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Fixed Income CFA® Preparation FIXED INCOME www.dbf-finance.com Reading Reading Title Study Session Number 32 The Term Structure and Interest Rate Dynamics 12 33 The Arbitrage-Free Valuation Framework Valuation and Analysis of Bonds with Embedded 34 Options 35 Credit Analysis Models 13 36 Credit Default Swaps Luis M. de Alfonso CFA® Preparation FIXED INCOME www.dbf-finance.com The Term Structure and Interest Rate Dynamics Study Session 12 Reading Number 32 Luis M. de Alfonso CFA® Preparation FI – The Term Structure and Interest Rate Dynamics www.dbf-finance.com LOS 32.a: Describe relationships among spot rates, forward rates, yield to maturity, expected and realized returns on bonds, and the shape of the yield curve SPOT RATES Ø Annualized market interest rate for a simple paymentto be received in the future Ø Normally we use spot ratesfor government securities to generatethespot rate curve Ø Spot ratescan be interpreted as the yieds on zero-coupon bonds (sometimes are referredas zero-couponrates) P = discount factor (price today of a 1$ par zero-coupon bond) # " Price of a zero-coupon bond P" = ' S" = spot rate (yield to maturity) #$ &' (discount factor) T = maturity The termstructure of spot rates(graphof the spot rate S" versus thematurity T) is known as the spot yield curve or spot curve Shape of spot curve changescontinously with the market prices of the bonds Luis M. de Alfonso CFA® Preparation FI – The Term Structure and Interest Rate Dynamics www.dbf-finance.com LOS 32.a: Describe relationships among spot rates, forward rates, yield to maturity, expected and realized returns on bonds, and the shape of the yield curve FORWARD RATES Ø Annualized interest rate on a loan to be initiated at a future period Ø The termstructure of forward ratesis called the forward curve Ø Forward curves and spot curves are mathematically related(we can derive one from the other) f (j,k) = the annualizedinterest rate applicableon a k-year loan starting in j years k # F(/,0) = t = 0 t = j t = j + k #$1 (/,0) 2 F(/,0) = forward price of a $1 par zero-coupon bond maturing at time j+k delivered at time j (discountfactor associatedwiththeforward rate) Luis M. de Alfonso CFA® Preparation FI – The Term Structure and Interest Rate Dynamics www.dbf-finance.com LOS 32.a: Describe relationships among spot rates, forward rates, yield to maturity, expected and realized returns on bonds, and the shape of the yield curve THE SPOT RATE FOR A GIVEN MATURITY CAN BE EXPRESSED AS A GEOMETRIC AVERAGE OF THE ONE PERIOD SPOT RATE AND A SERIES OF ONE PERIOD FORWARD RATES 3 S0 S# f (1, 1) f (2, 1) f (3, 1) f (� − 1, 1) …......................................................................................... t = 0 t = 1 t = 2 t = 3 t = 4 t = k - 1 t = k 3 S0 = S# x f (1,1) x f (2, 1) x f (3,1) x …...................x f (k − 1, 1) Luis M. de Alfonso CFA® Preparation FI – The Term Structure and Interest Rate Dynamics www.dbf-finance.com LOS 32.a: Describe relationships among spot rates, forward rates, yield to maturity, expected and realized returns on bonds, and the shape of the yield curve YIELD TO MATURITY (YTM) Ø Is the yield to maturity of a bond purchased at market price Ø If the spot rate curve is not flat, YTM will not be the same as the spot rate EXPECTED RETURN ON BOND Ø Ex-ante holding period return that a bond investor expect to earn Ø Will be equal to the bond´s yield only when: 1. The bond is held to maturity 2. All payments (coupon and principal) are made in time and in full 3. All coupons are reinvested at the original YTM REALIZED RETURN ON BOND Ø Actual return that the investor experiences over the investment´s holding period Luis M. de Alfonso CFA® Preparation FI – The Term Structure and Interest Rate Dynamics www.dbf-finance.com LOS 32.a: Describe relationships among spot rates, forward rates, yield to maturity, expected and realized returns on bonds, and the shape of the yield curve If spot rate curve would be flat (�# = �:= �;) YTM = Spot Rate Luis M. de Alfonso CFA® Preparation FI – The Term Structure and Interest Rate Dynamics www.dbf-finance.com LOS 32.b: Describe the forward pricing and forward rate models and calculate forward and spot prices and rates using those models FORWARD PRICING MODEL Forward pricing model values forward contractsbased on arbitrage-free pricing Knowing the spot curve, we can If there is no <(=>2) P P x F F calculate any forward price arbitrage then (/$0) = / (/,0) (/,0) = <= (considering no arbitrage) P(/$0) t = 0 t = j t = j + k P/ F(/,0) # # P = = Remember: / and F(/,0) = 2 #$ &= #$1 (/,0) Luis M. de Alfonso CFA® Preparation FI – The Term Structure and Interest Rate Dynamics www.dbf-finance.com LOS 32.b: