August 2014 © Dr. Enzo Mondello, CFA, FRM, CAIA
(Yields)
(Maturity) Normal Yield Curve
Inverted Yield Curve Flat Yield Curve
Treasury Yield Curve
Term Structure of Interest Rates
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The problems with using on-the-run Treasury issues are:
– On-the-run Treasury curve consists only of 6 points. Interpolation is needed
– Due to the strong dealer demand Treasury yields tend to be abnormally low, reducing their usefulness as a good benchmark
– They tend to have abnormally low reinvestment rate risk, and abnormally high interest rate risk
Despite the drawbacks of the on-the-run Treasury yield curve as the benchmark for valuing other fixed-income securities, it is the most widely used benchmark
An alternative to the on-the-run Treasury yield curve that is sometimes used is the yield curve for zero-coupon Treasury securities
The yield on zero-coupon bond securities is called the spot rate, with the yield on Treasury strips called the Treasury spot rate. When Treasury spot rates are plotted versus their maturities, the resulting curve is called the term
structure of interest rates
Term Structure of Interest Rates
August 2014 © Dr. Enzo Mondello, CFA, FRM, CAIA
A shift in the yield curve occurs when yields change.
In a parallel shift all yields change across the term structure by the same amount. A nonparallel shift occurs when the changes in yield are different for different maturities.
Yield Maturity Initial Curve Yield Maturity Initial Curve
Term Structure of Interest Rates
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A yield curve twist occurs when the curve (the slope of the curve) flattens or steepens due to a nonparallel shift. The yield curve can become flatter (less difference between long and short rates) or steeper (more difference between long and short rates).
A butterfly twist occurs when the curvature of the curve changes. – Positive butterfly: the curve becomes more straight (less humped).
– Negative butterfly: the curve becomes less of a straight line (more humped).
Term Structure of Interest Rates
August 2014 © Dr. Enzo Mondello, CFA, FRM, CAIA
Factors that Drive U.S. Treasury Security Returns
Researchers generally agree that three factors are responsible for changes in Treasury returns:
– Changes in the level of interest rates. This is by far the most important factor. It accounts for about 90% of historical returns and is measured by duration.
– Changes in the slope of the yield curve (distant second most influential factor). It accounts for about 8.5% of historical returns and is measured by key rate duration.
– Changes in the curvature of the yield curve (slight impact). It accounts for about 1.5% of historical bond returns.
Bond portfolio managers who want to hedge their interest rate risks, therefore, should be most concerned about protecting against the adverse effects of changes in the level of interest rates.
Term Structure of Interest Rates
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Various Universes of Treasury Securities used to Construct the Theoretical Spot Rate Curve
Constructing a theoretical spot rate yield curve is not simple. Ideally we would use the yield on default risk-free zero coupon bonds (to abstract from the coupon effect) for each maturity in the maturity spectrum.
There are several different combinations of Treasury securities that can be used to construct a default-free theoretical spot rate curve:
– On-the-run Treasury issues are the most recently auctioned issues of a given maturity. The Treasury is currently issuing bills with maturities of 1, 3, and 6 months, notes and bonds with maturities of 2, 5, 10, and 30 years. The bills are issued at a discount while the notes and bonds carry coupons. The resulting on-the-run yield curve is a par coupon curve because the notes and bonds are issued at par. Securities issued at par eliminate the tax effect that exist for securities issued at a discount or premium. The bootstrapping methodology is used to generate the theoretical spot rate curve. A potential criticism is that large maturity gaps exist, particularly after 5 years.
Term Structure of Interest Rates
August 2014 © Dr. Enzo Mondello, CFA, FRM, CAIA
Selected off-the-run Treasury issues can be added to the on-the-run issues to bridge the caps in on-the-run maturities. The par coupon yield curve is estimated and remaining gaps are filled by interpolation. Like with all on-the- run issues, bootstrapping is used to generate the theoretical spot rate curve.
All Treasury issue so that all coupon securities and bills are used. As a practical matter issues that have special circumstances such as tax advantages, illiquid markets, futures contract delivery are usually omitted to avoid yield distortions. Adjustments are made for taxes and call features. The advantage is that all information available in prices can be used.
