Part 09

Part-09 Fundamentals of Yield Curves & The Term Structure of Interest Rates 1

Introduction 



At any point in time an investor will have access to a wide variety of bonds. These will differ with respect to  

Their yields, and Their times to maturity

2

Introduction (Cont…) 



Investors and traders will be interested in the relationship between the time to maturity, and yield to maturity, for bonds belonging to a given risk class. A plot of yield versus time to maturity is termed as the 

Yield Curve 3

Introduction (Cont…) 

The Yield Curve 



Is an important indicator of the state of the bond market And provides valuable information

4

Introduction (Cont…) 

While constructing the yield curve it is important to ensure that  



The bonds belong to the same risk class And have a comparable degree of liquidity

For instance 



We may construct a curve for government securities Or for AAA rated corporate bonds 

But we cannot mix the data for the two categories 5

Introduction (Cont…) 

The primary yield curve in the domestic capital market is 



The Government or Treasury Bond Yield Curve

This is because 

Such instruments are free of default risk

6

Analyzing The Curve 

The Yield Curve 



Is an indication of where the bond market is trading currently It also has implications for what the market thinks will happen in the future

7

Analyzing…(Cont…) 





The curve sets the yield for all debt market instruments It fixes the price of money over the maturity structure Thus issuers of debt in the market use the yield curve to price debt securities. 8

Analyzing…(Cont…) 

The yields of government bonds set the benchmark for yields on other debt securities. 

For instance if the 5 year T-bond is trading at a yield of 5% 



All other bonds irrespective of the issuer will be trading at yields in excess of 5% The excess over the yield on a comparable T-bond is called the Spread. 9

Analyzing…(Cont…) 

The yield curve acts as an indicator of future yield levels. 



It assumes certain shapes in response to market expectations of future interest rates. Analysts therefore study the current shape of the curve in order to determine the direction of future interest rates. 10

Interest in the curve 

The curve is analyzed by    

Bond traders Fund managers Central bankers Corporate finance personnel

11

Interest…(Cont…) 

Central bankers and government Treasury departments analyze the curve for the information it provides  

Regarding forward rates Future interest levels 

This information is used to set rates for the economy as a whole 12

Interest…(Cont…) 

Portfolio managers use the curve to assess the relative values of investments across the maturity spectrum. 



The curve indicates the returns that are available at different points of time. Consequently it helps determine which bonds are cheap or costly. 13

YTM 



Consider a bond that makes an annual coupon of C on a semiannual basis. The face value is M, the price is P, and the number of coupons remaining is N.

14

YTM (Cont…) 

The YTM is the value of y that satisfies the following equation.

15

YTM (Cont…) 







The YTM is a solution to a non-linear equation. We generally require a financial calculator or a computer to calculate it. However it is fairly simple to compute the YTM in the case of a coupon paying bond with exactly two periods to maturity. In such a case it is simply a solution to a quadratic equation. 16

YTM for a Zero Coupon Bond 



 

The YTM is also easy to compute in the case of zero coupon bonds. Consider a ZCB with a face value of $1,000, maturing after 5 years. The current price is $500. The YTM is the solution to

17

Spot Rates 



The spot rate of interest for a particular time period, is the YTM of a zero coupon bond that is maturing at the end of the period. For instance assume that a six month zero coupon bond with a face value of $1,000 is selling for $961.54. 18

Spot Rates (Cont…) 

If we consider six months to be one period, then the one period spot rate is given by: 961.54 = 1,000 ⇒ s1 = 0.04 ≡ 4% --------(1 + s1)

19

Spot Rates (Cont…) 

Similarly, if a one year bond is selling for $873.44, then the two period spot rate is given by: 873.44 = 1,000 ⇒ s2 = .07 ≡ 7.00% ______ (1 + s2)2 20

Spot Rates & YTM 





A Plain Vanilla bond is a series of cash flows arising at six monthly intervals. Each cash flow can be perceived as a zero coupon bond maturing at that point in time. Thus a Plain Vanilla bond is essentially a portfolio of zero coupon bonds. 21

Spot Rates & YTM (Cont…) 



The correct way to price a Plain Vanilla bond is by discounting each cash flow at the spot rate that is applicable for that period. Take the case of a Plain Vanilla bond with a face value of $1,000 and one year to maturity, with a coupon of 7% per annum, paid on a semi-annual basis. 22

Spot Rates & YTM (Cont…) 

Using the spot rates calculated earlier the price of the bond can be calculated to be: P = 35 1,035 = 937.66 ____ + _____ (1.04) (1.07)2

23

Spot Rates & YTM (Cont…)  

What is the YTM of this bond? It is obviously the solution to: 937.66 = 35 + 1,035 ______ ______ (1+ y/2) (1 + y/2)2 ⇒ y/2 = 0.069454 ≡ 6.9454% 24

Spot Rates & YTM (Cont…) 

The YTM is therefore a complex average of the spot rates. 

