Yield to Maturity (YTM) • The interest rate implied by the payment structure of the bond • The interest rate at which the present value of the stream of payments (coupons and final face value redemption) is exactly equal to the current price • Denoted λ Bond Price Formula C F n P m = + k n λ n ∑ =1 (1+ ) (1+ λ ) m m Recall that n A A 1 P = = 1− ∑k =1 k n 1( + r) r 1( + r) Bond Price Formula F C 1 P = + 1− λ n λ n (1+ ) λ (1+ ) m m P = Price of the bond F = Face Value (set to 100) Lambda = yield to maturity m = number of coupons per year n = number of coupon periods remaining C = Annual coupon amount Interpretation of the Price-Yield Curve • Yields of bonds generally track each other and the prevailing interest rates of other fixed income securities • Price and yield are inversely related • As yields rise, prices fall and vice versa • Current yield on the 10 year T-note is 3.443% • This means that bonds maturing in 10 years should have yields at least this much, and much more as the quality of the bond decreases Coupon YTM 0% 5% 10% 15% 0.00% $100.00 $250.00 $400.00 $550.00 1.00% $74.14 $203.45 $332.77 $462.08 2.00% $55.04 $167.43 $279.82 $392.21 3.00% $40.93 $139.38 $237.83 $336.28 4.00% $30.48 $117.38 $204.28 $291.18 5.00% $22.73 $100.00 $177.27 $254.54 6.00% $16.97 $86.16 $155.35 $224.54 7.00% $12.69 $75.06 $137.42 $199.78 8.00% $9.51 $66.06 $122.62 $179.18 9.00% $7.13 $58.72 $110.32 $161.91 10.00% $5.35 $52.68 $100.00 $147.32 11.00% $4.03 $47.65 $91.28 $134.90 12.00% $3.03 $43.44 $83.84 $124.24 13.00% $2.29 $39.87 $77.45 $115.03 14.00% $1.73 $36.82 $71.92 $107.02 15.00% $1.30 $34.20 $67.10 $100.00 16.00% $0.99 $31.93 $62.87 $93.81 17.00% $0.75 $29.94 $59.13 $88.32 18.00% $0.57 $28.19 $55.81 $83.43 19.00% $0.43 $26.63 $52.84 $79.04 20.00% $0.33 $25.25 $50.16 $75.08 Price and YTM of 0% Bond 0% Bond $120.00 $100.00 $80.00 $60.00 $40.00 $20.00 $0.00 0.00% 5.00% 10.00% 15.00% 20.00% 25.00% Price and YTM of 5% Bond 5% Bond $300.00 $250.00 $200.00 $150.00 $100.00 $50.00 $0.00 0.00% 5.00% 10.00% 15.00% 20.00% 25.00% Price and Yield of 10% Bond 10% Bond $450.00 $400.00 $350.00 $300.00 $250.00 $200.00 $150.00 $100.00 $50.00 $0.00 0.00% 5.00% 10.00% 15.00% 20.00% 25.00% Price and Yield of 15% Bond 15% Bond $600.00 $500.00 $400.00 $300.00 $200.00 $100.00 $0.00 0.00% 5.00% 10.00% 15.00% 20.00% 25.00% Qualitative Pricing Facts • A bond with YTM = 0% has a price equal to the sum of all the coupons and face value (nothing is discounted) • If YTM is equal to the coupon, the price of the bond is 100 or par value. This is a par bond. • The price of a bond goes to zero as the YTM rises • The shape of the price yield curve is convex Influence of Time on Maturity • Interest rate risk is the danger that prevailing interests rise above the yield of your bond • Since yields must track the prevailing interest rate, existing yields must rise to match the yield that the market is offering, prices decline • The longer the term of the bond the greater the interest rate ris • Selling a bond before maturity Prices of 9% Coupon Bonds Yield TTM 5% 8% 9% 10% 15% 1 $103.85 $100.94 $100.00 $99.07 $94.61 5 $117.50 $104.06 $100.00 $96.14 $79.41 10 $131.18 $106.80 $100.00 $93.77 $69.42 20 $150.21 $109.90 $100.00 $91.42 $62.22 30 $161.82 $111.31 $100.00 $90.54 $60.52 Duration • A direct measure of interest rate sensitivity • Maturity of a bond is not enough to determine interest rate sensitvity • Long bonds are more sensitive to IR than short bonds • Duration is the weighted average of the times that payments (cash flows) are made • Units of time, between the time of the first cash flow and the last cash flow.