Post on 25-Dec-2015
Lesson 1
Essentialsof bond pricing
Financial Instruments
Bonds
Institute of Economic StudiesFaculty of Social SciencesCharles University in Prague
Essentials of bond pricing 2
Straight bond
1 2 3 T-1 T
๐ถ1 ๐ถ2
๐0
โฆ issuing price of the bond
โฆ maturity of the bondโฆ nominal value of the coupon (t = 1, 2, โฆ T)
๐ถ๐ โ1 ๐ถ๐
๐
โฆ principal (face or par value) of the bond๐ถ๐ก=๐ร๐
โฆ coupon rate (percentage)
Parties of a bond contract Issuer (borrower, debtor) obtains funds from the sale of the security and pays the interest rate called the coupon rate Holder (lender, creditor) invests funds in the purchase of the security and earns coupons Properties of a straight (plain vanilla) bond
Essentials of bond pricing 3
Diversities in bond contracts (1)
1 2 . . . Tsemi-annual bond zero-coupon bond Size of coupon payments
1 2 . . . T?
variable-coupon bond (inflation-linked bond, floating-rate bond) Redemption date
1 2 . . . perpetual bond, consol 1 2 . . . Tcallable bond
Frequency of coupon payments1 2 . . . T
?
โ
Essentials of bond pricing 4
Diversities in bond contracts (2)
Credit risk of the bond Currency denomination of the bond
Issuer of the bondgovernment bond (Treasuries, gilts)municipal bondcorporate bond (senior vs. junior, secured vs. unsecured)investment-grade bond (i.e. AAA, AA, A, BBB)speculative-grade bond (i.e. BB, B, CCC)junk (high-yielding) bonddomestic bond (issued by resident in resident currency)foreign bond (issued by non-resident in resident currency)Samurai (Japan), Yankee (USA), Bulldog (UK), Matador (Spain), Kiwi (New Zealand), Alpine (Switzerland) eurobond (issued in non-resident currency)
Essentials of bond pricing 5
Underlying principles of pricing
Fair pricing of financial contracts
Time and risk value of money
๐น๐ ๐ก (๐ถ๐น 0 )=๐ถ๐น 0ร (1+๐ )๐ก๐๐ (๐ถ๐น ๐ก )=๐ถ๐น ๐ก
(1+๐ )๐ก
A given nominal amount of money obtained at different times has different values (the longer the time until the monetary amount is obtained, the lower its present value) Time comparability is achieved by operations of discounting (against the direction of the passage of time) and augmenting (in the direction of the passage of time)
Otherwise one party of the contract (issuer, seller) would be favoured over the other or would be at a disadvantage vis-a-vis the other party of the contract (holder, buyer)
Present value of the stream of cash flows associated with given financial instrument must equal zero๐๐ (๐ถ๐น 1 ,๐ถ๐น 2 ,โฆ,๐ถ๐น๐)=0
Essentials of bond pricing 6
Annual discounting of annual coupons๐0=๐๐
(1+๐ )+๐๐
(1+๐ )2+โฆ+
๐๐+๐(1+๐ )๐
Semi-annual discounting of semi-annual coupons๐0=
12๐๐
(1+ 12 ๐ )+
12๐๐
(1+ 12 ๐ )2+โฆ+
12๐๐
(1+ 12 ๐ )2๐ โ1+
12๐๐+๐
(1+ 12 ๐ )2๐
Annual discounting of semi-annual coupons Semi-annual discounting of annual coupons๐0=
๐๐
(1+ 12๐ )
2+๐๐
(1+ 12 ๐ )4 +โฆ+
๐๐
(1+ 12 ๐)2(๐โ 1) +
๐๐+๐
(1+ 12 ๐ )2๐
๐0=
12๐๐
(1+๐ )1/2+
12๐๐
(1+๐ )1+โฆ+
12๐๐
(1+๐ )๐โ1 /2+
12๐๐+๐
(1+๐ )๐
Discounting conventions (1)
Essentials of bond pricing 7
Discounting conventions (2)
Valuation date differs from the issuance or the coupon payment date๐๐๐=๐๐+
๐๐(1+๐ )
+๐๐
(1+๐ )2+โฆ+
๐๐+๐(1+๐ )๐
๐0=1
(1+๐ )๐365
ร๐๐๐ number of days to the nearest coupon payment day
Price of the bond on the nearest coupon payment day () Price of the bond on the valuation day ()
Day/year conventions ACT/365, 30/360, ACT/ACT
ACT .. .actual number of calendar days in the period
8Essentials of bond pricing
๐ท๐๐๐ท๐๐๐ท๐ก ๐
Full (dirty) price = clean price + accrued coupon (it is the transaction price in principle equal to the fair price of a bond) Clean price is the quoted price (without the accrued coupon) Terminology Transaction takes place after ex-coupon day Seller receives the entire coupon and compensates the buyer for holding the bond for part of the coupon period by paying accrued coupon:
Transaction takes place prior to ex-coupon day Buyer receives the entire coupon and compensates the seller for holding the bond for part of the coupon period by paying accrued coupon:
Triad of coupon days
๐๐๐
๐๐๐
๐๐ก ๐
โฆ
๐ท๐๐ ๐ท๐๐
โฆ transaction dayrespective periods
Clean and full price
Essentials of bond pricing 9
Price-yield relationship
Summation formula
< 0 downward-sloping curve
c โฆ coupon ratem โฆ coupon frequencyr โฆ required yield Properties of price-yield relationship
> 0 convex curve (par bond)r
P
Essentials of bond pricing 10
Priceโmaturity relationship
Price of a bond may change simply because the bond is heading to maturity clean bond price remains constant
T
๐ premium bonddiscount bond
par bond๐
Economic jargon bond is selling at a premium bond is selling at par
clean price converges to the face value from above price converges to the face value from below
bond is selling at a discount
Essentials of bond pricing
Definition of YTM๐0=
๐๐(1+๐๐๐ )
+๐๐
(1+๐๐๐ )2+โฆ+
๐๐+๐(1+๐๐๐ )๐
= Interpretation of YTM
Disadvantages of YTM Ignoring reinvestment risk (in contrast with the term deposit) Implicit assumption that the bond is held to maturity
Discount rate at which the present value of discounted cash flow is equal to the market price of the bond
LHS โฆ terminal value at maturity of a bond that results from investing an amount needed to buy the bond at an interest rate equal to YTM RHS โฆ terminal value of accumulated cash flow from holding the bond under assumption that all coupons can be reinvested at a constant YTM
Other names: redemption yield, internal rate of return
11
Yield to maturity
Essentials of bond pricing 12
Other yield measures
= Holding period yield Current (flat, running) yield
Generalized YTM which takes into account buying price (, selling price ( and different rollover rates ( HPY must make assumptions about future values that are uncertain
= Exact YTM for perpetual bonds, approximate measure for other bonds (finite maturity, ignorance of capital gains and losses)
Used for bonds that are approaching maturity in order to ensure comparability with money market instruments
Simple yield to maturity = CY + Smoothing capital gains/losses over the remaining life of the bond Money market yield