This per se need not pose any problems. 



The problem is that the YTM is a function of the coupon rate. In other words, if we compare two bonds with the same time to maturity, but with different coupons, the YTMs for the two will differ. 25

Spot Rates & YTM (Cont…) 

This is despite the fact that both bonds have been priced correctly, using the appropriate spot rates. 



This is known as the Coupon Effect.

For instance let us consider a bond with a face value of $1,000 and one year to maturity, but with a coupon of 12% per annum. 26

Spot Rates & YTM (Cont…) 



Its price can be calculated to be: P= 60 + 1,060 = 983.54 ____ ______ (1.04) (1.07)2 The YTM is obviously given by: 983.54 = 60 + 1,060 ⇒ y/2 = 0.069092 ____ _____ (1+y/2) (1+y/2)2 ≡ y/2 = 6.9092%

27

Analyzing the Coupon Effect 



Quite obviously the YTM, which is a complex average of the spot rates, varies with the coupon rate when comparisons are sought to be made among bonds with an identical maturity. What is the reason for the 7% coupon bond to have a higher YTM than the 12% bond? 28

The Coupon Effect (Cont…) Take the 7% bond first.  It has a price of 937.66.  The present value of the first cash flow is 35 ____ = 33.65 (1.04)  Thus 33.65 = 0.0359 ≡ 3.59% of the value _____ of the bond is tied up in one29 

The Coupon Effect (Cont…) 



The balance 96.41% is obviously tied up in two period money. In the second case the price is 983.54 while the present value of the first cash flow is 60 _____ = 57.69, which is 5.87% (1.04) of the value of the bond.

30

The Coupon Effect (Cont…) 





The one period spot rate is 4% while the two period spot rate is 7%. Thus one period money is cheaper than two period money. Since the second bond has a greater percentage of its value tied up in one period money, it is obvious that its yield to maturity 31

Yield Curves and the Term Structure  



What is a Yield Curve? It is a graph depicting the YTM, which is plotted along the Y-axis, and the time to maturity, which is plotted along the X-axis. For the purpose of constructing the yield curve it is imperative that the bonds being compared belong to the same credit risk class. 32

Term Structure 





The expression `Term Structure of Interest Rates’ refers to the relationship between spot rates of interest, as depicted along the Y-axis, and the corresponding Time to Maturity, as depicted along the X-axis. Once again, to facilitate meaningful inferences the data should be applicable to bonds of the same risk class. The term structure is also referred to as the Zero Coupon Yield Curve. 33

Bootstrapping  





What is bootstrapping? In order to calculate the spot rates and thereby construct the term structure, we need access to the prices of zero coupon bonds maturing at different intervals. However, in real life most of the data that we have pertains to coupon paying bonds. Bootstrapping is a technique for 34

Illustration 

Assume that we have the following data. Time to Maturity 1 Year

Price

Coupon

1,000

6%

2 Years

975

8%

3 Years

950

9%

4 Years

925

10% 35

Illustration (Cont…)  



The one year bond is selling at par. So the one year spot rate must be 6%, which is the coupon rate on this bond. The two year spot rate can be determined as follows: 80 + 1080 975 = ____ ______ ⇒ s2 = 9.57% 2

36

Illustration (Cont…) 

Similarly the three year spot rate is given by: 90 90 1090 950 = ____ + ______ + ____ (1.06) (1.0957) (1+s3)3 ⇒ s3=11.32% 37

Illustration (Cont…) 

Using the same logic: 100 + 100 + 100 + 1100 925 = _____ ______ ______ ________ (1.06) (1.0957) (1.1132) (1+s4)4 ⇒

s4 = 12.99% 38

Practical Difficulties With Bootstrapping 

One problem that is often encountered in practice is that a bonds may not exist for certain maturities, for which spot rates are sought to be calculated. 



Or else, even if it were to exist, a bond may not be traded actively. And in the absence of active trading, the observed prices would be suspect. 39

Difficulties (Cont…) 

Secondly as our example shows, a combination of par, premium and discount bonds are likely to be used to construct the term structure. 

The problem while using bonds with different coupons is that we are exposed to the coupon effect and the liquidity effect. 40

Difficulties (Cont…) 

The Liquidity Effect: 



For a given maturity the more recently issued or on-the-run securities tend to be more liquid than the earlier issues or offthe-run securities. For instance at Treasury bill maturing on 16 May 1991 was trading at 6.31%, whereas an 8.125% Treasury note maturing on 15 May 1991 was trading at 6.37%.

