Date post: | 21-Dec-2015 |
Category: |
Documents |
View: | 221 times |
Download: | 0 times |
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).
Bond Markets 3
• 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).
Bond Markets 4
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.
Bond Markets 5
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
Bond Markets 6
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
Bond Markets 7
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.
Bond Markets 8
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
Bond Markets 9
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
Bond Markets 10
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?
Bond Markets 11
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
Bond Markets 12
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
Bond Markets 13
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
Bond Markets 14
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
Bond Markets 15
(One Period) Holding Period Return (HPR)
• Ex post measure of return
11
1
n nn t t tt n
t
P P CH
P
Bond Markets 16
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
Bond Markets 17
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
Bond Markets 18
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.
Bond Markets 19
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
Bond Markets 20
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
Bond Markets 21
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.
Bond Markets 22
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
Bond Markets 23
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
Bond Markets 24
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?
Bond Markets 25
Accrued Interest
• Cum-dividend: clean + accrued
• Ex-dividend: clean - rebate
Bond Markets 26
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
Bond Markets 27
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
Bond Markets 28
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.
Bond Markets 29
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.