Fixed Income
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 ) n1 (1  r ) t 1 (1  r ) P (1  r ) v P C M  (1  r )t  (1  r )v (1  r ) n1 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

109
103 3/4 105 3/8

1.0900000
1.0375000 1.0537500

1,000
100,000 25,000

1,090.00