41

Difficulties (Cont…) 



Off-the-run securities are less liquid since most of them tend to be in the hands of investors who intend to hold them until maturity. Thus the demand for on-the-run securities will be relatively higher and the yields will be lower, although both securities are virtually identical. 42

Difficulties (Cont…) 



Finally some of the earlier Treasury issues are callable in nature. It is not theoretically correct to treat bond with embedded options, such as callable bonds, on par with Plain Vanilla bonds, for the purpose of inferring the term structure. 43

The Coupon Yield Curve 



One of the problems with bootstrapping is that we typically have data for bonds with different coupons. At times however we may have data for bonds, all of which have the same coupon. 

The resulting yield curve is called the Coupon Yield Curve. 44

The Par Bond Yield Curve 

It is an estimate of the yield curve obtained by using data for bonds, which have different coupons, but all of which trade at par. 

In this case, the coupon for each bond will be equal to its YTM.

45

Illustration Time to Maturity 1 Year

Price in Dollars 1,000

Coupon

2 Years

1,000

8%

3 Years

1,000

9%

4 Years

1,000

10%

6%

46

Illustration (Cont…)  

The one year spot rate is 6%. The 2 year spot rate can be determined as follows. 80 1080 1000 = _____ + ______ (1.06) (1+s2)2 ⇒

S2 = 8.08%

47

Illustration (Cont…) 



Similarly the 3 year spot rate can be calculated as 9.16%, and the 4 year spot rate as 10.30%. The par bond yield curve is often used by primary market analysts. 

Since new bonds are always issued at par, such a curve can be used to estimate the coupon to be offered on a new bond whose issue is being contemplated. 48

Deriving the par bond curve in the absence of par bonds 



In the absence of data on par bonds, the par bond yield curve can still be derived. Assume that we have the following vector of spot rates.

49

Deriving…(Cont…) Time to Maturity

Spot Rate

1 Year

6%

2 Years

9.57%

3 Years

11.32%

4 Years

12.99% 50

Deriving…(Cont…) The yield for a one year par bond is obviously 6%.  The yield or coupon for a two year par bond may be calculated as: C + 1000+C 1,000 = _____ ________ (1.06) (1.0957)2 ⇒ C = 94.0441 ⇒ c = 9.4044% 51 

Deriving…(Cont…) 

Similarly: C 1000+C 1000 =

+

C

+

_____ _____ ________ (1.06) (1.0957)2

(1.1132)3 ⇒

C = 109.9837 ⇒ c= 10.9984%

52

Patterns 



Having derived the spot rates, we can plot them versus the time to maturity. In practice the graph may be upward sloping, or inverted, or else may be humped.

53

Upward Sloping Curve

54

Inverted Yield Curve

55

Humped Yield Curve

56

Forward Rates 



Let 1f1 be the one period forward rate one period from now. That is, it represents the applicable rate for a forward contract that is made today, at time 0, to extend a one period loan next period, that is, at time 1. 57

Forward Rates (Cont…) 



Consider an investor who is contemplating making a loan for two periods. He will be indifferent between making a two-period loan at the two-period spot rate, and a oneperiod loan at the one-period spot rate with a forward contract to rollover the proceeds for one 58

Forward Rates (Cont…) 





That is, if arbitrage opportunities were to be non-existent, we would require that: (1+s2)2 = (1+s1)(1+1f1) f is known as the one period implied forward rate. 1 1

In general, if we have an m period spot rate and an n period spot rate, where m>n, then (1+sm)m = (1+sn)n(1+nfm-n)m-n59

Illustration 



Assume that the five year spot rate is 10% and that the four year spot rate is 9%. The one year forward rate four years from now is given by: 1 + 4f1 = (1.10)5 = 1.1409 ⇒ 4f1 = 14.09% ________ (1.09)4 60

Theories of the Term Structure 



 

The first theory that we shall look at is called the Expectations Hypothesis. As per this theory, forward rates are nothing but unbiased expectations of future spot rates. Thus nfm-n = E0[nsm-n] In other forwards the (m-n) period forward rate, n periods from now is the current expectation of the (m-n) period spot rate that is expected to prevail n periods from now. 61

Theories (Cont…) 



The Expectations hypothesis can explain any shape of the yield curve. For instance an expectation that future short term interest rates will be above the current level would lead to an upward sloping yield curve. 62

Illustration (Cont…) 

 



Assume that s1 = 5.50%; E[1s1] = 6%;E[2s1] = 7.5%, and that E[3s1] = 8.5% S2 = [(1.055)(1.06)]1/2 – 1 = 5.75% S3 = [(1.055)(1.06)(1.075)]1/3 – 1 = 6.33% S4 = [(1.055)(1.06)(1.075)(1.085)]1/4 63

Theories (Cont…) 



If the unbiased expectations theory is true, then the yield curve is an important forecasting tool, since it is an indicator of the direction of future short term interest rates. According to the unbiased expectations hypothesis, investors care only about expected returns 64

Theories (Cont…) 

Take the two period case. 