Describe the forward pricing and forward rate models and calculate forward and spot prices and rates using those models P(?) # P/ = = # $ &= t = 0 t = 2 t = 5 P: F(:,;) <(=>2) F(/,0) = <= <@ F(:,;) = <A Luis M. de Alfonso CFA® Preparation FI – The Term Structure and Interest Rate Dynamics www.dbf-finance.com LOS 32.b: Describe the forward pricing and forward rate models and calculate forward and spot prices and rates using those models FORWARD RATE MODEL Relates forward and spot rates (=>2) This equations derive (/$0) 3 3 #$& 1 + S (1 + S )/ 1 + � (=>2) from the equations of (/$0) = / x (/,0) 1 + �(/,0) = = (#$&=) the previous slide S(/$0) t = 0 t = j t = j + k S/ �(/,0) If the yield curve is upward sloping S(/$0) > S/ �(/,0) > S(/$0) Luis M. de Alfonso CFA® Preparation FI – The Term Structure and Interest Rate Dynamics www.dbf-finance.com LOS 32.b: Describe the forward pricing and forward rate models and calculate forward and spot prices and rates using those models S? t = 0 t = 2 t = 5 S: f(:,;) (=>2) 3 # $&(=>2) 1 + �(/,0) = = (#$&=) @ ; # $&@ 1 + �(:,;) = A (# $&A) Luis M. de Alfonso CFA® Preparation FI – The Term Structure and Interest Rate Dynamics www.dbf-finance.com LOS 32.c: Describe how zero-coupon rates (spot rates) may be obtained from the par curve by bootstrapping PAR RATE Ø Yield to maturity of a bond trading at par (by definition, par rate = coupon rate of the bond) Ø Par ratesfor bonds with different maturities make up the par rate curve By bootstrapping, spots rates (or zero-coupon rates) can be derived from the par curve Luis M. de Alfonso CFA® Preparation FI – The Term Structure and Interest Rate Dynamics www.dbf-finance.com LOS 32.c: Describe how zero-coupon rates (spot rates) may be obtained from the par curve by bootstrapping 1.- The one year spot rate = one year par rate �� = 1,00% #,:? #G#,:? 2.- We can value the 2 years bond using par rates: 100 = + (#,G#:?) (#,G#:? )A Alternatively, we can alsovalue the two years #,:? #G#,:? � � 100 = + A introducing # = 1,00% we get � = 1,252% bond using spot rates (#$HI) (#$HA) #,?G #,?G #G#,?G � � � 3.- Similarlyforthe three years bond: 100 = + A + K introducing # and : we get � = 1,51% (#$HI) (#$HA) (#$HK) Luis M. de Alfonso CFA® Preparation FI – The Term Structure and Interest Rate Dynamics www.dbf-finance.com LOS 32.d: Describe the assumptions concerning the evolution of spot rates in relation to forward rates implicit in active bond portfolio management RELATIONSHIP BETWEEN SPOT AND FORWARD RATES Upwardsloping spot curve Forward rates rises as j increases Ex. Spot and forward curves as of July 2013 Downward sloping spot curve Forward rates declines as j increases Upwardsloping spot curve Forward curve will be above spot curve Downward sloping spot curve Forward curve will be below spot curve From the forward rate model: M 1 + S" = (1 + S# ) x (1 + f (1,1)) x (1 + f (2,1)) x …...................x (1 + f (T − 1, 1)) Spot rate for a long maturity equal the geometric mean of the M 1+ S" one period spot rate and a series of one-year forward rates 1+S# (1 + f (1,1)) (1 + f (2,1)) (1 + f (T− 1,1)) ….............................................. 0 1 2 3 T - 1 T Luis M. de Alfonso CFA® Preparation FI – The Term Structure and Interest Rate Dynamics www.dbf-finance.com LOS 32.d: Describe the assumptions concerning the evolution of spot rates in relation to forward rates implicit in active bond portfolio management FORWARD PRICE EVOLUTION Ø If the future spot ratesactuallyevolve as forecasted by theforward curve, the forward price will remain unchanged Ø Therefore, a change in the forward price, indicatesthat the future spot rate(s) did not conform the forward curve OPQRS UWXSRYZR When spot ratesturn out to be than implied by the forward curve forward price will TUVTRS [RXSRYZR A trader expecting lower future spot rates (than impliedby the current forward rates) would purchase the forward contract to profit from its operation An active portfolio manager will try to outperform the overall bond market by predicting how the future spot rates will differ from those predicted by the current forward curve “For a bond investor, the return of a bond over one year horizon (independently from the maturity of the bond) is alwaysequal to the one year risk free rate if the spot ratesevolve as predicted by today´s forward curve. If not, the returnover the one year period will differ depending on the bond´s maturity” Luis M. de Alfonso CFA® Preparation FI – The Term Structure and Interest Rate Dynamics www.dbf-finance.com LOS 32.d: Describe the assumptions concerning the evolution of spot rates in relation to forward rates implicit in active bond portfolio management Luis M.
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