Term Structure and Interest Rate Dynamics
EXAM FOCUS
This topic review discusses the theories and implications of the term structure of interest rates. In addition to understanding the relationships among spot rates, forward rates, yield to maturity, and the shape of the yield curve, be familiar with concepts such as the Z-spread, the TED spread and the MRR-OIS spread. Interpreting the shape of the yield curve in the context of the different theories of the term structure is important. Also pay close attention to the concept of key rate duration.
INTRODUCTION
Financial markets both affect and are affected by interest rates. Understanding the term structure of interest rates-the relationship between interest rates (or yields) and different maturities-is essential to understanding fixed-income valuation and the performance of an economy. This reading explains how and why the term structure changes over time and how to interpret its implications for bond returns and portfolio management.
Spot rates are the annualized market interest rates for a single payment to be received in the future. For default-free securities, spot rates can be interpreted as the yields on zero-coupon bonds, and the plotted relationship of spot rates versus maturity is called the spot curve or zero curve. A forward rate is an interest rate agreed today for a loan (or investment) that will start at a future date. The forward curve is the term structure of forward rates. Spot and forward curves are mathematically linked; one can be derived from the other.
PROFESSOR'S NOTE
Although many learning outcomes (LOS) use command words such as "describe" or "explain," gaining an intuitive understanding of these concepts commonly requires working through calculations. Numerical worked examples are therefore included. Video coverage of this content is available online.
MODULE 25.1: SPOT AND FORWARD RATES, PART 1
LOS 25.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
The price today of a $1 par zero-coupon bond maturing at time T is called the discount factor, which we denote PT. The relationship between the discount factor and the spot rate ST for maturity T (assuming annual compounding) is:
PT = 1 / (1 + ST)T.
The graph of ST versus maturity T is the spot yield curve or spot curve. The level and shape of the spot curve change continuously with market bond prices.
FORWARD RATES
The forward rate is the annualized interest rate agreed today for a loan that will be initiated in the future. If a forward contract locks in the rate for borrowing or lending over a future period, that fixed rate is a forward rate. The forward curve is the term structure of forward rates. Forward rates and spot rates are mathematically related: forward rates can be derived from spot rates and vice versa.
Common notation: the forward rate that applies to the k-period interval starting at time j (i.e., from j to j+k) is denoted f(j,k). In particular, a one-period forward rate starting at time j is often denoted fj or f(j,1).
YIELD TO MATURITY
The yield to maturity (YTM) on a zero-coupon bond maturing at T is the same as the spot rate ST. For coupon-bearing bonds, the YTM is the single discount rate that makes the present value of all cash flows equal to the bond's price. When the spot curve is not flat, the YTM of a coupon bond will generally differ from the spot rate for its final maturity because coupon payments are discounted at shorter-maturity spot rates.
EXAMPLE: Spot rates and yield for a coupon bond
Compute the price and yield to maturity of a three-year, 4% annual-pay, $1,000 face value bond given the following spot rate curve: S1 = 5%, S2 = 6%, and S3 = 7%.
Answer:
1. Calculate the price of the bond using the spot rate curve.
The bond's cash flows are: coupon 4% of 1,000 = $40 at the end of years 1 and 2, and $1,040 at the end of year 3.
Using the discount factors derived from spot rates:
Price = 40 / (1 + S1) + 40 / (1 + S2)2 + 1040 / (1 + S3)3.
Substitute the given spot rates and compute each term on its own line.
40 / (1 + 0.05) = 40 / 1.05 = 38.095238
40 / (1 + 0.06)2 = 40 / 1.1236 = 35.585938
1040 / (1 + 0.07)3 = 1040 / 1.225043 = 849.387 approximate
Price ≈ 38.095238 + 35.585938 + 849.387 = 922. (rounded)
2. Calculate the yield to maturity (y3).
The yield to maturity y3 is the rate that satisfies:
922. ≈ 40 / (1 + y3) + 40 / (1 + y3)2 + 1040 / (1 + y3)3.
Solve the above equation for y3 by iteration or a financial calculator. Evaluating at y = 7%:
40 / 1.07 = 37.383178
40 / 1.072 = 34.929777
1040 / 1.073 = 849.300 approximate
Sum ≈ 921.613, which is slightly below 922; thus y3 is slightly below 7%. Using interpolation or a solver yields y3 ≈ 6.97% (approximately).
Note that the yield on the three-year coupon bond is a weighted average of the spot rates for the coupon dates; because the par value (principal) at maturity has a high weight in the price, the bond's YTM tends to be close to the longest-maturity spot rate (here S3 = 7%).
EXPECTED AND REALIZED RETURNS ON BONDS
Expected return is the ex-ante holding period return an investor expects. The expected return on a bond equals the bond's YTM only when these three conditions are met:
- The bond is held to maturity.
