Fi8000 Bonds, Interest Rates Fixed Income Portfolios Milind Shrikhande.

Fi8000Fi8000Bonds, Interest RatesBonds, Interest Rates

Fixed Income PortfoliosFixed Income Portfolios

Milind ShrikhandeMilind Shrikhande

Debt instrumentsDebt instruments

☺Types of bondsTypes of bonds

☺Ratings of bonds (default risk)Ratings of bonds (default risk)

☺Spot and forward interest ratesSpot and forward interest rates

☺The yield curveThe yield curve

☺DurationDuration

Bond CharacteristicsBond Characteristics

☺ A bond is a security issued to the lender (buyer) by the A bond is a security issued to the lender (buyer) by the borrower (seller) for some amount of cash.borrower (seller) for some amount of cash.

☺ The bond obligates the issuer to make specified The bond obligates the issuer to make specified payments of interest and principal to the lender, on payments of interest and principal to the lender, on specified dates.specified dates.

☺ The typical The typical coupon bondcoupon bond obligates the issuer to make obligates the issuer to make coupon payments, which are determined by the coupon payments, which are determined by the coupon coupon raterate as a percentage of the as a percentage of the par valuepar value ( (face valueface value). ). When the bond matures, the issuer repays the par value.When the bond matures, the issuer repays the par value.

☺ Zero-coupon bondsZero-coupon bonds are issued at discount (sold for a are issued at discount (sold for a price below par value), make no coupon payments and price below par value), make no coupon payments and pay the par value at the maturity date.pay the par value at the maturity date.

Bond Pricing - ExamplesBond Pricing - Examples

☺ The par value of a risk-free zero coupon bond is The par value of a risk-free zero coupon bond is $100. If the continuously compounded risk-free $100. If the continuously compounded risk-free rate is 4% per annum and the bond matures in rate is 4% per annum and the bond matures in three months, what is the price of the bond three months, what is the price of the bond today? - $99.005today? - $99.005

☺ A risky bond with par value of $1,000 has an A risky bond with par value of $1,000 has an annual coupon rate of 8% with semiannual annual coupon rate of 8% with semiannual installments. If the bond matures 10 year from installments. If the bond matures 10 year from now and the risk-adjusted cost of capital is 10% now and the risk-adjusted cost of capital is 10% per annum compounded semiannually, what is per annum compounded semiannually, what is the price of the bond today? - $875.3779the price of the bond today? - $875.3779

Yield to Maturity - ExamplesYield to Maturity - Examples

☺ What is the What is the yield to maturityyield to maturity (annual, (annual, compounded semiannually) of the risky coupon-compounded semiannually) of the risky coupon-bond, if it is selling at $1,200? – 5.387%bond, if it is selling at $1,200? – 5.387%

☺ What is the What is the expected yield to maturityexpected yield to maturity of the of the risky coupon-bond, if we are certain that the risky coupon-bond, if we are certain that the issuer is able to make all coupon payments but issuer is able to make all coupon payments but we are uncertain about his ability to pay the par we are uncertain about his ability to pay the par value. We believe that he will pay it all with value. We believe that he will pay it all with probability 0.6, pay only $800 with probability probability 0.6, pay only $800 with probability 0.35 and won’t be able to pay at all with 0.35 and won’t be able to pay at all with probability 0.05. – 4.529%probability 0.05. – 4.529%

Default Risk and Bond RatingDefault Risk and Bond Rating

☺ Although bonds generally promise a fixed flow of income, Although bonds generally promise a fixed flow of income, in most cases this cash-flow stream is uncertain since in most cases this cash-flow stream is uncertain since the issuer may default on his obligation.the issuer may default on his obligation.

☺ US government bonds are usually treated as free of US government bonds are usually treated as free of default (credit) risk. Corporate and municipal bonds are default (credit) risk. Corporate and municipal bonds are considered risky.considered risky.

☺ Providers of bond quality rating:Providers of bond quality rating:☺ Moody’s Investor ServicesMoody’s Investor Services☺ Standard and Poor’s CorporationStandard and Poor’s Corporation☺ Duff & PhelpsDuff & Phelps☺ Fitch Investor ServiceFitch Investor Service

Default Risk and Bond RatingDefault Risk and Bond Rating

☺ AAA (Aaa) is the top rating.AAA (Aaa) is the top rating.☺ Bonds rated BBB (Baa) and above are Bonds rated BBB (Baa) and above are

considered considered investment-grade bondsinvestment-grade bonds..☺ Bonds rated lower than BBB are considered Bonds rated lower than BBB are considered

speculative-gradespeculative-grade or or junk bondsjunk bonds..☺ Risky bonds offer a risk-premium. The greater Risky bonds offer a risk-premium. The greater

the default risk the higher the the default risk the higher the default risk-default risk-premiumpremium..

