1. The Term Structure and Interest Rate Dynamics Flashcards
Explain ‘Forward Pricing Model’.
The forward pricing model values forward contracts based on arbitrage-free pricing.
F(j,k) = P(j+k)/Pj
Investor enter in a ‘j-year’ forward contract maturing in ‘k’ years
Explain the ‘Forward Rate Model’.
[1+S(j+k)]^(j+k) = (1+S(j))^j * (1+f(j,k))^k
This equation suggests that the forward rate f(j,k) should make investors indifferent between buying a (j+k)-year-zero coupon bond versus buying a j-year-zero-coupon bond and at maturity reinvesting the principal for ‘k’ additional years.
How zero-coupon rates (spot rates) may be obtained from the par curve by bootstrapping?
A par rate is the yield to maturity of a bond trading at par. By using bootstrapping, spot rates can be derived from the par curve using the output of one step as an input to the next step.
FV = PMT/(1+S1) + PMT/(1+S2)^2+….
Example: Given the following (annual-pay) par curve, compute the corresponding spot rate curve:
Maturity 1: Par Rate = 1,00%
Maturity 2: Par Rate = 1,25%
Maturity 3: Par Rate = 1,5%
S1=1,00% (given directly)
If we discount each cash flow of the bond using its yield, we get the market price of the bond. Here, the market price is the par value. Consider the 2-year bond.
100 = 1/1,0125 + 101,25/(1,0125)^2
Alternatively, we can also value the 2-year bond USING SPOT RATES:
100 = 1,25/(1+S1) + 101,25/(1+S2)^2
= 1,25/1,01 + 101,25/(1+S2)^2
S2 = 1,252%
Explain ‘Forward Price Evolution’.
If the future spot rates actually evolve as forecasted by the forward curve, the forward price will remain unchanged. Therefore, a change in the forward price indicates that the future spot rates did not conform to the forward curve.
When spot rates turn out to be lower(higher) than implied by the forward curve, the forward price will increase (decrease).
A trader expecting lower future spot rates (than implied by the current forward rates) would purchase the forward contract to profit from its appreciation.
Describe the strategy of riding the yield curve.
The most straightforward strategy for a bond investor is maturity matching - purchasing bonds that have a maturity equal to the investor’s investment horizon. However, with an upward-sloping interest rate term structure, investors seeking superior returns may pursue a strategy called ‘riding the yield curve’ (also known as ‘rolling down the yield curve’).
Under this strategy, an investor will purchase bonds with maturities longer than his investment horizon. In an upward-sloping yield curve, shorter maturity bonds have lower yields than longer maturity bonds. As the bond approaches maturity (i.e. rolls down the yield curve), it is valued using sucessively lower yields and therefore, at sucessively higher prices.
If the yield curve remains unchanged over the investment horizon, riding the yield curve strategy will produce higher returns than a simple maturity matching strategy, increasing the total return of a bond portfolio. The greater the difference between the forward rate and the spot rate, and the longer the maturity of the bond, the higher the total return.
The risk of such a leveraged strategy is the possibility of an increase in spot rates.
Explain the swap rate curve and why and how market participants use it in valuation.
In a plain vanilla interest rate swap, one party makes payments based on a fixed rate while the counterparty makes payments based on a floating rate. The fixed rate in an interest rate swap is called the swap fixed rate or swap rate.
If we consider how swap rates vary for various maturities, we get the swap rate curve, which has become an important interest-rate benchmark for credit markets.
Given a notional principal of $1 and a swap fixed rate SFR(t), the value of the fixed rate payments on a swap can be computed using the relevant (e.g. LIBOR) spot rate curve. For a given swap tenor T, we can solve for SFR in the following equation.
SOMA[SFR(t)/(1+S(t))^t + 1/(1+S(t))^T= 1, t=1 to t=T
In the equation, SFR can be thought of as the coupon rate of a $1 par value bond given the underlying spot rate curve.
How to calculate and interpret the swap spread for a given maturity.
Swap spread refers to the amount by which the swap rate exceeds the yield of a government bond with the same maturity.
SwapSpread(t) = SwapRate(t) - TreasuryYield(t)
For example, if the fixed rate of a one-year fixed-for-floating LIBOR swap is 0,57 and the one-year Treasury is yielding 0,11%, the 1-year swap spread is 0,57-0,11=0,46%.
Explain the ‘I-Spread’.
The I-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. In a case where the swap rate for a specific maturity is not available, the missing swap rate can be estimated from the swap rate curve using linear interpolation (hen
ce the ‘I’ in the I-spread).
Describe the ‘Z-Spread’.
The Z-spread is the spread 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 the bond’s market price. Therefore, the Z-Spread is a spread over the entire spot rate curve.