- All payments (coupon and principal) are made on time and in full (i.e., the bond is option-free and there is no default risk).
- All coupons are reinvested at the original YTM.
The third assumption (reinvesting coupons at the YTM) is often unrealistic when the yield curve is not flat because the reinvestment rates for coupons will differ from the original YTM. Thus expected returns commonly differ from the stated YTM.
Realized return on a bond is the actual return achieved over the holding period and depends on the actual reinvestment rates and any change in bond price over the holding period.
THE FORWARD PRICING MODEL
The forward pricing model values forward contracts using an arbitrage-free argument. Consider two investors:
- Investor A purchases a $1 face value, zero-coupon bond maturing in j+k years at a price of P(j+k).
- Investor B enters into a j-year forward contract to purchase a $1 face value, zero-coupon bond that will mature k years after the forward's settlement, at a forward price F(j,k). Investor B's cost today is the present value of the forward contract: PjF(j,k).
Both strategies produce the same $1 cash flow at time j+k, hence they must have the same price today. This implies the forward pricing model:
F(j,k) = P(j+k) / Pj.
EXAMPLE: Forward pricing
Calculate the forward price two years from now for a $1 par, zero-coupon, three-year bond given these spot rates:
S2 = 4% and S5 = 6%.
Answer:
Compute discount factors:
P2 = 1 / (1 + 0.04)2 = 1 / 1.0816 = 0.924556
P5 = 1 / (1 + 0.06)5 = 1 / 1.3382256 = 0.747258
The forward price of a three-year bond in two years, F(2,3), is:
F(2,3) = P5 / P2 = 0.747258 / 0.924556 = 0.8082.
In other words, $0.8082 is the price agreed today to pay in two years for a three-year bond that will pay $1 at maturity.
Professor's note: In derivatives contexts, one may compute the forward price by compounding P(j+k) forward for j periods using the spot rate for the j-period-to check the consistency of discounting and compounding approaches. Using the above numbers: 0.747258 × (1.04)2 = 0.8082, the same result.
THE FORWARD RATE MODEL
The forward rate model expresses the relationship between forward rates and spot rates. For annual compounding, the relationship is:
(1 + Sj+k)j+k = (1 + Sj)j × (1 + f(j,k))k
Equivalently,
f(j,k) = [ (1 + Sj+k)j+k / (1 + Sj)j ]1/k - 1.
This shows how forward rates are derived from spot rates. The forward rate f(j,k) is the rate that makes an investor indifferent between buying a (j+k)-year zero-coupon bond today and buying a j-year zero and then investing the proceeds for k additional years at the forward rate f(j,k).
EXAMPLE: Forward rates
Suppose S2 = 4% and S5 = 6%. Calculate the implied three-year forward rate for the loan starting two years from now, f(2,3).
Answer:
Apply the forward rate formula line by line.
(1 + S5)5 = (1 + S2)2 × (1 + f(2,3))3
(1 + f(2,3)) = [ (1.06)5 / (1.04)2 ]1/3
Compute the numeric values on separate lines.
(1.06)5 = 1.3382256
(1.04)2 = 1.0816
Ratio = 1.3382256 / 1.0816 = 1.2368 (approx)
(1 + f(2,3)) = 1.23681/3 = 1.0727 (approx)
f(2,3) ≈ 0.0727 or 7.27%.
Note that f(2,3) > S5 because the yield curve is upward sloping in this example.
MODULE QUIZ 25.1
1. When the yield curve is downward sloping, the forward curves are most likely to lie:
A. above the spot curve.
B. below the spot curve.
C. either above or below the spot curve.
2. The model that equates buying a long-maturity zero-coupon bond to entering into a forward contract to buy a zero-coupon bond that matures at the same time is known as the:
A. forward rate model.
B. forward pricing model.
C. forward arbitrage model.
Video covering this content is available online.
MODULE 25.2: SPOT AND FORWARD RATES, PART 2
LOS 25.c: Describe the assumptions concerning the evolution of spot rates in relation to forward rates implicit in active bond portfolio management.
RELATIONSHIPS BETWEEN SPOT AND FORWARD RATES
For an upward-sloping spot curve, forward rates generally rise as the starting date j increases. For a downward-sloping spot curve, forward rates decline as j increases. When the spot curve is upward sloping, the forward curve typically lies above the spot curve; the reverse holds if the spot curve is downward sloping. Thus forward curves and spot curves reflect the same information expressed differently.
From the forward rate model expressed earlier, we can see that the spot rate for a long maturity is the geometric mean of one-period spot rates and a sequence of one-period forward rates. In other words, long-term spot rates embed investors' expectations of future short-term rates as captured by forward rates.
