Fixed Income Flashcards
Price of a bond at a certain node
value = {[.5(Vup + coupon/2)] + [.5(Vdown + coupon/2)]}/ (1 + interest rate/2)
How does the value of the embedded call option change when interest rate volatility increases?
All option values increase when the volatility of the underlying asset increases.
How will an increase in volatility affect the value of the callable bond?
The value of the callable bond decreases if the interest rate volatility increases because the value of the embedded call option increases. Since the value of the callable bond is the difference between the value of the non-callable bond and the value of the embedded call option, its value has to decrease.
How does the value of the embedded call option react to an increase in interest rates? The value of the embedded call is most likely to:
decrease. There are two different effects that an increase in interest rate will cause in this situation. The first (and primary) impact stems from the relationship between interest rates and bond values: when interest rates increase, bond values decrease. Since the underlying asset to the option (the bond) decreases in value, the option will decrease in value also. The second (and much smaller) effect stems from the fact that when interest rates are higher, call option prices are generally higher because holding a call (rather than the underlying) is more attractive when interest rates are high. However, this secondary effect is likely to be smaller than the impact of the change in bond value.
Value of a callable bond
Vcallable = Vnoncallable − Vcall
If the volatility of interest rates decreases, the call option is less valuable, which increases the value of the callable bond.
Z spread
is the spread that when added to each spot rate on the default free spot curve makes the present value of a bonds cash flows equal to the bonds market price.
Example: 1yr spot rate 4% 2yr: 5% Price 104.12 coupon 8%.
104.12= 8/(1+.04+Z) + 108/(1+.05+Z)^2
not appropriate to use with bonds that have embedded options
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.
Example 2.35% yield, maturity 1.6 years.
1.5 swap rate = 1.35%
2 yr swap rate= 1.5
1.35 + ([1.6-1.5)(1.5-1.35)/.5]= 1.38
then
2.35 - 1.38 = I spread .97
TED spread
difference between 3-month libor and 3-treasury
*a rising TED spread indicates that market participants believe banks are increasingly likely to default on loans
LIBOR-OIS spread
amount by which the LIBOR rate exceeds the OIS rate (overnight indexed swap).
A low LIBOR-OIS spread is a sign of high market liquidity. A high spread is a sign that banks are unwilling to lend due to credit concerns
Unbiased Expectations Theory
Pure Expectations Theory
it is the investors expectations that determine the shape of the interest rate term structure
Liquidity Preference Theory
address short comings of the Unbiased Theory, by adding a liquidity premium to compensate investors for exposure to interest rate risk
Segmented market Theory
yields are not determined by liquidity premiums and expected spot rates, rather determined by the preferences of borrowers and lenders which drives the balance between supply/demand
Preferred Habitat Theory
rates represent expected future spot rates plus a premium. It suggests that the existence of an imbalance between the supply and demand for funds in a given maturity range will induce lenders and borrowers to shift from their preferred habitats (maturity range).
CIR Model - Cox-Ingersoll-Ross
is based on the idea that interest rate movements are driven by individuals choosing between consumption today verses investing and consuming at a later time.
dr = a(b-r)dt + vol * √rdz
The Vasicek Model
like the CIR model, it suggest that interest rates are mean reverting to some long-run value.
The difference from the CIR model is this model does not increase as the level of interest rates increase.
Disadvantage: the model does not force interest rates to be non-negative
Ho-Lee Model
The model assumes that changes in the yield curve are consistent with a no-arbitrage condition.
Can be used to price zero-coupon bonds and to determine the spot curve.
Model produces normal distribution of future rates
Key Rate duration vs Effective
Key Rate - more precise, better at measuring the impact of nonparallel yield curve shifts.
The binomial interest rate tree framework is a lognormal random walk model with 2 desirable properties
higher volatility at higher rates
non-negative interest rates
The upfront payment made/received by the protection buyer
Upfront payment = (CDS spread − CDS coupon) × duration × notional principal
The price of a 1-year $1 par, zero-coupon bond to be issued in two years is closest to:
(1+S3)^3/(1+S2)^2
How to calculate effective duration and effective convexity
with up and down values with changes in rates
ED = (V- − V+) / (2V0(∆y))
EC = (V- + V+ − 2V0) / (V0(∆y)2)
OAS is interpreted as the average spread over the Treasury spot rate curve. The nominal spread is measured relative to the Treasury yield curve.
OAS is interpreted as the average spread over the Treasury spot rate curve.
The nominal spread is measured relative to the Treasury yield curve.
