Bond Markets

Investments Chapter 7

QFE Section 1.1—1.2 & 20.1

Bond Markets 2

Bond Markets

• Payments: Redemption value, M, paid at maturity, n, and Coupons, Ct, paid at specified dates, t, until t = n.

• Ct a form of interest and typically expressed as a percentage of M.

• Typically longer term to maturity compared to money markets (1<n<30yrs)

• Possiblilty of Capital gain/loss (trade at discount/premium).

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• Issued by government (state/ county/ municipality) or (large) corporations.

• Domestic currency issued bonds assumed risk-free (No exchange risk + right to print money).

• Assumed risk-premium, rp, above safe rate, rf, commensurate in size to preceived risk level of income stream (Security of Ct + time-to-maturity related interest rate risk).

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Prices & Rates of Return?• From the redemtion value, M, the size and

number of coupon payments, C, and time-to-maturity, n, markets determine a price, P, given other instruments.

• P should provide a rate of return commensurate with the return on similar assets.

• Many ways to calculate a return, depending on needs.

• Conventionally involves compounding.

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Pure Discount Bonds• Recall,

• Thus, the price today of money to be received in n periods time should be commensurate with the discount rate:

• rs(n) is the price of n-period money (annual)

• Spot Rate for n-period money: rsn = f(P, M, n)

(1 ) 1

nn nn n

FV MDPV P

rs rs

1

1 1n

nM P Mrs rs

P P

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Example: Zeroes & Spot Rates

• P = 62,321.30, M = 100,000, n = 6years

• Spot rate?

16

6 100,0001 0.082 8.2% . .

62,321.3rs p a

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Coupon Paying Bonds?

• Stream of coupon payments Ct, which are ‘known’ at issue.

• Government bonds’ coupons are generally fixed. May actually be indexed or variable [in corporate bond case].

• Redeemable at €M at some specified time in the future [perpetuities aside].

• Prices quoted clean. Prices paid involve accrued interest, i.e. dirty price.

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Current Yield

• A.k.a. running/flat/interest yield:

• Quick summary of simple interest, i.e. annual income relative to expenditure

• Caveats:– No capital gain– Time to Maturity & Face Value?– Interest on coupons?

Annual

Clean

C

P

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Yield to Maturity (YTM)• YTM can be calculated ex ante:YTM = y = f(P,M,C,n)

• YTM the discount rate (rate of return) that sets the price of a n-period bond, P(n), equal to the PDV of its income stream.

• YTM is the internal rate of return of the bond’s cash-flow.

( )1 2 1

...(1 ) (1 ) (1 ) (1 ) (1 )

nn n n

C C C C MP

y y y y y

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YTM: Assumptions & Caveats

• Assumes bond held to maturity.• Assumes reinvest C at rate y. Why?• The rate of return is constant at y.• If several payments per period (year) the rate is

grossed-up in simple annual terms. [See examples on semi-annual coupons.]

• Inverse relationship among P and y.• P = f(y), and that function is convex in y (non-

linear).• If y = C => P = M. Why?

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Aside: Pricing an Annuity• Recall, the price of a perpetuity:

• Thus, an annuity in n periods time should cost:

• The difference should be the price of a T-period annuity.

0Perpetuity

CP y

1

1nPerpetuity n

CP

y y

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Breaking-Up a Coupon Paying Bond

• For simplification, we can view the coupon paying bond as having a one-off lump-sum payment at redemption, M, and a series of periodic payments, the coupons, which can be priced as an annuity.

1

11 1

n n

C MP

y y y

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Example: YTM of Semi-Annual Coupon Paying Bond

• P = 900 C = 10% of M = 1,000, 3 years to maturity semi-annual. y?

• y = 0.142 = 14.2% p.a.

1 2 3 4 5 6 6

900

50 50 50 50 50 50 1000

1 1 1 1 1 1 12 2 2 2 2 2 2y y y y y y y

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Example: Price when YTM given?

• 20-year, C = 10% of M = 1,000, YTM = 0.11 p.a. semi-annual

• P?

• or 40 40

50 1 10001 802.31

0.055 1 0.055 1 0.055P

2 40 40

50 50 50 1000...

0.11 0.11 0.11 0.111 1 1 12 2 2 2

P

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(One Period) Holding Period Return (HPR)

• Ex post measure of return

11

1

n nn t t tt n

t

P P CH

P

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Realised Compound Yield (RCY)

• A.k.a. Total return/effective holding period return.