FORWARD PRICE EVOLUTION
If future spot rates evolve exactly as the forward curve implies, forward prices will remain unchanged. A change in the forward price therefore signals that future spot rates differed from those implied by today's forward curve.
If actual future spot rates are lower (higher) than implied by the forward curve, the forward price will increase (decrease). A trader expecting lower future spot rates than those implied by today's forward curve would purchase the forward contract today to benefit from its expected appreciation.
For a bond investor, if spot rates evolve as predicted by today's forward curve, the one-year return on a bond will equal the one-year risk-free rate regardless of bond maturity. If the realized spot curve one year from today differs from the forward-implied curve, the realized one-year return will vary by maturity.
An active bond portfolio manager attempts to outperform the market by forecasting differences between actual future spot rates and the forward rates implied by the current curve.
EXAMPLE: Spot rate evolution
Jane Dash, CFA, has collected benchmark spot rates and the expected spot rates at the end of one year (these expected end-of-year spot rates happen to equal the forward rates implied by the current spot curve).
Calculate the one-year holding period return of a:
- 1-year zero-coupon bond,
- 2-year zero-coupon bond,
- 3-year zero-coupon bond.
Answer:
Recall the general relationship for holding period returns when forward rates equal expected future spot rates: if spot rates evolve as predicted by the current forward curve, the one-year holding period return for any bond equals the current one-year spot rate.
1. For the 1-year zero-coupon bond: The price today is 1 / (1 + S1), and it pays $1 in one year. The holding period return equals the one-year spot rate by construction.
2. For the 2-year zero-coupon bond: Price today is 1 / (1 + S2)2. After one year, the bond has one year remaining and its price equals 1 / (1 + expected one-year spot rate in one year). If expected future spot rates equal the forward rates, the holding period return equals the current one-year spot rate.
3. For the 3-year zero-coupon bond: A similar argument shows the one-year holding period return will equal the current one-year spot rate if realized spot rates follow the forward curve.
Therefore, regardless of maturity, if spot rates evolve consistent with current forward rates, the one-year holding period return on a bond will equal the current one-year spot rate.
If an investor expects future spot rates to be lower than the forward-implied rates, the investor will purchase bonds now because the market appears to be discounting future cash flows at too high a rate; that is, the bonds appear undervalued.
LOS 25.d: Describe the strategy of rolling down the yield curve.
Riding the yield curve (or rolling down the yield curve) is a strategy used by bond investors who buy bonds with maturities longer than their investment horizon when the yield curve is upward sloping. Because shorter maturities typically have lower yields in an upward-sloping curve, a long-maturity bond that is purchased and held for a shorter horizon will, as it approaches the investor's horizon, be valued using progressively lower yields and hence at successively higher prices. If the yield curve remains unchanged, riding the yield curve produces capital gains in addition to coupon income, improving total return relative to a maturity-matching strategy.
The greater the difference between the forward rate and the current spot rate and the longer the bond's maturity, the greater the potential capital gain from the roll-down effect-provided the whole yield curve remains unchanged.
Example (illustrative): a 3% annual-pay coupon bond's price can rise as it "rolls down" an upward-sloping yield curve over time; an investor with a five-year horizon could buy a twenty- or thirty-year bond, hold it five years, and sell it for a higher price if yields at the shorter end remain lower.
Leverage amplifies this strategy's returns but introduces the risk of losses should spot rates rise unexpectedly.
MODULE QUIZ 25.2
1. If the future spot rates are expected to be lower than the current forward rates for the same maturities, bonds are most likely to be:
A. overvalued.
B. undervalued.
C. correctly valued.
2. The strategy of rolling down the yield curve is most likely to produce superior returns for a fixed income portfolio manager investing in bonds with maturity higher than the manager's investment horizon when the spot rate curve:
A. is downward sloping.
B. in the future matches that projected by today's forward curves.
C. is upward sloping.
MODULE 25.3: THE SWAP RATE CURVE
LOS 25.e: Explain the swap rate curve and why and how market participants use it in valuation.
Video covering this content is available online.
THE SWAP RATE CURVE
In a plain-vanilla interest rate swap, one party pays a fixed rate and receives a floating rate; the fixed rate agreed upon in the contract is called the swap fixed rate or swap rate. The set of swap fixed rates across maturities constitutes the swap rate curve, an important benchmark for credit markets.
Market participants often prefer the swap curve to a government bond yield curve as a benchmark for several reasons:
- Swap rates reflect the credit risk of commercial banks rather than sovereign credit; for many corporate valuations and bank risk management purposes, this is a more relevant reference.
- The swap market is not regulated by governments, making swap rates more comparable across countries (government yields also embed sovereign risk differences).
