Term Structure of Interest Rate<br />Debt Markets<br />Group Members<br />JigarJani (9)<br />DhawalKantwala (12)<br />Prolin B. Nandu (20)<br />GowriNayak (21)<br />Prasad Shahane (30)<br />
Introduction<br /><ul><li>End of the Bretton Woods System and initiation of Floating rate exchange regime
Traders began looking at bond yields with a different perspective
Not considering individual maturities as a separate market
Arrive at a curve through all the yields</li></ul>The Beginning <br />Definition<br /><ul><li>Relationship between interest rates on bonds of different maturities, usually depicted in the form of a graph often called a yield curve.
Measure of the market's expectations of future interest rates given the current market conditions.</li></ul>Importance<br /><ul><li>Economists and investors believe that the shape of the yield curve reflects the market's future expectation for interest rates and the conditions for monetary policy.
Forms the basis for valuation of all fixed income instruments</li></li></ul><li>Understanding the Yield Curve<br />- Upward Sloping<br />- Risk Premium<br /><ul><li>Liquidity Spread
Attractiveness of Debt Markets</li></ul>- Treasury is selected as the benchmark <br /><ul><li>Default Risk
Liquidity</li></li></ul><li>Deriving the Yield Curve and Benchmark for Valuation<br /><ul><li>If the yield curve is positively slopped, then the theoretical spot curve will lie above the yield curve
This is due to the fact that the greater the maturity, the greater the yield.
to maintain the yield to maturity/zero coupon value equilibrium
The opposite is true when the yield curve is inverted: the spot curve will lie below the yield curve.
The yield for each cash flow that treats the cash flow as a zero-coupon bond. A coupon-paying bond is a set of zero-coupon bonds.
Allow s the investor to utilize the discount rate appropriate to the specific date associated with each cash flow</li></li></ul><li>Types of Yield Curve<br />Flat Yield Curve<br />Normal Yield Curve<br />Inverted Yield Curve<br />Steep Yield Curve<br />
Theories of yield curve<br />Yield curve theories explains the relationship between the yield and the maturity.<br />Expectation theories<br />Pure expectation theory<br />Liquidity preference theory<br />Preferred habitat theory<br />Market – Segmentation theory<br />
Liquidity preference theory<br />yield<br />yield<br />yield<br />maturity<br />maturity<br />maturity<br />Liquidity premium<br />Liquidity premium<br />Future Short term rate expected to rise.<br />Future Short term rate expected to remain flat.<br />Liquidity premium<br />Future Short term rate expected to fall.<br />
Preferred Habitat<br />The theory states thatthe term structure reflects the expectation of the future path of interest rates as well as a risk premium, but the risk premium must not necessarily rise uniformly with maturity. <br />Investors have distinct investment horizons and require a meaningful premium to buy bonds with maturities outside their "preferred" maturity, or habitat. <br />When the long term interest rates are higher then the short term interest rates we can interpret that short-term investors are more prevalent in the fixed-income market.<br />
Segmented market theory<br />The market-segmentation theory recognizes that investors have preferred habitats dictated by the nature of their liabilities. <br />This theory also proposes that the major reason for the shape of the yield curve lies in asset/liability management constraint (either regulatory or self-imposed) and creditors (borrowers) restricting their lending (financing) to specific maturity sectors. <br />However, the market-segmentation theory differs from the preferred-habitat theory in that it assumes that neither investors nor borrowers are willing to shift from one maturity sector to another to take advantage of opportunities arising from difference between expectations and forward rates. <br />Thus, for the segmentation theory, the shape of the yield curve is determined by supply of and demand for securities within each maturity sector.<br />
Segmented market theory<br />Yield<br />S<br />D<br />S<br />D<br />S<br />D<br />Long Term Maturity<br />Short Term Maturity<br />Medium Term Maturity<br />Maturity<br />
Constructing Yield Curve<br />Zero coupon rates are often not observable in the market hence estimation methodology is required to derive the zero coupon curves from observable data<br />On the other hand, the availability of market data such as YTM and YTM derived prices means par yield curves may be modelled and constructed directly<br />
Constructing Yield Curve<br />Chose Proper<br />Tenure<br />Universe of Securities<br />On-The-Run Securities<br />On-The-Run Securities + Off-The-Run Securities<br />Complete Universe of Securities<br />Calculate YTM Yield Curve and Interpolate for all the Maturity Points<br />Calculate Spot Rate Curve<br />
Bootstrapping<br />Consider Par YTM Curve<br />Spot Rate Curve to be Constructed for Tenure of 5 Years<br />Year 1 Treasury Bill available in the market gives 1 year Spot Rate or Zero Coupon Rate<br />Treating the Cash Flows from the Bonds as individual Zero Coupon Bonds to calculate Spot Rates<br />