Treasury coupon strips are observable zero-coupon securities that can be used directly to create an actual spot rate curve. The relative illiquidity in the strips market implies that strip rates include a premium.
Term Structure of Interest Rates
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Swap Rate Curve (LIBOR Curve)
LIBOR is the rate at which high quality banks will borrow or lend U.S. dollars outside the U.S. amongst themselves, and 3 months LIBOR is the most common floating rate used in interest rate swap agreements. The LIBOR spot rate curve is calculated using the same bootstrapping procedure used to calculate Treasury spot rates.
The swap rate curve represents the swap rates available at various future time periods to convert fixed rates to floating rates and vice versa
The swap rate curve is used to hedge interest rates, to value bonds, and for performance evaluation
Term Structure of Interest Rates
August 2014 © Dr. Enzo Mondello, CFA, FRM, CAIA
The swap rate curve tends to be better benchmark than the government bond curve for the following reasons: – There is little or no government regulation of the swaps market.
– A large demand for government bonds in the repo market can unrealistically change the yield curve. The swaps market does not have these yield problems.
– The swap curve has the credit risk of the underlying banks. Credit risks are thus more similar in the swaps market (LIBOR) than when comparing various government bond market.
– The swaps market has more bond maturities to construct a yield curve than the government bond market. Swap rates quoted in the swap market have maturities of 2, 3, 4, 5, 6, 7, 8, 9, 10, 15, and 30 years.
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Expectations Theory Biased Expectations Theory Pure Expectations Theory Broadest
Interpretation Expectations Local Liquidity Theory Habitat Theory Preferred Segmentation
Theory
Theories of the Term Structure of Interest Rates
Term Structure of Interest Rates
August 2014 © Dr. Enzo Mondello, CFA, FRM, CAIA
Pure (unbiased) expectations theory says the investor’s expectations of future interest rates alone creates the shape of the yield curve. Forward rates are the expected future spot rates. This implies that if the yield curve is upward (downward) sloping, short-term rates are expected to rise (fall), and if the yield curve is flat, the market expects short- term rates to be constant. The drawback is that it fails to consider price risk and reinvestment risk, but interest risk increases as the term to maturity increases.
– The broadest interpretation is that given any investment horizon, investors expect the same return, regardless of the maturity of the investment vehicle selected. This ignores the price risk associated with selling a bond prior to its maturity.
– The local expectations form of the pure expectation theory is an interpretation that suggests that the return on bonds with different maturities will be identical over a short-term investment period, commencing immediately. This is the only interpretation of the pure expectation theory that can be sustained in equilibrium.
Term Structure of Interest Rates
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The two form of biased expectations theory are the liquidity theory and the preferred habitat theory.
– Liquidity theory: duration measures the price risk of holding a bond. Duration increases as the bond’s maturity lengthens. Liquidity theory says that investors will demand a risk premium for holding bonds with long maturities because the risk of this bonds is higher. - The yield curve will typically be upward sloping as investors demand higher yields on longer bonds. The yield curve could slope downwards, however, if expectations for lower rates in the future overwhelm the risk premium.
– Preferred habitat theory also proposes that forward rates represent expected future spot rates plus a premium, but it does no support the view that this premium is directly related to maturity. The existence of an imbalance between supply and demand for funds in a given maturity range will induce lenders and borrowers to shift from their preferred habitat (maturity range) to one that has the opposite imbalance. To do so, they must be offered a risk premium to compensate for the price and/or reinvestment risk.
Term Structure of Interest Rates
August 2014 © Dr. Enzo Mondello, CFA, FRM, CAIA
Market segmentation theory is similar to the preferred habitat theory in that it agrees that lenders and borrowers have preferred maturity ranges and there is no premium (or discount) large enough to induce investors out of their preferred maturity range.
Instead, the shape of the yield curve is proposed to be determined by the supply and demand for securities within a given maturity range. In the extreme, the segmentation theory implies that rates for a given maturity segment will be determined independently of all other maturities.
The shape of the yield curve depends exclusively on the supply and demand within maturity segments. Under this theory the yield curve can take any shape.