☺ The The yield spreadyield spread is the difference between the is the difference between the yield to maturity of high and lower grade bond. yield to maturity of high and lower grade bond.

Estimation of Default RiskEstimation of Default Risk

☺ The determinants of the The determinants of the bond default riskbond default risk (the (the probability of bankruptcy) and probability of bankruptcy) and debt quality ratingsdebt quality ratings are based on measures of financial stability:are based on measures of financial stability:

☺ Ratios of earnings to fixed costs;Ratios of earnings to fixed costs;☺ Leverage ratios;Leverage ratios;☺ Liquidity ratios;Liquidity ratios;☺ Profitability measures;Profitability measures;☺ Cash-flow to debt ratios.Cash-flow to debt ratios.

☺ A complimentary measure is the A complimentary measure is the transition matrixtransition matrix – – estimates the probability of a change in the rating of estimates the probability of a change in the rating of the bond.the bond.

The Term-Structure of Interest RatesThe Term-Structure of Interest Rates

☺ The The short interest rateshort interest rate is the interest rate is the interest rate for a specific time interval (say one year, for a specific time interval (say one year, which does not have to start today).which does not have to start today).

☺ The The yield to maturityyield to maturity ( (spot ratespot rate) is the ) is the internal rate of return (say annual) of a zero internal rate of return (say annual) of a zero coupon bond, that prevails today and coupon bond, that prevails today and corresponds to the maturity of the bond.corresponds to the maturity of the bond.

ExampleExample

In our previous calculations we’ve assumed In our previous calculations we’ve assumed that all the that all the short interest ratesshort interest rates are equal. Let are equal. Let us assume the following:us assume the following:

DateDate Short Short Interest rateInterest rate

For the time intervalFor the time interval

00 rr11 = 8% = 8% t = 0 to t = 1t = 0 to t = 1

11 rr22 = 10% = 10% t = 1 to t = 2t = 1 to t = 2

22 rr33 = 11% = 11% t = 2 to t = 3t = 2 to t = 3

33 rr44 = 11% = 11% t = 3 to t = 4t = 3 to t = 4

ExampleExample

What is the price of the 1, 2, 3 and 4 years What is the price of the 1, 2, 3 and 4 years zero-coupon bonds paying $1,000 at zero-coupon bonds paying $1,000 at maturity?maturity?

MaturityMaturity Zero-Coupon Bond PriceZero-Coupon Bond Price

11 $1,000/1.08 = $925.93$1,000/1.08 = $925.93

22 $1,000/(1.08*1.10) = $841.75$1,000/(1.08*1.10) = $841.75

33 $1,000/(1.08*1.10*1.11) = $758.33$1,000/(1.08*1.10*1.11) = $758.33

44 $1,000/(1.08*1.10*1.11$1,000/(1.08*1.10*1.1122) = $683.18) = $683.18

ExampleExample

What is the yield-to-maturity of the 1, 2, 3 and What is the yield-to-maturity of the 1, 2, 3 and 4 years zero-coupon bonds paying $1,000 at 4 years zero-coupon bonds paying $1,000 at maturity?maturity?

MaturityMaturity PricePrice Yield to MaturityYield to Maturity

11 $925.93$925.93 yy11 = 8.000% = 8.000%

22 $841.75$841.75 yy22 = 8.995% = 8.995%

33 $758.33$758.33 yy33 = 9.660% = 9.660%

44 $683.18$683.18 yy44 = 9.993% = 9.993%


The price of the zero-coupon bond is calculated The price of the zero-coupon bond is calculated using the using the short interest ratesshort interest rates (r (rtt, t = 1,2…,T). For , t = 1,2…,T). For a bond that matures in T years there may be up to a bond that matures in T years there may be up to T different short annual rates.T different short annual rates.

Price = FV / [(1+rPrice = FV / [(1+r11)(1+r)(1+r22)…(1+r)…(1+rTT)])]

The The yield-to-maturityyield-to-maturity (y (yTT) of the zero-coupon ) of the zero-coupon bond that matures in T years, is the internal rate of bond that matures in T years, is the internal rate of return of the bond cash flow stream.return of the bond cash flow stream.