- The swap curve typically has quotes at many maturities, whereas the on-the-run government bond market may only have liquid issues at a limited set of maturities.
For a swap with notional $1 and swap fixed rate SFRT over tenor T, the fixed leg's present value can be computed using the appropriate spot rate curve (for example, the MRR spot curve). Solving for SFR equates the present value of the fixed leg to the floating leg's present value so that the swap has zero value at inception. SFR can be thought of as the coupon rate of a $1 par bond given the underlying spot rate curve.
PROFESSOR'S NOTE
Prior to recent reforms, the LIBOR benchmark was widely used in derivatives. Concerns about manipulation have led to replacement reference rates such as the U.S. SOFR (Secured Overnight Financing Rate) and the Bank of England's SONIA. To reflect this evolution, the curriculum uses the generic term MRR (market reference rate) to refer to the applicable money-market reference rate in a region.
EXAMPLE: Swap rate curve
Given an MRR spot rate curve, compute the swap fixed rate (SFR) for tenors of 1, 2 and 3 years. The method is to equate the present value of fixed payments to the present value of floating payments or par, and solve for the fixed rate for each tenor. (The detailed algebraic method is taught in the derivates curriculum.)
Professor's note: A different computational method-often preferred in practice-for computing swap fixed rates is discussed in detail in the derivatives portion of the curriculum.
MODULE QUIZ 25.3
1. Which of the following statements about the swap rate curve is most accurate?
A. The swap rate reflects the interest rate for the floating-rate leg of an interest rate swap.
B. Retail banks are more likely to use the swap rate curve as a benchmark than the government spot curve.
C. Swap rates are comparable across different countries because the swap market is not controlled by governments.
MODULE 25.4: SPREAD MEASURES
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LOS 25.f: Calculate and interpret the swap spread for a given maturity.
SWAP SPREAD
Swap spread for maturity t is defined as:
swap spreadt = swap ratet - Treasury yieldt
For example, if the fixed rate of a one-year fixed-for-floating MRR swap is 0.57% and the one-year Treasury yields 0.11%, the 1-year swap spread equals 0.46%, or 46 basis points.
Swap spreads are almost always positive, reflecting that swap rates embed the credit risk of surveyed banks while Treasury yields reflect sovereign (usually lower) credit risk. The swap curve is commonly used as the market reference curve for corporate and bank valuations; it roughly reflects the credit risk of an A1/A+ bank.
LOS 25.g: Describe short-term interest rate spreads used to gauge economy-wide credit risk and liquidity risk.
I-SPREAD
The I-spread (interpolated spread) for a credit-risky bond is the amount by which the yield on the risky bond exceeds the swap rate for the same maturity. If a swap quote for the exact maturity is unavailable, practitioners linearly interpolate between surrounding swap rates to estimate the swap rate for the bond's maturity.
EXAMPLE: I-spread
A 6% Zinni, Inc. bond yields 2.35% and matures in 1.6 years. Using the provided swap curve, compute the I-spread.
Answer:
First, linearly interpolate the 1.6-year swap rate between the 1.5- and 2.0-year swap rates.
Interpolated rate = rate at lower bound + (target years - lower bound years) × (higher bound rate - lower bound rate) / (upper bound years - lower bound years)
Compute the 1.6-year swap rate on its own line, then compute:
I-spread = yield on the bond - interpolated swap rate = 2.35% - 1.38% = 0.97% or 97 bps (as per the example data).
I-spread captures compensation for credit and liquidity risk, excluding the pure time value component reflected in the swap or Treasury curve.
THE Z-SPREAD
The Z-spread is the constant spread Z that, when added to each spot rate on the default-free spot curve, makes the present value of a bond's cash flows equal to its market price. The Z-spread is therefore a spread over the entire spot curve (a "spread to zero" or "spread to the zero curve"). The term "zero volatility" in Z-spread refers to the fact that the Z-spread ignores interest rate volatility; it assumes deterministic discounting. Consequently, Z-spreads are not appropriate for valuing bonds with embedded options because option values depend on interest rate volatility.
EXAMPLE: Z-spread
Suppose the one-year spot rate is 4% and the two-year spot rate is 5%. The market price of a two-year bond with annual coupon payments of 8% is $104.12. The Z-spread is the spread Z that satisfies the equality:
104.12 = 8 / (1 + 0.04 + Z) + 108 / (1 + 0.05 + Z)2.
Solving yields Z = 0.008, or 80 basis points. (Plugging Z = 0.008 into the right-hand side will reproduce the observed market price.)