Term Structure of Interest Rates
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Spot Rate Curve, Yield Curve on Coupon Bonds, Par Curve, and Forward Rate Curve
A yield curve shows the term structure of interest rates by displaying yields across different maturities (i.e., yields of U.S. Treasury coupon bonds). Yields are calculated for several maturities and yields for bonds with maturities between these are estimated by linear interpolation.
The spot rate curve is a yield curve for single payments in the future, such as 0%-bonds or stripped Treasury par bonds. Yields on zero-coupon government bonds are spot rates.
The par curve shows the coupon rates for bonds of various maturities that would result in bond prices equal to their par values. It is not calculated from yields on actual bonds but is constructed from the spot rate curve.
Term Structure of Interest Rates
August 2014 © Dr. Enzo Mondello, CFA, FRM, CAIA
With spot rates of 1%, 2%, and 3%, a 3-year annual par bond will have payments that are:
Thus, the payment is 2,96 and the par bond coupon rate is 2.96%
A forward curve is a yield curve composed of forward rates, such as 1-year rates available at each year over a future period
(1.03)
100
PMT
2.96
100
PMT
)
02
.
1
(
PMT
01
.
1
PMT
3 2
Term Structure of Interest Rates
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Forward Rates
The general formula to calculate a semi-annual forward is:
where:
1fm forward rate that starts in m semi-annual periods for 6 months zm spot rate for a period of m semi-annual periods
zm+1 spot rate for a period of m semi-annual periods plus 6 months
Doubling the forward rate 1fm gives the bond equivalent yield for the forward rate that starts in m months for 6 months
1
)
z
1
(
)
z
1
(
f
m m 1 m 1 m m 1
Term Structure of Interest Rates
August 2014 © Dr. Enzo Mondello, CFA, FRM, CAIA
Example
The spot rates for 6-month Treasury bills and 1-year Treasury bills are 2.50% and 2.80% respectively, expressed as bond equivalent yields. The 6-month forward rate expressed as bond equivalent yields is closest to:
A) 2.96% B) 3.00% C) 3.10%
Term Structure of Interest Rates
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Example
Given the following spot rate curve, the implied forward rate in 12 months for 6 months is closest to:
Maturity Spot Rate
6 months 3.00% 12 months 4.00% 18 months 5.00% 24 months 6.00% A) 6.55% B) 7.02% C) 7.54%
Term Structure of Interest Rates
August 2014 © Dr. Enzo Mondello, CFA, FRM, CAIA
The relationship between a T-period spot rate (zT), the current 6-month spot rate (z1), and the 6-month forward rates is stated as:
where:
1f1 forward rate that starts in 6 months for 6 months 1f2 forward rate that starts in 12 months for 6 months
Just know that a spot rate is a package of forward rates and that discounting at either the forward rates or the spot rate will give the same present value.
(1
z
)(1
f
)(1
f
)...(1
f
1
z
T
1
1 1
1 2
1 T1 1/T
Term Structure of Interest Rates
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Example
The following forward rates are given:
Semi-annual Periods Notation Forward Rate
1 1f0 4.00%
2 1f1 4.60%
3 1f2 5.00%
4 1f3 5.20%
The 2-year spot rate is closest to: A) 4.40%
B) 4.70% C) 4.95%
Term Structure of Interest Rates
August 2014 © Dr. Enzo Mondello, CFA, FRM, CAIA
Example
The following forward rates are given:
Semi-annual Periods Notation Forward Rate
1 1f0 4.00%
2 1f1 4.60%
3 1f2 5.00%
4 1f3 5.20%
A 6% coupon bond pays the coupons semi-annually and has a remaining maturity of 1.5 years. The price of the bond is
closest to: A) 101.56% B) 102.00% C) 102.12%
Term Structure of Interest Rates
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Risks Associated with Investing in Bonds
Fixed-Income Valuation
Term Structure of Interest Rates
Yield Measures
Interest Rate Risk: Duration and Convexity
Credit Risk: Fundamentals of Credit Analysis
Managing Bond Portfolio
Content
Relative-Value Methodologies for Global
Corporate Bond Portfolio Management
Exchange Rate Risk: International Bond Investing
Managing Interest Rate Risk with Derivatives
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August 2014 © Dr. Enzo Mondello, CFA, FRM, CAIA