Price = FV / (1+yPrice = FV / (1+yTT))TT


The price of the zero-coupon bond paying $1,000 The price of the zero-coupon bond paying $1,000 in 3 years is calculated using the short term rates:in 3 years is calculated using the short term rates:

Price = $1,000 / [1.08*1.10*1.11] = Price = $1,000 / [1.08*1.10*1.11] = $758.33$758.33

The The yield-to-maturityyield-to-maturity (y (y33) of the zero-coupon ) of the zero-coupon

bond that matures in 3 years solves the equationbond that matures in 3 years solves the equation

$758.33$758.33 = $1,000 / (1+y = $1,000 / (1+y33))33

yy33 = = 9.660%.9.660%.


Thus the Thus the yieldsyields are in fact are in fact geometric geometric averagesaverages of the of the short interest ratesshort interest rates in in each periodeach period

(1+y(1+yTT))TT = (1+r = (1+r11)(1+r)(1+r22)…(1+r)…(1+rTT))

(1+y(1+yTT) = [(1+r) = [(1+r11)(1+r)(1+r22)…(1+r)…(1+rTT)])](1/T)(1/T)

The The yield curveyield curve is a graph of bond yield-to- is a graph of bond yield-to-maturity as a function of time-to-maturity.maturity as a function of time-to-maturity.

The Yield Curve (Example)The Yield Curve (Example)

YTM

Time to Maturity2 4

8.000%

8.995%

9.660%

9.993%

1 3


If we assume that all the If we assume that all the short interest ratesshort interest rates (r (rtt, t , t = 1, 2…,T) are equal, then all the = 1, 2…,T) are equal, then all the yieldsyields (y (yTT) of ) of zero-coupon bonds with different maturities (T = 1, zero-coupon bonds with different maturities (T = 1, 2…) are also equal and the yield curve is flat.2…) are also equal and the yield curve is flat.

A A flatflat yield curve is associated with an expected yield curve is associated with an expected constant interest rates in the future;constant interest rates in the future;An An upward slopingupward sloping yield curve is associated with yield curve is associated with an expected increase in the future interest rates;an expected increase in the future interest rates;A A downward slopingdownward sloping yield curve is associated yield curve is associated with an expected decrease in the future interest with an expected decrease in the future interest rates.rates.

The Forward Interest RateThe Forward Interest Rate

☺ The The yield to maturityyield to maturity is the internal rate of is the internal rate of return of a zero coupon bond, that prevails return of a zero coupon bond, that prevails today and corresponds to the maturity of the today and corresponds to the maturity of the bond.bond.

☺ The The forward interest rateforward interest rate is the rate of return a is the rate of return a borrower will pay the lender, for a specific loan, borrower will pay the lender, for a specific loan, taken at a specific date in the future, for a taken at a specific date in the future, for a specific time period. If the principal and the specific time period. If the principal and the interest are paid at the end of the period, this interest are paid at the end of the period, this loan is equivalent to a forward zero coupon loan is equivalent to a forward zero coupon bond.bond.


Suppose the price of 1-year maturity zero-coupon Suppose the price of 1-year maturity zero-coupon bond with face value $1,000 is $925.93, and the bond with face value $1,000 is $925.93, and the price of the 2-year zero-coupon bond with $1,000 price of the 2-year zero-coupon bond with $1,000 face value is $841.68. face value is $841.68.

If there is no opportunity to make arbitrage profits, If there is no opportunity to make arbitrage profits, what is the 1-year forward interest rate for the what is the 1-year forward interest rate for the second year?second year?

How will you construct a synthetic 1-year forward How will you construct a synthetic 1-year forward zero-coupon bond (loan of $1,000) that zero-coupon bond (loan of $1,000) that commences at t = 1 and matures at t = 2?commences at t = 1 and matures at t = 2?


If there is no opportunity to make arbitrage profits, If there is no opportunity to make arbitrage profits, the 1-year forward interest rate for the second year the 1-year forward interest rate for the second year must be the solution of the following equation:must be the solution of the following equation:

(1+y(1+y22))22 = (1+y = (1+y11)(1+f)(1+f22),),

wherewhere

yyTT = yield to maturity of a T-year zero-coupon bond = yield to maturity of a T-year zero-coupon bond

fftt = = 1-year forward rate for year t1-year forward rate for year t


In our example, In our example, yy11 = 8% and y = 8% and y22 = 9%. Thus, = 9%. Thus,

(1+0.09)(1+0.09)22 = (1+0.08)(1+f = (1+0.08)(1+f22))

ff22 = 0.1 = 10%. = 0.1 = 10%.