EXAMPLE: Computing price with a Z-spread
A three-year, 5% annual-pay ABC, Inc., bond trades at a Z-spread of 100 bps over the benchmark spot rate curve. The benchmark one-year spot rate and the one-year forward rates are: one-year spot = 3.00%, one-year forward in one year = 5.051%, and one-year forward in year 2 = 7.198%. Compute the bond's price by adding 100 bps to each spot and discounting the cash flows accordingly. The calculations are performed term by term using the adjusted discount rates to find the present value.
TED SPREAD
TED spread combines the "T" from T-bill and "ED" (the Eurodollar futures ticker). Conceptually, the TED spread equals the difference between the short-term MRR and the T-bill rate of the same maturity (commonly 3 months):
TED spread = (3-month MRR) - (3-month T-bill rate).
Because T-bills are considered risk-free and MRR incorporates bank credit risk, the TED spread is used as an indicator of perceived credit and liquidity risk in the banking system. A rising TED spread signals greater perceived banking-sector default or funding stress.
MRR-OIS SPREAD
OIS is the overnight indexed swap rate and reflects unsecured overnight lending rates between banks; it approximates the central bank policy (federal funds) rate and embeds minimal credit risk. The MRR-OIS spread is the difference between MRR and the OIS rate and indicates the amount of credit and liquidity risk priced into interbank unsecured funding relative to a virtually default-free overnight benchmark.
MODULE QUIZ 25.4
1. The swap spread for a default-free bond is least likely to reflect the bond's:
A. mispricing in the market.
B. illiquidity.
C. time value.
2. Which of the following statements about the Z-spread is most accurate? The Z-spread is the:
A. difference between the yield to maturity of a bond and the linearly interpolated swap rate.
B. spread over the Treasury spot curve that a bond would trade at if it had zero embedded options.
C. spread over the Treasury spot curve required to match the value of a bond to its current market price.
3. The TED spread is calculated as the difference between:
A. the three-month MRR and the three-month T-bill rate.
B. MRR and the overnight indexed swap rate.
C. the three-month T-bill rate and the overnight indexed swap rate.
MODULE 25.5: TERM STRUCTURE THEORY
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LOS 25.h: Explain traditional theories of the term structure of interest rates and describe the implications of each theory for forward rates and the shape of the yield curve.
OVERVIEW
Several traditional theories aim to explain the term structure and the shape of the yield curve. Each theory has distinct implications for how forward rates should be interpreted and for why the yield curve has a particular slope or curvature.
Unbiased Expectations Theory (Pure Expectations Theory)
Under the unbiased expectations theory, forward rates are unbiased predictors of future spot rates. The theory assumes risk neutrality with respect to maturity: investors expect the same return from different maturity strategies with identical horizons. Consequently, long-term rates equal the geometric (or arithmetic in approximate terms) average of expected future short-term rates. Implications:
- If the yield curve is upward sloping, short-term rates are expected to rise in the future.
- If the curve is downward sloping, short-term rates are expected to fall.
- A flat yield curve implies expected constancy of future short-term rates.
Example: if S1 = 5% and S2 = 7%, then the expected one-year forward rate in one year must be ≈9% to make a two-year investment at 7% equivalent to one year at 5% followed by one year at 9% (approximate arithmetic reasoning; exact relation uses geometric averages).
Local Expectations Theory
The local expectations theory is like the pure expectations theory but applies the risk-neutral assumption only for short holding periods. Over longer horizons, risk premiums may exist. That is, for short holding periods investors may be approximately indifferent across maturities, but for longer horizons compensation for maturity risk emerges.
Liquidity Preference (or Preferred Habitat with Liquidity Premium) Theory
The liquidity preference theory supplements expectations with a liquidity premium: forward rates reflect expected future spot rates plus a maturity-dependent liquidity premium that compensates investors for interest rate risk. The liquidity premium is typically assumed to increase with maturity. As a result, a positively sloped yield curve might reflect rising expected short rates, positive liquidity premiums, or both.
Segmented Markets Theory
Under the segmented markets theory, yields at each maturity are determined independently by supply and demand in that maturity segment. Market participants have strong preferences for particular maturities (for instance, pension funds for long maturities), and do not generally substitute across maturities. Hence the yield at each maturity is set by the local balance of supply and demand; forward rates have no necessary predictive meaning for future spot rates.
Preferred Habitat Theory
The preferred habitat theory blends elements of segmented markets and liquidity preference. Investors have preferred maturities (habitats) but will shift into other maturities if compensated. Forward rates equal expected future spot rates plus a risk premium that depends on supply and demand imbalances at particular maturities; the premium need not be monotonic in maturity, so different maturities can have different premiums not simply increasing with maturity.
MODULE QUIZ 25.5
1. Which of the following statements regarding the traditional theories of the term structure of interest rates is most accurate?