An investor can buy a two year bond yielding a rate of s2. Or else he can buy a one period bond yielding s1 and then roll over into another one period bond at maturity. According to the expectations hypothesis he will be indifferent if the expected returns from the two strategies are equal. 65

Theories (Cont…) 





In other words the market will be in equilibrium if: (1+s2)2 = E[(1+s1)(1+1s1)] But if arbitrage is to be ruled out we require that: (1+s2)2 = (1+s1)(1+1f1) Thus if the expectations hypothesis is valid, then 1f1 = E(1s1) 66

Theories (Cont…) 





How can expectations of rising interest rates lead to an upward sloping yield curve? If rates are expected to rise then investors in long term bonds will indeed be perturbed. Rising interest rates imply falling bond prices, and long term bonds are more vulnerable to changing interest rates than short term bonds.

67

Theories (Cont…) 





In such a scenario investors will start selling long term bonds and buying short term bonds. This will push up the yield on long term bonds, and lead to a decline in yields on short term bonds. The overall effect will manifest itself as an upward sloping yield curve. 68

Policy Implications 



As per this theory changes in the relative amounts of long term and short term bonds will not affect the shape of the yield curve, unless investor’s expectations of the future were to be affected. For instance the central banks of countries conduct open market operations on a regular basis by buying and selling Treasury securities. 69

Policy Implications (Cont…) 



As per the expectations hypothesis the central bank cannot influence the shape of the yield curve by buying securities of one maturity and selling another. This is because as per this theory, investors regard all securities, whatever their maturity, as perfect substitutes. 70

Liquidity Premium Hypothesis 

 



We know that long term bonds are more vulnerable to interest changes than short term bonds. Most investors prefer to lend short-term. Take the case of an investor who intends to invest for one period. He can buy a one period bond and get a rate of s1. 71

Liquidity…(Cont…) 





Or else he can by a two period bond and sell it after one period. In this case the rate of return will be uncertain since the price of the bond at the end of the period will be uncertain. Take the case of a zero coupon bond with a face value of $1,000. 72

Liquidity…(Cont…) 

Its current price is: 1000 ____________ (1+s1)(1+1f1)

The expected price after a period is: E[ 1000] 1000 ______ ≥ _________ (1+1s1) [1 + E(1s1)] 73

Liquidity…(Cont…) 

The rate of return from the two period bond over the first year is: E[ 1000] 1000 1000 1000 ______ - __________ ________ - __________ (1 + 1s1) (1+s1)(1+1f1) [1+E(1s1)] (1+s1)(1+1f1)

______________________ ≥______________________ 1000 1000 ________ ___________ (1+s1)(1+1f1) (1+s1)(1+1f1)

74

Liquidity…(Cont…)

≥ (1+s1)(1+1f1) __________ - 1 1 + E(1s1) 

This will be greater than s1 only if f > E(1s1)

1 1

75

Liquidity…(Cont…) 



In other words an investor with a one period horizon will hold a two period bond only if its expected return is greater than the rate of return on a one period bond, which implies that the forward rate must be greater than the expected spot rate. Thus if investors are risk averse, the forward rate will embody a risk or liquidity premium. In other words, the forward rate will exceed the expected future spot rate by the amount of the premium. 76

The Liquidity Premium Theory and the Term Structure 

We know that: (1+s2)(1+s2) = (1+s1)(1+1f1)



According to the liquidity preference theory: (1+s1)(1+1f1) > (1+s1)[1+E(1s1)]



Therefore: (1+s2)(1+s2) > (1+s1)[1+E(1s1)] 77

Liquidity…(Cont…) 





Consider a downward sloping term structure, that is, s1 > s2. The inequality will hold only if E(1s1) is substantially smaller than s1. Thus a downward sloping yield curve will be observed only if the market expects spot rates to decline significantly. 78

Illustration  





Assume that s1 = 7% and that s2 = 6%. The term structure is obviously downward sloping. 1f1 = (1.06)(1.06) ______________ - 1 = 5.01% (1.07) If we assume that the liquidity premium is .41%, then E(1s1) = 4.60%, which implies that the spot rate is expected to decline significantly. 79