Constructing the loan (borrowing):Constructing the loan (borrowing):

1. Time t = 0 CF should be zero;1. Time t = 0 CF should be zero;

2. Time t = 1 CF should be +$1,000;2. Time t = 1 CF should be +$1,000;

3. Time t = 2 CF should be -$1,000(1+f3. Time t = 2 CF should be -$1,000(1+f22) = -$1,100.) = -$1,100.


Constructing the loan:Constructing the loan:we would like to borrow $1,000 a year from now we would like to borrow $1,000 a year from now for a forward interest rate of 10%.for a forward interest rate of 10%.

1.1. (#3) CF(#3) CF00 = $925.93 but it should be zero. We offset that = $925.93 but it should be zero. We offset that cash flow if we buy the 1-year zero coupon bond for cash flow if we buy the 1-year zero coupon bond for $925.93. That is, if we buy $925.93/$925.93 = 1 units of $925.93. That is, if we buy $925.93/$925.93 = 1 units of the 1-year zero coupon bond;the 1-year zero coupon bond;

2.2. (#1) CF(#1) CF11 should be equal to $1,000; should be equal to $1,000;

3.3. (#2) CF(#2) CF22 = -$1,000*1.1 = -$1,100. We generate that cash = -$1,000*1.1 = -$1,100. We generate that cash flow if we sell 1.1 units of the 2-year zero-coupon bond flow if we sell 1.1 units of the 2-year zero-coupon bond for 1.1 * $841.68 = $925.93.for 1.1 * $841.68 = $925.93.

Bond Price SensitivityBond Price Sensitivity

☺ Bond prices and yields are inversely related.Bond prices and yields are inversely related.

☺ Prices of Prices of long-term bondslong-term bonds tend to be more tend to be more sensitive to changes in the interest rate sensitive to changes in the interest rate (required rate of return / cost of capital) than (required rate of return / cost of capital) than those of short-term bonds (compare two zero those of short-term bonds (compare two zero coupon bonds with different maturities).coupon bonds with different maturities).

☺ Prices of Prices of high coupon-ratehigh coupon-rate bondsbonds are less are less sensitive to changes in interest rates than sensitive to changes in interest rates than prices of low coupon-rate bonds (compare a prices of low coupon-rate bonds (compare a zero-coupon bond and a coupon-paying bond zero-coupon bond and a coupon-paying bond of the same maturity).of the same maturity).

DurationDuration

The observed bond price properties suggest that the The observed bond price properties suggest that the timingtiming and and magnitudemagnitude of of all cash flowsall cash flows affect bond affect bond prices, not only time-to-maturity. prices, not only time-to-maturity. Macaulay’s durationMacaulay’s duration is a is a measure that summarizes the timing and magnitude effects measure that summarizes the timing and magnitude effects of all promised cash flows.of all promised cash flows.

1

Cash flow weight:/ 1

Macauley's Duration:

t

tt

T

tt

CF yw

BondPrice

D t w

Example (Textbook, Page 524)Example (Textbook, Page 524)

Calculate the duration of the following bonds:Calculate the duration of the following bonds:

1.1. 8% coupon bond; $1,000 par value; 8% coupon bond; $1,000 par value; semiannual installments; Two years to semiannual installments; Two years to maturity; The annual discount rate is maturity; The annual discount rate is 10%, compounded semi-annually.10%, compounded semi-annually.

2.2. Zero-coupon bond; $1,000 par value; Zero-coupon bond; $1,000 par value; Two year to maturity; The annual Two year to maturity; The annual discount rate is 10%, compounded semi-discount rate is 10%, compounded semi-annually.annually.


t = Years t = Years to CFto CF CFCF

PV(CF)PV(CF)

(k = 5%)(k = 5%)WeightWeight Weight * tWeight * t

0.50.5 4040 38.09538.095 0.03950.0395 0.01970.0197

1.01.0 4040 36.28136.281 0.03760.0376 0.03760.0376

1.51.5 4040 34.55434.554 0.03580.0358 0.05370.0537

2.02.0 1,0401,040 855.611855.611 0.88710.8871 1.77411.7741

Sum Sum 964.540964.540 1.00001.0000 1.88521.8852


Calculation of the durations:Calculation of the durations:

For the coupon bond:For the coupon bond:

D = Sum { wD = Sum { wtt * t } = 1.8852 years * t } = 1.8852 years

For the zero coupon bond:For the zero coupon bond:

D = Time to maturity = 2 yearsD = Time to maturity = 2 years

Properties of the DurationProperties of the Duration

☺ The The duration of a zero-coupon bond duration of a zero-coupon bond equals its time to maturity;equals its time to maturity;

☺ Holding maturity and par value constant, Holding maturity and par value constant, the bond’s the bond’s duration is lowerduration is lower when the when the coupon rate is highercoupon rate is higher;;

☺ Holding coupon-rate and par value Holding coupon-rate and par value constant, the bond’s constant, the bond’s duration generally duration generally increasesincreases with its with its time to maturitytime to maturity..