• Ex post measure of return.• Assumes the interest earned on each coupon is

known, plus resale price.• The RCY of a bond held for n-periods:

1

1 1n

n

RCY RCY

TV TVr r

P P

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Example: RCY• 5 year bond with C = 10% pf M = 1,000, trading

at par.• RCY assuing 2 year horizon, interest rate r = 8%

and a YTM after 2 years of 9%?• TV of Coupons: 100 100(1.08) 208

32 2 3 3

100 100 100 10001025.31

1.09 1.09 1.09 1.09P

2 1233.311 0.1105

1000RCY RCYr r

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Pricing a Bond• A coupon paying bond must be priced

such that each its payments is discounted by the pertinent spot rate.

• Deviations from this policy will result in arbitrage opportunities from coupon stripping.

• Hence, if arbitrage opportunities exist traders will exploit these, thus exerting pressure on prices. This behaviour will eliminate the arbitrage opportunities.

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Bond Pricing

• Each coupon represents a single payoff at a certain time in the future.

• Each payment can thus be treated as comparable to a zero of equal maturity.

• If provided with spot rates you should be able to find a price for a bond.

21 2

...1 1 1

n

n

C C C MP

r r r

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Bond Pricing: Spots

• Bond A: coupon 8¾% of FV = 100 annual, 2 years to maturity

• Bond B: coupon 12% of FV = 100 annual, 2 years to maturity

• Spots: r1 = 0.05 r2 = 0.06

2

2

8.75 108.75105.12

1.05 1.0612 112

111.111.05 1.06

A

B

P

P

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Calculating Spot (Bootstrapping)

• Riskless deep discount securities only have short maturities. Spot rates of longer maturities have to be imputed.

• Take the spot rates you have, say up to a year, then calculate the spot rate for the next period (e.g. six months, year) using comparable (riskless) instruments, such as coupon paying government bonds of that maturity.

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Example: Bootstrapping• Spot rates (annual return) given for first six

months, r1 = 8%, and year, r2 = 8.3%.

• Calculate the 18-month spot rate given an 18-month coupon paying bond with C = 8.5% of M = 100 semi-annual.

1 2 3

1 2 3

3

3

3

4.25 4.25 104.2599.45

1 1 12 2 2

104.2599.45 4.0865 3.9180

1 2

0.0893

r r r

r

r

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Coupon Stripping

• C = 12.5% of FV = 100 = P, semi-annual, 20 yrs• YTM?

• Spots: r6months = 0.08 and r12months = 0.083

• PV(C1) = 6.25/1.04 = 6.0096

• PV(C2) = 6.25/(1.0415)2 = 5.7618

• Profits?

2 20

6.25 6.25 106.25...

1.0625 1.0625 1.06255.88 5.54 ... 31.61 100

P

P

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Equilibrium Price

• Spot 1-year r1 = 0.1• Spot 2-year r2 = 0.11• Consider 2-year coupon bond, C= 9% of

M = 1000 & P = 966.4866• Stripping coupons:• PV(C1)= 90/1.1 = 81.8182• PV(C1)= 1090/1.112 = 884.668• What is your guess as to the YTM?

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Accrued Interest

• Cum-dividend: clean + accrued

• Ex-dividend: clean - rebate

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Example: Accrued Interest (Cum Dividend)

• 31.03.1993 a 9% T-Bill 2012 quoted at £106(3/16) for settlement 1.04.1993.

• Last coupon on 6.02.1993

• Accrued Interest?

• 22 days in February + 31 in March + 1 April.

• N = 54 9(54/365) = 1.3315

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Example: Accrued Interest (Rebate)

• 31.03.1993 a 9% Treasury 2004 quoted at £111(5/32)xd

• Next coupon date is 25.04.1993 (i.e. 24 days)

• Rebate? 9(24/365) = 0.592

• Dirty Price? £111(5/32) – 0.592 = 110.56

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Convertible Bonds

• A bond that can be converted to a specified number of shares from a certain date on.

• Allows for a lower initial cost of capital, since the option to convert provides the holder with upside potential.

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Call Provisions

• Bonds are described as callable if they can be redeemed from a certain date on at (above) a specified strike price.

• The bond will tend not to trade above the strike price.

• Implies that if interest rates fall the company can refinance.