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Econ173A_topic8

Econ173A_topic8 - Ec 173A FINANCIAL MARKETS Foster UCSD...

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Ec 173A – FINANCIAL MARKETS LECTURE NOTES Foster, UCSD 22-Nov-08 TOPIC 8 – BONDS & INTEREST RATES A. Bond Prices and Yields [BKM, Ch. 14] 1. Review: a) Fixed-income securities -- medium and long-term bonds or debt instruments involving a promise to pay a certain fixed amount (principal) to the holder at a certain future date (maturity), often with fixed “coupon” interest payments along the way. Examples: Corporate bonds. US Treasury Bonds and Notes. State and local govt municipal bonds (“munis”). Federal agency debt (bonds issued by Fannie Mae, Ginnie Mae, Freddie Mac, etc.) b) Basic bond value equation and definitions. 1) Annual interest I = r c M. Most bonds pay semiannually ($I/2 every 6 months). 2) Yield to maturity r y is the discount rate that equates DCF with current price B 0 . It is like the annual rate of return on the bond. 3) Basis point -- 1/100 of a percentage point; if interest rates rise from 4.0% to 4.5%, they have increased by 50 basis points. 2. Yield to Maturity: 1 a) YTM -- pure discount bonds. 1) If B 0 grows to M in T years, r y is found by solving the following growth equation: M = B 0 (1+r y ) T Ln(1+r y ) = Ln(M/B 0 )/T = γ, 1+r y = e γ 2) Numerical example. B 0 = $887, M = $1,000, T = 34 months (2.83 years) γ = Ln(1000/887)/2.83 = 0.0423, e 0.0423 = 1.0432, r y = 4.32% b) YTM -- annual coupon interest. 1) If the bond pays $I interest at the end of each year, a financial calculator finds r y by solving the equation below (corresponding to the bond value equation above) using an iterative algorithm: B 0 = I × PVA r,T + M × PV r,T 2) A formula for approximating YTM is available: 1 See the Appendix for E XCEL spreadsheet formulas to compute YTM, accrued interest, and bond duration. Bond Notation B 0 current price of bond ($) M par or face value (usually $1,000) T term to maturity (years) r c coupon interest rate (%/yr) I annual coupon interest ($) r y yield to maturity (YTM, %/yr) T y y y r M I r I r I B ) 1 ( ) 1 ( ) 1 ( 2 0 + + + + + + + = 3 2 0 0 B M T B M I r y + - +
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Ec 173A BONDS & INTEREST RATES p. 2 3) Numerical example. Given: B 0 = $982, M = $1,000, I = $60/yr, T = 15 years Using E XCEL , r y = 6.188%. Using the approximation, r y ≈ 6.194%. c) YTM -- semi-annual coupon interest. 1) If the bond pays $I/2 every 6 months, a financial calculator finds r y by solving the following equation using an iterative algorithm: B 0 = I/2 × PVIFA r/2,2T + M × PVIF r/2,2T 2) This corresponds to the bond value equation below. The algorithm computes r/2, then report r y = 2 (r/2), an annual percentage rate (APR). 3) Numerical example. Given: B 0 = $982, M = $1,000, I = $30 per 6 months, T = 15 years Using E XCEL , r y = 6.186%. 3. Interpretation of YTM: a) We interpret r y as the compound annual rate of return on the investment of $B 0 in the bond if it is held to maturity . b) This is easy to see with zero-coupon bonds. 1) If B 0 = M/(1+r y ) T , then B 0 (1+r y ) T = M. That is, B 0 grows to M in T years, so r y is the annual rate of return (rate of growth of investment value).
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