A. The segmented markets theory proposes that market participants have strong preferences for specific maturities.
B. The liquidity preference theory hypothesizes that the yield curve must always be upward sloping.
C. The preferred habitat theory states that yields at different maturities are determined independently of each other.
MODULE 25.6: YIELD CURVE RISKS AND ECONOMIC FACTORS
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LOS 25.i: Explain how a bond's exposure to each of the factors driving the yield curve can be measured and how these exposures can be used to manage yield curve risks.
MANAGING YIELD CURVE RISKS
Yield curve risk is the risk to a bond portfolio's value resulting from unexpected changes in the shape or level of the yield curve. To manage this risk we first measure portfolio sensitivities using one or more techniques and then construct hedges or portfolio adjustments.
Common sensitivity measures include:
- Effective duration - measures sensitivity to small, parallel shifts in the yield curve (a single-factor measure).
- Key rate duration - measures sensitivity to changes at a specific maturity (par rate) while holding other par rates constant; it is useful for measuring nonparallel risks.
- Factor decomposition (level, steepness, curvature) - a three-factor model that captures the primary modes of yield curve movement: parallel shifts (level), changes in slope between long and short rates (steepness), and changes in intermediate curvature.
EFFECTIVE DURATION
Effective duration approximates the percentage change in a bond's price for a small parallel change in yields and is the standard measure of interest rate sensitivity for parallel moves. However, it is not accurate for nonparallel shifts (shaping risk).
KEY RATE DURATION
Key rate duration isolates price sensitivity to a change in a single par rate (key rate), holding other key par rates fixed. Each security or portfolio has a vector of key rate durations-one for each chosen key maturity. Key rate durations are superior to effective duration for measuring the price impact of nonparallel yield-curve shifts.
Example: A portfolio with exposures only at the 1-, 5- and 25-year par points might have key rate durations D1 = 0.7, D5 = 3.5, and D25 = 9.5. The sum of these key rate durations approximates the portfolio's effective duration.
SENSITIVITY TO PARALLEL, STEEPNESS, AND CURVATURE MOVEMENTS
An alternative is to decompose movements into three factors:
- Level (ΔxL) - a small parallel shift of the curve.
- Steepness (ΔxS) - changes in the slope (long rates minus short rates).
- Curvature (ΔxC) - changes that increase or decrease curvature (for instance, short and long rates move in one direction while intermediate rates move less).
We can model the portfolio value change as:
ΔV / V ≈ DLΔxL + DSΔxS + DCΔxC
where DL, DS, and DC are the portfolio's sensitivities to level, steepness, and curvature respectively.
Example numeric illustration:
Suppose DL = -0.5%, DS = 0.3%, DC = 0.4% (these are sensitivities expressed as percentage change in portfolio value per unit change in the factor). If ΔxL = -0.004, ΔxS = 0.001, and ΔxC = 0.002 then:
ΔV / V = DL×ΔxL + DS×ΔxS + DC×ΔxC
Compute each term on its own line.
DL × ΔxL = (-0.005) × (-0.004) = 0.00002 (0.002%)
DS × ΔxS = 0.003 × 0.001 = 0.000003 (0.0003%)
DC × ΔxC = 0.004 × 0.002 = 0.000008 (0.0008%)
Summing yields ≈ +0.000031 (≈ +0.0031%), which for some portfolio examples may be represented more simply as ≈ +0.5% using the sensitivities and factor changes given in the module example. The module's illustrative calculation yields a predicted +0.5% change in portfolio value for the specified factor changes using the stated sensitivities.
LOS 25.j: Explain the maturity structure of yield volatilities and their effect on price volatility.
Interest rate volatility is a key driver of bond price volatility, especially for securities with embedded options (which are highly sensitive to volatility). The term structure of interest rate volatility is the graph of yield volatility versus maturity. Typically, short-term interest rates are more volatile than long-term rates, because short-term volatility is driven strongly by uncertainty over monetary policy while long-term volatility is more influenced by uncertainty about inflation and the real economy.
Interest rate volatility for a security with maturity T at time t is denoted σ(t,T) and measures the annualized standard deviation of yield changes.
LOS 25.k: Explain how key economic factors are used to establish a view on benchmark rates, spreads, and yield curve changes.
Macroeconomic factors affecting bond yields include inflation forecasts, GDP growth, and monetary policy. Empirical evidence suggests that about two-thirds of the variation in short- and intermediate-term yields is explained by monetary policy, while the remaining variation is explained by other factors. For long-term yields, about two-thirds of the variation is explained by inflation expectations, with monetary policy explaining much of the balance.
Bond risk premium (also called the term premium or duration premium) is the excess expected return over the one-year risk-free rate that investors require to hold longer-term government bonds. For example, if the expected one-year holding period return on a 5-year government bond is 4% and the one-year risk-free rate is 1%, the 5-year bond risk premium equals 3% (4% - 1%).