The Use of DurationThe Use of Duration

☺ It is a simple It is a simple summary statisticsummary statistic of the effective of the effective average maturity of the bond (or portfolio of average maturity of the bond (or portfolio of fixed income instruments);fixed income instruments);

☺ Duration can be presented as a Duration can be presented as a measure of measure of bond (portfolio) price sensitivitybond (portfolio) price sensitivity to changes to changes in the interest rate (cost of capital);in the interest rate (cost of capital);

☺ Duration is an essential Duration is an essential tool in immunizingtool in immunizing portfolios against interest rate risk.portfolios against interest rate risk.

Macaulay’s DurationMacaulay’s Duration

Bond price (p) changes as the bond’s yield Bond price (p) changes as the bond’s yield to maturity (y) changes. We can show that to maturity (y) changes. We can show that the proportional price change is equal to the the proportional price change is equal to the proportional change in the yield times the proportional change in the yield times the duration.duration.

(1 )

(1 )

P yD

P y

Modified DurationModified Duration

Practitioners commonly use the modified Practitioners commonly use the modified duration measure duration measure D*=D/(1+y),D*=D/(1+y), which can be which can be presented as a measure of the bond price presented as a measure of the bond price sensitivity to changes in the interest rate. sensitivity to changes in the interest rate.

*PD y

P

ExampleExample

Calculate the percentage price change for the following Calculate the percentage price change for the following bonds, if the semi-annual interest rate increases from 5% to bonds, if the semi-annual interest rate increases from 5% to 5.01%:5.01%:

1.1. 8% coupon bond; $1,000 par value; semiannual 8% coupon bond; $1,000 par value; semiannual installments; Two years to maturity; The annual installments; Two years to maturity; The annual discount rate is 10%, compounded semi-annually.discount rate is 10%, compounded semi-annually.

2.2. Zero-coupon bond; $1,000 par value; Two year to Zero-coupon bond; $1,000 par value; Two year to maturity; The annual discount rate is 10%, maturity; The annual discount rate is 10%, compounded semi-annually.compounded semi-annually.

3.3. A zero-coupon bond with the same duration as the 8% A zero-coupon bond with the same duration as the 8% coupon bond (1.8852 years or 3.7704 6-months coupon bond (1.8852 years or 3.7704 6-months periods. The modified duration is 3.7704/1.05 = 3.591 periods. The modified duration is 3.7704/1.05 = 3.591 period of 6-months).period of 6-months).

ExampleExample

The percentage price change for the following bonds as a The percentage price change for the following bonds as a result of an increase in the interest rate (from 5% to 5.01%):result of an increase in the interest rate (from 5% to 5.01%):

1.1. ∆∆P/P = -D*·∆y = -(3.7704/1.05)·0.01% = -0.03591%P/P = -D*·∆y = -(3.7704/1.05)·0.01% = -0.03591%

2.2. ∆∆P/P = -D*·∆y = -(4/1.05)·0.01% = -0.03810%P/P = -D*·∆y = -(4/1.05)·0.01% = -0.03810%

3.3. ∆∆P/P = -D*·∆y = -(3.7704/1.05)·0.01% = -0.03591%P/P = -D*·∆y = -(3.7704/1.05)·0.01% = -0.03591%

Note that:Note that:When two bonds have the same duration (not time to When two bonds have the same duration (not time to maturity) they also have the same price sensitivity to maturity) they also have the same price sensitivity to changes in the interest rate: 1 vs. 3.changes in the interest rate: 1 vs. 3.When the duration (not time-to-maturity) of one bond is When the duration (not time-to-maturity) of one bond is higher then the other, it’s price sensitivity is also high: higher then the other, it’s price sensitivity is also high: 2 vs. 1 or 3.2 vs. 1 or 3.

Practice ProblemsPractice Problems

BKM Ch. 14: 1-2, 11-12.BKM Ch. 14: 1-2, 11-12.

BKM Ch. 15:BKM Ch. 15:

Concept check: 8-9;Concept check: 8-9;

End of chapter: 6, 10, 23-24.End of chapter: 6, 10, 23-24.

BKM Ch. 16: 1-3.BKM Ch. 16: 1-3.

Fi8000 Bonds, Interest Rates Fixed Income Portfolios Milind Shrikhande.

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