March 13, 2013 Kris Boudt
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Plan
• Pricing of bonds; • Price quotation of bonds; • Measuring the yield of a bond investment.
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PRICING OF BONDS: GIVEN THE CASHFLOWS AND REQUIRED YIELD, WHAT IS THE PRICE OF THE BOND?
Chapter 2, Fabozzi
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Pricing of bonds
• Suppose: – Cash flows from the issuer to the investor are known (excludes bonds with implied options) – The required yield is known Then the bond price can be computed as the present value of the future cash flows. • To simplify the problem further, suppose a fixed periodic interest rate such that the interest rate payments are an annuity. • If we further suppose that the first investment occurs one period from now, it is referred to as an ordinary annuity. • Recall: present value of a $1 ordinay annuity over n periods and required rate or return r:
an|r
1 1 (1 r ) n j r j 1 (1 r ) n 4
Pricing of bonds
Then the bond price (P) can be computed using the following formula: n C M P t (1 r ) (1 r ) n t 1
M P a n| r C (1 r ) n
P = price (in dollars) n = number of periods (if semiannual: number of years times 2) t = time period when the payment is to be received C = (semiannual) coupon payment (in dollars) r = periodic interest rate (if semiannual: required annual yield divided by 2) M = maturity value
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Exercise: Zero-coupon bond
• What’s the price of a zero-coupon bond (P) that matures 15 years from now, if the maturity value is $1,000 and the required yield is 9.4%.
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Exercise: Zero-coupon bond
Given M = $1,000, r = 0.094 / 2 = 0.047, and n = 2(15) = 30, we have:
M $1000 P $252.12 n 30 (1 r ) (1 0.047)
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Exercise: Fixed rate bullet bond
• What’s the price of a 20-year 10% coupon bond with a par value of $1,000 and a required yield of 11%.
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Exercise: Fixed rate bullet bond
• Semi-annual coupon is C = 0.1($1,000) / 2 = $50 • Coupon is received for n = 2(20) = 40 periods • Semi-annual discount rate is r = 0.11 / 2 = 0.055.
1000 P a40|0.05550 (1 0.055) 40
$16.046131* 50 117.46 $919.77.
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Bond pricing
• Complications: the last interest rate payment does not have the same periodicity as the previous interest payments.
• Example:
– Determine the price for a 6% four and a half year bond with coupon payments that are annual except for the last coupon which is paid at maturity? Required yield is 7%.
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•
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Cashflows?
60 USD annuity, n=4, r=7%.
PV a4|0.07 60 $203.23
At maturity: 1030, n=4.5, r=7%.
1030 PV $759.44 4 .5 (1 0.07)
Price: 962.67 USD
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Bond pricing
• Complications: the next coupon payment is not exactly six months/1 year away.
• Let v=(days between settlement and next coupon)/(days in the regular period between two coupons)
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C M P v t 1 (1 r ) v (1 r ) n 1 t 1 (1 r ) (1 r ) (1 r ) n C M P (1 r )v (1 r )t 1 (1 r )v (1 r ) n1 (1 r ) t 1 (1 r ) P (1 r ) v P C M (1 r )t (1 r )v (1 r ) n1 t 1 n n
1 M a n| r C (1 r ) v 1 (1 r ) v (1 r ) n 1
• Note: if v=1, we have the regular equation.
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1 M P a C v 1 n|r n (1 r ) (1 r )
HOW ARE BOND PRICES QUOTED ON THE MARKET?
Chapter 2, Fabozzi
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Price quotes
When quoting bond prices, traders quote the price as a percentage of par value. • A bond selling at par is quoted as 100, meaning 100% of its par value. • A bond selling at a discount will be selling for less than 100. • A bond selling at a premium will be selling for more than 100.
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Price Quotes Converted into a Dollar Price
(1) Price Quote (2) Converted to a Decimal [= (1)/100] (3) Par Value (4) Dollar Price [= (2) × (3)]
80 1/8 76 5/32 86 11/64 100
0.8012500 0.7615625 0.8617188 1.0000000
10,000 1,000,000 100,000 50,000
8,012.50 761,562.50 86,171.88 50,000.00
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103 3/4 105 3/8
1.0900000
1.0375000 1.0537500
1,000
100,000 25,000
1,090.00