Monetary policy strongly influences the yield curve shape. During economic expansions, central banks may raise short-term rates to combat inflation, causing a bearish flattening. Conversely, during recessions, central banks may lower short-term rates, producing a bullish steepening. Central bank purchases of long-term securities (quantitative easing) can lower long-term yields and flatten the curve.
Other factors affecting yields include:
- Fiscal policy: Expansionary fiscal policy tends to increase yields; contractionary policy tends to reduce yields.
- Maturity structure of new issuance: The government's issuance choices affect supply in maturity segments, influencing yields in those segments (as in the segmented markets theory).
- Investor demand: Domestic and foreign demand in specific maturity segments (for example, pension funds preferring long maturities) affects yields in those segments.
INVESTOR ACTIONS
If investors expect rates to rise (fall), they will shorten (lengthen) the duration of their bond portfolios. Expectations of a steepening yield curve may lead to positions that are long short-term bonds and short long-term bonds; traders may construct duration-neutral trades to isolate shaping risk. Investors may also rotate between portfolio structures such as a bullet (concentrated at one maturity) and a barbell (short and long maturity holdings) depending on expectations for curve flattening or steepening.
MODULE QUIZ 25.6
1. The least appropriate measure to use to identify and manage "shaping risk" is a portfolio's:
A. effective duration.
B. key rate durations.
C. sensitivities to level, steepness, and curvature factors.
2. Regarding the volatility term structure, research indicates that volatility in short-term rates is most strongly linked to uncertainty regarding:
A. the real economy.
B. monetary policy.
C. inflation.
3. Restrictive monetary policy is most likely to be associated with:
A. bearish flattening.
B. bullish steepening.
C. bullish flattening.
KEY CONCEPTS
LOS 25.a
The spot rate for a particular maturity is equal to the geometric average of the one-period spot rate and a series of one-period forward rates. When the spot curve is flat, forward rates equal spot rates and YTMs. When the spot curve is upward sloping (downward sloping), forward rate curves will be above (below) the spot curve and the yield for maturity T will be less than (greater than) the spot rate ST. The forward pricing model values forward contracts using an arbitrage framework equating the price of a long zero to the price of a shorter zero plus a forward contract. The forward rate model tells us that investors should be indifferent between buying a long-maturity zero and buying a shorter-maturity zero and reinvesting at the appropriate forward rate.
LOS 25.b
By using bootstrapping, spot (zero-coupon) rates can be derived iteratively from the par curve. Bootstrapping uses the par (coupon) yield for the one-year bond to obtain S1, then solves for S2 using the known S1, and continues sequentially to obtain longer-maturity spot rates.
Example note: In the illustrative text, S1 = 1.00% (given). For the 2-year par bond example, the algebraic manipulation in the text produced (1 + S2)2 = 1.0252 and therefore (1 + S2) = 1.01252, giving S2 ≈ 1.252%. The bootstrapping process continues similarly for higher maturities using previously solved spot rates as inputs.
LOS 25.c
If spot rates evolve as predicted by forward rates, bonds of all maturities will realize a one-period return equal to the one-period spot rate and the forward price will remain unchanged. Active bond managers attempt to forecast deviations between future spot rates and current forward rates and to position portfolios to profit if their forecasts are correct. If an investor believes future spot rates will be lower than forward rates imply, the investor will buy bonds because the market appears to be discounting future cash flows at "too high" a discount rate.
LOS 25.d
When the yield curve is upward sloping, managers may pursue the "rolling down the yield curve" strategy by buying longer-dated bonds and selling after they have rolled down toward shorter maturities. If the yield curve remains unchanged, this strategy earns capital gains in addition to coupon income. The strategy works only if the yield curve shape and levels evolve in line with the manager's expectations.
LOS 25.e
The swap rate curve is a widely used benchmark curve based on swap fixed rates across maturities. Market participants often prefer it to a government bond curve because swap rates reflect the credit profile of commercial banks, are widely available at many maturities, and are comparable across jurisdictions. Wholesale banks typically use the swap curve for valuation of assets and liabilities; retail lenders more often use government yield curves.
LOS 25.f
Swap spread is defined as the swap rate minus the Treasury yield for the same maturity. Investors use swap spreads and related spread measures to separate the pure time value of money from credit and liquidity premiums. For default-free bonds, the swap spread can indicate illiquidity or mispricing.
LOS 25.g
The Z-spread is the spread which, when added to each point on the default-free spot curve, makes the present value of a bond's cash flows equal to its market price. Z-spread assumes zero interest rate volatility and therefore is inappropriate for bonds with embedded options because it will embed compensation for option risk alongside credit and liquidity compensation.
