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Topic 7 Yield Curve and the Term Structure
Yield Curve • Yield curve is a term used to describe the plot of yield to maturity against time to maturity or against a risk measure, such as the modified duration of debt securities in a certain market segment (such as Treasury or corporate bonds). • By incorporating the expectations of diverse participants in the marketplace, the shape of the yield curve succinctly captures and summarizes the cost of credit for various maturities of different issuers. • Duration/convexity strictly valid for parallel shifts only.
Implied Zero Yields • Term structure – based on zero’s. True TVM. • The goal is to be able to specify a zero coupon yield at every maturity based on the coupon yield curve plotted earlier. • This zero curve is estimated from the YTM of the coupon notes and bonds. Hence it is termed as the implied zero curve. • Implied zero curve is a very critical concept in fixed-income markets.
Getting Implied Zeroes Note that the price of bond 1 is 99.50 and the coupon is 5. So, we can use the pricing formula to solve for the yield of zero 1 as follows:
Par Bond Yield Curve • The yields on implied zeroes have been referred to as the spot rates of interest. • Once implied zeroes are extracted, we can determine the coupons at which a bond should be issued at each maturity so that it may sell at par. This concept is known as the par bond yield curve.
Estimating Implied Zeroes • In practice we estimate the zeroes simultaneously by matrix inversion. This procedure will work if there are coupon bonds for every maturity interval for which we want to estimate zeroes. • More commonly due to gaps in the yield curve statistical procedures are used to estimate the implied zero curve.
Estimating Implied Zeroes Build cash flow matrix (y x y) and invert (must mark off y x y space with miniverse) – use dirty price. This gives the securities to be bought/sold to synthetically recreate a zero. Press Control+shift+Enter Keys simultaneously.
Estimating Implied Zeroes To get the vector of implied zero prices, multiply the vector of dirty prices and the Inverse of the cash flow matrix: Press Control+shift+Enter Keys simultaneously.
Estimating Implied Zeroes The implied zero prices also provide discount factors for future cash flows: Recipe for creating zero coupon bonds from coupon bonds
Estimating Implied Zeroes Recipe for synthetically creating zeroes • Short 0.00054 of the first couponbonds (from inverse) • Go long 0.009434 of the second coupon bond. • Investment: -0.00054 * 100 + 0.009434 *101 = 0.8989 (bid for two year zero) • Cash flow in Year 1: -0.00054 * 105 + 0.009434 * 6 = 0
Recipe for creating zeroes • Cash flow in Year 2: -0.00054 * 0 + 0.009434 * 106 = 1 . • This creates a two-year zero coupon bond. • Bottom line: The inverse matrix contains the recipe for creating zeroes from coupon bonds. Each row represents the recipe for a specific maturity. The first row for the one-period zero, and so on.
Estimating Implied Zeroes • In practice we estimate the zeroes simultaneously by matrix inversion. This procedure will work if there are coupon bonds for every maturity interval for which we want to estimate zeroes. • More commonly due to gaps in the yield curve statistical procedures are used to estimate the implied zero curve.
Estimating Implied Zeroes • In practice we can estimate the implied zero curve with many bonds. In the next few pages we show how to extract the implied zeroes using 13 bonds in the US Treasury Market. • This procedure will work if there are coupon bonds for every maturity interval for which we want to estimate zeroes.
Estimating Implied Zeroes Inverse:
Estimating Implied Zeroes • The procedure that we used to get implied zeroes is known as bootstrapping. • The par bond yield curve is the pricing benchmark for issuers in other sectors of the fixed-income markets. Corporate issuers, Municipals, Agencies, etc. base their pricing on the Treasury par bond curve.
Estimating Implied Zeroes Multiply Vector of dirty prices by inverse:
Forward Rates Determination of a rate beginning on T1 through T2, given the yield curve. T2 t T1
Forward Rates • Forward agreements are prevalent in dealer markets and listed markets. The pricing in these markets is based on the theory that we have developed. • In discounting cash flows, the choice of the discount rates imply a reinvestment rate assumption. Using spot rates leads to a more plausible reinvestment assumption.
Forward Rates • We can lock in forward rates based on the implied zero curve that we extract from the market as of today. • Such forward rates allow one to eliminate future reinvestment risk.
Forward Rates 100 Zeros – pure TVM 100 2 0 1
Forward Rates • The right hand side is the sum of two things: • The first term is the reinvested return at the end of year 2. The reinvestment rate is the forward rate between year 1 and 2 that was known at date 0. • The second term is the balloon payment at year 2. • The implied reinvestment rate is the forward rate.
STRIPS • There is a markets where zero coupon Treasury securities are created and traded. This is known as the STRIPS market. • Since 1985, Treasury has directly helped to improve liquidity in this market. • Since 1987, bonds and notes can be reconstituted from STRIPS. • Strips are not implied zeros – actual securities
Strips versus Implied Zeroes • The existence of strips market allows us to compare the pricing inefficiencies between the coupon sector and the strips market. • The implied zeroes from the coupon sector can be compared with the strips to see whether there are profitable arbitrage opportunities between the strips market and the coupon curve.
Strips versus Implied Zeroes • We first extract implied zeroes from the coupon curve. • Then, we compare this implied zero curve with the strips curve. It is possible that the strips sector is expensive in certain maturity sectors and cheap in other maturity sectors. • This information may then be used to decide whether stripping or reconstitution is profitable in certain maturity sectors.
Strips versus Implied Zeroes • Strips tend to be illiquid. Principal Only (PO) strips are not fungible: two PO strips stripped from two different securities expiring on the same day are not viewed as substitutes. • Interest Only (IO) strips are fungible. • In extracting implied zeroes, we need to work with coupon securities that sell close to par to avoid tax effects.
Synthetic Forwards • Our goal is to create a loan [outflow] of $100 at t=1 and obtain the principal and interest back at time t=2. • The rate on this loan is fixed at time t=0 and is known as the forward rate of interest. • Step 1: Since we want an outflow at t=1, we short the one-year zero. • Step 2: We use the proceeds from the short-sale to finance a long position in the two-year zero. • Step 3: The position at time t=0 must be self-financing: this will determine the number of two-year zeroes that we must buy.
Create a Forward Loan at t=1 maturing on t=2. • We loaned $100 at t=1 and received 95/90 * 100 at time t=2. This implied a loan rate of Note that this rate is only dependent on the zero prices at time t=0 and hence can be contracted at time t=0. This rate is the forward rate of interest.
Relative value between strips and coupon securities • Stripping coupon securities in order to create long-dated zeroes to meet the duration needs of insurance companies and pension funds to meet their duration gap needs. • Reconstituting coupon bonds from strips in order to exploit the demand for certain coupon bonds that are deliverable to futures contracts. • 3. Relative value opportunities between coupon securities and strips.