For example, 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 that balances the following equality:
104,12 = 8/(1+0,04+Z) + 108/(1+0,05+Z)^2
In this case, the Z-spread is 0,008. (Plug Z = 0,008 into the right-hand-side of the equation above to reassure yourself that the present value of the bond’s cash flows equals 104,12.
The term zero volatility in the Z-Spread refers to the assumption of zero interest rate volatility. Z-Spread is not appropriate to use to value bonds with embedded options; without any interest rate volatility options are meaningless. If we ignore the embedded options for a bond and estimate the Z-Spread, the estimated Z-Spread will include the cost of the option.
Explain the TED-Spread.
The ‘TED’ in TED spread is an acronym that combines the ‘T’ in ‘T-bill’ with ‘ED’ (the ticker symbol for the Eurodollar futures contract).
Conceptually, the TED spread is the amount by which the interest rate on loans between banks (formally, three-month LIBOR) exceeds the interest rate on short-term US government debt (three-month T bills).
For example, if three-month LIBOR is 0,33% and the three-month T-bill rate is 0,03%, then:
TED Spread = (3-month LIBOR rate) - (3-month T-bill rate) = 0,33% - 0,03% = 0,30%
Because T-bills are considered to be risk free while LIBOR reflects the risk of lending to commercial banks, the TED spread is seen as an indication of the risk of interbank loans. A rising TED spread indicates that market participants believe banks are increasingly likely to default on loans and that risk-free T-bills are becoming more valuable in comparison. The TED spread captures the risk in the banking system more accurately than dows the 10-year swap spread.
Explain Libor-OIS Spread.
OIS stands for overnight indexed swap. The OIS rate roughly reflects the federal funds rate and includes minimal counterparty risk.
The LIBOR-OIS spread is the amount by which the LIBOR rate (which includes credit risk) exceeds the OIS rate (which includes only minimal credit risk). This makes the LIBOR-OIS spread a useful measure of credit risk and an indication of the overall wellbeing of the banking system.
A low LIBOR-OIS spread is a sign of high market liquidity while a high LIBOR-OIS spread is a sign that banks are unwilling to lend due to concerns about creditworthiness.
Explain the ‘Unbiased Expectations Theory’.
Under the unbiased expectations theory, we hypothesize that it is investors’ expectations that determine the shape of the interest rate term structure.
Specifically, this theory suggests that forward rates are
solely a function of expected future spot rates, and that every maturity strategy has the same expected return over a given investment horizon.
In other words, long-term interest rates equal the mean of future expected short-term rates. This implies that an investor should earn the same return by investing in a five-year bond or by investing in a three-year bond an then a two-year bond after the three-year bond matures.
Similarly, an investor with a three-year investment horizon would be indifferent between investing in a three-year bond or in a five-year bond that will be sold two years prior to maturity.
The underlying principle behind the pure expectations theory is risk neutrality: investors don’t demand a risk premium for maturity strategies that differ from their investment horizon.
Explain the ‘Local Expectations Theory’.
The Local Expectations Theory is similar to the Unbiased Expectations Theory with one major difference: the local expectations theory preserves the risk-neutrality assumption only for short holding periods.
In other words, over longer periods, risk premiums should exist. This implies that over short time periods, every bond (even long-maturity risky bonds) should earn the risk-free rate.
The local expectations theory can be shown not to hold because the short-holding-period returns of long-maturity bonds can be shown to be higher than short-holding-period returns on short-maturity bonds due to liquidity premiums and hedging concerns.
Explain the ‘Liquidity Preference Theory’.
The Liquidity Preference Theory of the term structure addresses the shortcomings of the Pure Expectations Theory by proposing that forward rates reflect investors’ expectations of future spot rates, PLUS a liquidity premium to compensate investors for exposure to interest rate risk. Furthermore, the theory suggests that this liquidity premium is positively related to maturity: a 25-year bond should have a larger liquidity premium than a five-year bond.
Thus, the liquidity preference theory states that forward rates are biased estimates of the market’s expectation of future rates because they include a liquidity premium. Therefore, a positive-sloping yield curve may indicate either: a) the market expects future interest rates to rise or b) rates are expected to remain constant (or even fall), but the addition of the liquidity premium results in a positive slope. A downward-sloping yield curve indicates steeply falling short-term rates according to the liquidity theory.
Explain ‘Segmented Markets Theory’.
Under the Segmented Markets Theory, yields are not determined by liquidity premiums and expected spot rates. Rather, the shape of the yield curve is determined by the preferences of borrowers and lenders, which drives the balance between supply of and demand for loans of different maturities. This is called the segmented markets theory because the theory suggests that the yield at each maturity is determined independently of the yields at other maturities; we can think of each maturity to be essentially unrelated to other maturities.
The segmented markets theory supposes that various market participants only deal in securities of a particular maturity because they are prevented from operating at different maturities. For example, pension plans and insurance companies primarily purchase long-maturity bonds for asset-liability matching reasons and are unlikely to participate in the market for short-term funds.