Illustration (Cont…) 



The expectations hypothesis would also say that the spot rate is expected to decline significantly. However, according to it, E(1s1) = 5.01%

80

Liquidity…(Cont…)  





What about a flat term structure? If s1 = s2, then according to the liquidity premium hypothesis, E(1s1) < s1. Thus according to this hypothesis a flat term structure is an indication that spot rates are likely to decline. In contrast a flat term structure would imply no change in the one period spot rate, if the expectations hypothesis were to be valid. 81

Illustration (Cont…) 



For instance if s1 = s2 = 7% and the liquidity premium is .41%, the liquidity premium would imply that E(1s1) = 6.59%. In contrast according to the expectations hypothesis E(1s1) = 7%. 82

Liquidity…(Cont…) 





What about an upward sloping term structure? If s1 < s2 and the curve is slightly upward sloping, then the liquidity premium hypothesis would be consistent with an expectation that interest rates are going to slightly decline. However if the curve were to be steeply sloped upward it would be consistent with the expectation that rates are 83 going to rise.

Illustration 

  



Assume that s1 = 7% and s2 = 7.1%. Let the liquidity premium be .41%. If so, 1f1 = 7.2% ⇒ E(1s1) = 6.79% However if s2 = 7.3%, then 1f1 = 7.6% ⇒ E(1s1) = 7.19%. However in both cases the expectations hypothesis would predict a rise in the spot rate. 84

Market Segmentation 





The market segmentation or the hedging pressure hypothesis argues that securities are not perfect substitutes for each other. Different investor groups have their own maturity preferences. A group will not stray from its desired maturity range unless it is induced to do so by higher yields or other favourable terms.

85

Market Segmentation (Cont…) 



This theory argues that financial intermediaries like banks, pension funds, and mutual funds, often act like risk minimizers rather than profit maximizers as assumed by the expectations hypothesis. That is they prefer to hedge against the risk of fluctuations in prices and yields by balancing the maturity structures of their assets with that of their liabilities. 86

Market Segmentation (Cont…) 

 



For instance pension funds have stable and predictable long term liabilities. Thus they prefer long term assets. Banks have relatively short term liabilities and hence tend to prefer short term assets. Thus financial markets are not one large pool of loanable funds. 87

Market Segmentation (Cont…) 





Thus the debt market is essentially a number of sub markets. Demand and supply within each group is the dominant reason for the level and structure of interest rates within that maturity range. However rates prevailing in a group are relatively unaffected by rates prevailing elsewhere. 88

Policy Implications 





If submarkets are relatively isolated from each other than government policymakers can alter the shape of the curve by influencing supply and demand in one or more market segments. For instance if a positively sloped yield curve were to be desired, the market can flood the market with long term bonds. It could simultaneously purchase short 89 term bonds.

The Preferred Habitat or Composite Theory 



This theory attempts to unifies all the earlier theories. It argues that investors seek out their preferred habitat along the scale of varying maturities, that matches their risk preferences, tax exposure, liquidity needs, regulatory requirements, and planned holding periods. 90

Preferred Habitat (Cont…) 

An investor will not stray from his preferred habitat unless the return on another segment of the market is high enough to overcome his preferences.

91

Solution 

Econometric models may be specified to calculate the rate that best fits the market prices of bonds in the sample, using a technique like non-linear least squares.

92

The Nelson-Siegel Technique It is a parametric approach for deriving the zero-coupon yield curve.  According to this model, the m period spot rate is given by: s(m,β) = β0 + β1 x [1 – e-m/τ] + ________ m/τ β2 x 1 – e-m/τ [________ – e-m/τ] m/τ 

93

Nelson-Sigel (Cont…) 

 



The parameters β0, β1, β2 and τ have to be empirically estimated. Advantages of the method: It can handle a wide variety of term structure shapes that are observed in the market. It avoids the need for interpolation to determine the spot rates between discrete points in time 94

Nelson-Siegel (Cont…) 







Thus spot rates can be derived at any point in time and not just at discrete points in time. The parameters can be interpreted as follows. β0 must be positive and is the asymptotic instantaneous forward rate. It is a function of the term to 95 maturity.

Nelson-Siegel (Cont…) 





β1 measures the deviation from the asymptote. It measures the speed with which the curve tends towards its long term value. If it is positive, the curve will have a negative slope and vice-versa. 96

Nelson-Siegel (Cont…) 







τ must be positive and is the position of the hump or the ushape on the curve. β2 measures the magnitude and direction of the hump. If it is positive there will be a hump. Else there will be a u-shape. 97