually and 2)repayment to occur in three equal installments over the next three years, then what is the annual installment payment needed to amortize the loan?What are the cash flows over the three-year period for:a) stable interest rates, r1 = r2 = 10%b) rising interest rates r1 = 11%, r2 = 12%c) falling interest rates r1 = 9%, r2 = 8%

Market Value Risk

The bank want to sell the loan to another bank after two years. The value of the loan after two years, V2, will depend on the prevailing interest rate on one year loans, i.e. r2, as follows

V2 = $40211.48 / (1+r2)r2 V2

Stable 10%Rising 12%Falling 8%

Credit Risk

Consider a one year, pure discount bond with a $100 face value.Probability distribution of payoffsState Probability PayoffGrowth .80 $100Mild Recession .10 $95Moderate .05 $90Severe .05 $85What is the expected payment in one year?E = .8(100) + .1(95) + .05(90) + .05(85)

= $98.25

Default Risk

Default Risk — the chance that a borrower will not be able to make promised payments. Counterparty Risk-- the possibility of default in on

e firm will cause defaults in other firms/institutions holding its assets. This will depend on the exposure of an institution to one type of firm as well as the probability of default.

The difference in promised interest rates between otherwise identical securities (differing only in risk) is known as credit spread. For example, a lower rated bond such as a BBB has higher yields than a AAA bond with the same cash flows.

Measuring Default Risk: Analysts’ Assessments

Credit Rating Agencies

Four firms: Moody’s, Standard & Poor’s, Duff & Phelps, Fitch

Evaluate default risk of firms and rank. Investment grade, speculative grade, high-yield or junk bond

Operating leverage refers to the cost structure of a business and the degree to which fixed costs vs. variable costs dominate a firm.

Higher fixed costs imply the firm is more susceptible to fluctuations in revenues.

A measure of a firm’s debt vs. equity is known as financial leverage.

Higher debt/equity ratio implies the firm is more susceptible to fluctuations in earnings.

Measuring Default Risk: Using Historical Data

One can look at which bonds have defaulted in the past and try to estimate the probabilities of default for the different ranked bonds.

Data is almost impossible to get. Many institutions have the data from historical relationships with customers.

The default probabilities often depend on the horizon you intend to hold the asset. A longer horizon often means there is a larger chance of default and it also means the estimate of the default is more difficult to interpret because of uncertainties in the future.

You are also interested in whether there was an event that lead to or contributed to the default. Default probabilities may change under certain macroeconomic conditions.

Illiquidity Risk 流動性風險 A liquid financial instrument can be converted to c

ash quickly, in sizable volume, without influencing the price appreciably.

One indication of high liquidity is a narrow bid-ask spread.

Bid-ask spreads tend to be larger on longer maturity and higher risk assets