Bootstrapping - Limitations<br />Two bonds with same maturity but different coupon rates may have different yields<br />If bond prices for relevant tenure are not available, the curve plotted may not be accurate<br />
Regression<br />Y is the bond price, and (x1 . . . xn) are the cash flows on dates 1 to n<br />If α is set to zero, and β1 . . . βn can be estimated and will be the discount factors.<br />If we make certain statistical assumptions, we can measure how good the estimates are<br />Yi = α +β1x1i + β2x2i + . . . + βnxni + εi<br />Price = C1D1 + C2D2 + . . . + CnDn + εi<br />
Regression - Limitations<br />Assumption of Normal distribution of error terms must hold true<br />Assumptions such as Nonlinearity, Heteroscedasticity (variance is not constant), Multicollinearity (Independent variables are correlated) should hold true to apply regression methods<br />
Yield Curves in Emerging Markets<br />Lack of Securities for different maturities<br />Lack of Liquidity and Price Discovery<br />Lack of Data<br />Iterative Methods<br />Cubic Spline<br />Nelson Siegel<br />
Estimation of ZCYC in India<br />Estimation of the ZCYC involves estimation of the appropriate set of interest rates that go into deriving the present value of cash flows of a bond. <br />This is done by specifying a functional form of the interest rate-maturity relation/discount function/forward rate function. <br />ZCYC estimation in India is done using the ‘Nelson-Siegel’ (NS) functional form<br />Data on secondary market trades in Government securities reported on the Wholesale Debt Market segment of the National Stock Exchange (NSE-WDM)is used as input for the ZCYC estimated by NSE<br />
Advantages of a ZCYC<br />ZCYC based on default-free government securities can be used to price all non-sovereign fixed income instruments after adding an appropriate credit spread<br />It can be used to value government securities that do not trade on a given day, or to provide default-free valuations for corporate bonds<br />Estimates of the ZCYC at regular intervals over a period of time provides us with a time-series of the interest rate structure in the economy, which can be used to analyze the extent of impact of monetary policy<br />This also forms an input for VaR systems for fixed income systems and portfolios<br />
Nelson – Siegel Method<br />Parametric model developed by Nelson and Siegel in 1987 for estimating the Zero Coupon Yield Curve (ZCYC) using past data<br />Gives a forward rate function<br />f(m,b)=β0 + β1 * exp(-m/τ) + β2 [(m/τ) * exp(-m/τ)]<br />where,<br />m denotes maturity<br />b = [β0, β1, β2, τ] are parameters that needs to be estimated<br />
Nelson – Siegel Method<br />The forward rate function can be integrated to obtain the relevant spot rate function or the term structure of interest rate r(m,b)<br />r(m,b) = β0 + (β1 + β2) * [1 - exp(-m/τ)]/(m/τ) - β2 * exp(-m/τ)<br />β0 depicts the long term component of interest rate since r(m,b) tends to β0 as m tends to infinity<br />β0 + β1 depicts the short term component of interest rate since r(m,b) tends to β0 + β1 as m tends to zero<br />τ represents the decay factor<br />
Nelson – Siegel Method<br />Using the r(m,b) function we can estimate the model price (p_mod) for any fixed income security<br />The market price of the bond (p_mkt) usually deviates from the model price derived using r(m,b).<br />Assume that is a error term εi which gives us the relation<br />p_mkti = p_modi + εi<br />
Nelson – Siegel Method<br />The above equation is estimated so that the error term εiis minimized<br />Steps involved in estimation are<br />Vector [β0, β1, β2, τ] is selected<br />The term structure is determined using the r(m,b) function<br />This is used to estimating PV of cash flows of the bond and hence starting model bond prices<br />
Nelson-Siegel Method<br />Assuming non-negativity of long term and short term interest rates, numerical optimization techniques are used to minimize the sum of squared price errors<br />The estimated set of parameters are used to determine the spot rate function r(m,b) and the model prices for each bond<br />The model prices are used to compute the associated model YTM for each bond<br />
Issues with Nelson Siegel Method<br />This method gives the price of the bond if the term structure is the only factor that influences the pricing of the bond. <br />In practice, however, observed prices differ from this ‘average’ price due to factors other than the term structure that affect the price of a bond include <br /><ul><li>Differential tax rates for income and capital gains that affect the relative valuations of bonds with different cash flows.
Illiquid bonds trade at a premium relative to liquid bonds of the same residual maturity.
Transaction costs that vary with the size of the trade
An intra-day effect on account of new developments during the day
Expectations about the directionality of the term structure that have not been explicitly accounted for in the estimation.</li>