Ch-03 - Fixed Income Securities and Markets Chapter 3...

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Unformatted text preview: Fixed Income Securities and Markets Chapter 3 Yield-to-Maturity Yield- to- Each cash flow of a fixed income security Each must be discounted at the factor or rate appropriate for the term of that cash flow. Yield-to-Maturity Yield- toYield-to-maturity, or yield, is the single Yield- torate that when used to discount a bond's cash flows, produces the bond's market price. Yield is not a good measure of relative Yield value or of realized return to maturity. If a bond's YTM remains unchanged over a If short time period, that bond's realized total rate of return equals its yield. 1 102 + 18.125 3.125 3.125 3.125 103.125 = + + + 2 3 4 y⎛ 32 y⎞ ⎛ y⎞ ⎛ y⎞ 1+ 2 ⎜1 + 2 ⎟ ⎜1 + 2 ⎟ ⎜1 + 2 ⎟ ⎝ ⎠⎝ ⎠⎝ ⎠ YTM=4.8875% YTM=4.8875% The price of a T-year security making Tsemiannual payments of c/2 and a final principal payment of F is: 2T P (T ) = ∑ t =1 c/2 (1 + y / 2 ) c⎡ ⎛ 1 ⎞ P(T ) = ⎢1 − ⎜ ⎟ y ⎢ ⎝ 1+ y / 2 ⎠ ⎣ t 2T + F (1 + y / 2 ) 2T 2T ⎤ ⎛1⎞ ⎥+F⎜ ⎟ ⎥ ⎝ 1+ y / 2 ⎠ ⎦ From the formula When Coupon Rate = YTM. Par bonds. When When Coupon Rate > YTM, Premium When bonds. The bonds are selling for more than face value. When Coupon Rate < YTM, Discount When bonds, or the bond sells at a discount to par. 2 Bond prices approach par as they Bond approach maturity: pull to par. If a bond's YTM over a six-month period sixremains unchanged, the annual total return of the bond over that period equals its YTM. P0 = c/2 c/2 1+ c / 2 + +L + 2 2T 1 + y / 2 (1 + y / 2 ) (1 + y / 2 ) P2= 1/ c c/2 1+ c / 2 + +L + 2T −1 2 1+ y / 2 (1 + y / 2 ) P 2 = (1 + y / 2) P0 1/ 6.25s of Feb. 15, 2003 ⎛P ⎞ y = 2 ⎜ 1/ 2 − 1⎟ P0 ⎝ ⎠ YTM=4.8875% Spot Rates: 5.008%,4.929%,4.864%,4.886% 102 + 18.125 3.125 3.125 3.125 103.125 = + + + 2 3 4 y⎛ 32 y⎞ ⎛ y⎞ ⎛ y⎞ 1+ 1 + ⎟ ⎜1 + ⎟ ⎜1 + ⎟ 2 ⎜ 2⎠ ⎝ 2⎠ ⎝ 2⎠ ⎝ 3.125 3.125 3.125 103.125 = + + + ˆ r.5 ⎛ r ⎞ 2 ⎛ r ⎞3 ⎛ r ⎞ 4 ˆ ˆ ˆ2 1 1.5 1+ 2 ⎜1 + 2 ⎟ ⎜1 + 2 ⎟ ⎜1 + 2 ⎟ ⎝ ⎠⎝ ⎠ ⎠⎝ Flat term structure of spot rates, YTM=Spot Flat Upward sloping, YTM is below the last spot Upward Downward sloping, YTM above last spot Downward 3 Yield, cash flow, and spot rates Yield of a zero coupon bond of a Yield particular maturity equals the spot rate of that maturity. Par coupon bonds: coupon bonds Par selling at par, the yield equals the coupon rate c. Par Coupon Bonds 100c 2T ∑ d (t / 2) + 100d (T ) = 100 2 t =1 c= 2 [1 − d (T ) ] 2T ∑ d (t / 2) t =1 4 Par Nonprepayable Mortgage: present value=amount borrowed=par yield discounts future cash flows into PV 2T X ∑ d (t / 2) = 100 t =1 ⎛ 2T ⎞ X = 100 / ⎜ ∑ d (t / 2) ⎟ ⎝ t =1 ⎠ 2T 100 = X ∑ t =1 1 (1 + yT / 2 ) t Discussion of Figure 3.2 All equal at .5 years All Short-end, downward sloping, par yields Shortexceed zero yields, negligible Medium-term, upward sloping, zero yields Mediumexceed par yields, spread increasing Long-end, down sloping, spread narrows LongQualitatively, relative to zero yield, Qualitatively, mortgage yield is just like par yield, except more pronounced YTM and relative value: the coupon effect The impact of coupon level on the The YTM of coupon bonds with the same maturity is called the coupon effect. The size of the coupon effect depends The very much on the shape of the term structure of interest rates and the cash flow structure of the securities. 5 YTM and realized return 102 + 18.125 3.125 3.125 3.125 103.125 = + + + 2 3 4 y⎛ 32 y⎞ ⎛ y⎞ ⎛ y⎞ 1+ 2 ⎜1 + 2 ⎟ ⎜1 + 2 ⎟ ⎜1 + 2 ⎟ ⎝ ⎠⎝ ⎠⎝ ⎠ 4 3 2 ⎡⎛ y⎞ y⎞ ⎛ y⎞ ⎛ y ⎞⎤ ⎛ 102.5664 ⎜1 + ⎟ = 3.125 ⎢⎜1 + ⎟ + ⎜ 1 + ⎟ + ⎜ 1 + ⎟ ⎥ + 103.125 ⎝ 2⎠ ⎢⎝ 2 ⎠ ⎝ 2 ⎠ ⎝ 2 ⎠ ⎥ ⎣ ⎦ YTM is not the bond's return if held to maturity YTM Coupons are invested at uncertain future rates Coupons 6 ...
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