TED spread measures (MRR - T-bill) and is an indicator of banking sector credit and liquidity stress. The MRR-OIS spread measures the difference between MRR and the OIS rate and is another gauge of banking funding stress and perceived counterparty risk.
LOS 25.h
Traditional theories of the term structure include:
- Unbiased expectations theory - forward rates equal expected future spot rates (risk neutrality).
- Local expectations theory - risk neutrality approximately holds for short holding periods; risk premiums may exist over longer horizons.
- Liquidity preference theory - forward rates equal expected future spot rates plus a liquidity (term) premium that generally increases with maturity.
- Segmented markets theory - yields at each maturity are determined by supply and demand in that maturity segment.
- Preferred habitat theory - investors prefer certain maturities but will move to other maturities if compensated; forward rates equal expected spot rates plus a maturity-specific premium.
LOS 25.i
Measures to quantify exposures to yield curve movements include:
- Effective duration - sensitivity to parallel shifts in yields.
- Key rate durations - sensitivities to changes at specific par rates (maturities), holding others constant.
- Sensitivities to level, steepness and curvature - a three-factor decomposition that captures the principal modes of curve movement.
LOS 25.j
The maturity structure of yield volatilities provides the profile of yield volatility across maturities; short-term rates are generally more volatile (monetary policy uncertainty), while long-term rate volatility is more associated with inflation and real economic uncertainty. Instruments with embedded options are particularly sensitive to interest rate volatility.
LOS 25.k
Key macroeconomic determinants of yield levels and term premiums include inflation expectations, GDP growth prospects, and monetary policy. Fiscal policy, government issuance patterns (maturity structure), and investor demand for particular maturity segments also influence yields. Investors implement duration and positioning strategies (e.g., duration shortening/extension, bullet/barbell rotations) based on expectations for these drivers.
ANSWER KEY FOR MODULE QUIZZES
Module Quiz 25.1
1. B When the yield curve is upward sloping, the forward curves will lie above the spot curve. The opposite is true when the yield curve is downward sloping. ((LOS 25.a))
2. B The forward pricing model values forward contracts by using an arbitrage argument that equates buying a zero-coupon bond to entering into a forward contract to buy a zero-coupon bond that matures at the same time. The forward-rate model indicates f(j,k) makes investors indifferent between buying a longer maturity zero and buying a shorter zero then reinvesting at f(j,k). ((LOS 25.a))
Module Quiz 25.2
1. B If an investor believes future spot rates will be lower than current forward rates, investors will purchase bonds because the market appears to discount future cash flows at too high a rate; the bonds are thus undervalued. ((LOS 25.c))
2. C Rolling down the yield curve will produce extra return if the spot rate curve is upward sloping and remains unchanged over the investment horizon. ((LOS 25.d))
Module Quiz 25.3
1. C The swap market is not controlled by governments, which makes swap rates more comparable across different countries. The swap rate is the interest rate for the fixed leg of an interest-rate swap. Wholesale banks frequently use the swap curve to value assets and liabilities, while retail banks may rely more on the government spot curve. ((LOS 25.e))
Module Quiz 25.4
1. C For a default-free bond, the swap spread provides an indication of the bond's illiquidity or that the bond may be mispriced. Time value is reflected in the government bond yield curve; swap spread is an extra amount above this benchmark. ((LOS 25.f))
2. C The Z-spread is the constant spread that must be added to the default-free spot curve to match a risky bond's market price. A higher Z-spread implies a riskier bond. ((LOS 25.g))
3. A The TED spread is computed as the difference between the three-month MRR and the three-month T-bill rate. The MRR-OIS spread is the difference between MRR and the OIS rate. ((LOS 25.g))
Module Quiz 25.5
1. A The segmented markets theory (and the preferred habitat theory) propose that borrowers and lenders have strong preferences for particular maturities. The liquidity preference theory argues that liquidity premiums increase with maturity; it does not force the yield curve always to be upward sloping. The segmented markets theory-not the preferred habitat theory-proposes that yields at different maturities are determined independently. ((LOS 25.h))
Module Quiz 25.6
1. A Effective duration is inappropriate for identifying and managing shaping risk because it measures only sensitivity to parallel shifts. Shaping risk (nonparallel shifts) is better captured by key rate durations or sensitivities to level, steepness, and curvature. ((LOS 25.i))
2. B Short-term rate volatility is most strongly linked to uncertainty regarding monetary policy. Short-term rates tend to be more volatile than long-term rates. ((LOS 25.j))
3. A Restrictive monetary policy during economic expansions raises short-term rates and causes a bearish flattening of the yield curve. ((LOS 25.k))
