Chapter 11 Flashcards
(139 cards)
1
Q
What is the primary objective of the chapter on valuation of investments?
A
To describe how to value several different types of interest rate derivatives
Derivatives include forward rate agreements, interest rate swaps, interest rate caps and floors, and swaptions.
2
Q
What does the clean price of a bond exclude?
A
Accrued interest
The clean price is the quoted market price that appears on dealing screens.
3
Q
What is the dirty price of a bond?
A
The sum of the clean price and accrued interest
This is the price at which the bond is actually traded.
4
Q
Define zero-coupon spot yields.
A
The theoretical price of a bond obtained by discounting individual payments by appropriate zero-coupon spot yields
Zero-coupon spot yields are also referred to as zero-coupon rates, spot rates, or simply zero rates.
5
Q
What is the equation for determining the yield on a bond?
A
The yield is determined by discounting the sequence of payments at a single interest rate to give the market value
This yield is also referred to as the gross redemption yield.
6
Q
What does the par yield represent?
A
The coupon rate required to make the theoretical value of the bond equal to its nominal value
This is under the prevailing pattern of zero-coupon interest rates.
7
Q
What is bootstrapping in the context of yield curves?
A
A technique to construct the zero-coupon yield curve from the observed prices of coupon-bearing bonds
It involves finding zero-coupon yields recursively.
8
Q
Fill in the blank: The clean price of a bond is the price excluding _______.
A
accrued interest
9
Q
True or False: The dirty price of a bond is always lower than its clean price.
A
False
10
Q
What is the first step in the bootstrapping process?
A
Determine the zero-coupon yield for the bond with the shortest outstanding term
This information is then used to determine the spot yield for the next shortest bond.
11
Q
What is the formula for calculating the price of a bond?
A
Price = redemption yield x (N - 1 + D) / (1 + freq) + L / (100 x rate freq)
Where rate yld is the coupon rate, redemption yield, redemption value, and various other terms are defined in the text.
12
Q
What do we assume about interest rates and yields throughout Section I?
A
They are compounded continuously, unless explicitly stated otherwise.
13
Q
What are the derivatives valued in this chapter?
A
- Forward rate agreements
- Interest rate swaps
- Interest rate caps and floors
- Swaptions
14
Q
What is the market price of a bond if its theoretical price is 95.99?
A
The market price is equal to the theoretical price of 95.99.
15
Q
What is the significance of the zero curve?
A
It is a plot of the zero-coupon yields obtained by bootstrapping.
16
Q
How do you calculate accrued interest for a bond?
A
Accrued interest = (coupon rate x number of days since last payment) / total days in coupon period
This calculation is necessary for determining the dirty price.
17
Q
What does the term ‘empirical characteristics of asset prices’ refer to?
A
It refers to the observed behaviors and patterns of asset prices identified through empirical studies.
18
Q
What is the role of self-assessment questions in this chapter?
A
To help students grasp the ideas being discussed.
19
Q
Fill in the blank: The theoretical price of a bond is calculated using _______.
A
zero-coupon spot yields
20
Q
What is the relationship between the market price and the theoretical price of a bond?
A
They should be equal, indicating the bond is fairly priced.
21
Q
What is the 4-year zero-coupon yield?
A
0.055281
This yield is denoted as s4.
22
Q
What is a plot of zero-coupon yields obtained by bootstrapping called?
A
Zero curve.
The zero curve visually represents the relationship between term and spot yield.
23
Q
How is the zero curve assumed to behave by convention?
A
- Linear between bootstrapped points
- Horizontal for terms shorter than the first point and longer than the last point.
24
Q
What is a forward rate?
A
The interest rate implied by current zero-coupon rates for a specified future time period.
Forward rates are derived from zero-coupon yields.
25
What does the term structure of forward rates represent?
It can be found directly from the term structure of zero rates.
## Footnote
Conversely, the term structure of zero rates can also be derived from forward rates.
26
What is the 1-year forward rate for the nth year equal to?
The rate at which the investor can agree now to lend or borrow over the 1-year period ending at time n.
27
What is the forward rate for the 2nd year derived from?
The present value of a 2-year zero-coupon bond using the 2-year zero-coupon rate.
## Footnote
It can also be calculated by discounting over two years using forward rates.
28
What happens when R1 is less than R2 in the context of the zero curve?
Rp > R2.
## Footnote
This indicates an upward-sloping zero curve.
29
What is the instantaneous forward rate for a maturity of T?
The value of RF obtained as T2 approaches T1 with a common value of T.
## Footnote
It applies to an infinitesimal time interval beginning at time T.
30
What is a Forward Rate Agreement (FRA)?
A contract specifying that an interest rate RK will be earned for a specified principal over a time period between T1 and T2.
31
How is the value V of an FRA evaluated?
As the present value of the difference between the interest payments.
## Footnote
RK is the agreed interest rate, and Rp is the current forward rate.
32
What is the relationship between RK and RF when RK equals RF?
The FRA must have a value of zero.
33
How do interest rate futures differ from FRAs?
Interest rate futures are standardized and exchange-tradable contracts.
34
What do three-month interest rate future contracts typically involve?
They are available in various currencies and trade with specified delivery months.
35
What is the typical contract size for most interest rate futures?
1 million currency units.
36
What is the quoted price for interest rate futures?
Prices are quoted per 100 nominal.
37
Calculate the continuously-compounded forward rate for Year 2 given 1-year and 2-year zero rates of 7.5% and 7% respectively.
To be calculated based on the provided rates.
38
What is the value of a forward rate agreement that pays a rate of 7.1% pa compounded annually on a principal of $2 million from the end of Year 1 to the end of Year 2?
To be calculated based on the specific conditions of the agreement.
39
What is the principal amount equal in value to in a single Eurodollar contract?
About $1111.
40
How are interest rate futures prices quoted?
Per 100 nominal.
41
What is the formula to calculate the contract price for a Eurodollar futures contract?
10,000[100 - 0.2s(100 - z)]
42
If the quoted price for a 3-month Eurodollar interest rate future is 95.00, what is the actual price paid?
$987,500.
43
How much does a change of one basis point (0.01) in a Eurodollar futures quote correspond to in terms of contract price change?
$25.
44
What is the tick size for Eurodollar futures?
0.01.
45
What is the Eurodollar futures interest rate for a quoted price of Z = 95.00?
5% per annum.
46
What happens when the actual interest rate on the settlement date is known?
The contract is settled in cash.
47
What is the relationship between the quoted price (Z) and the actual interest rate (R) at settlement?
Z = 100 - R.
48
What is the profit or loss per contract if the price at settlement is 10 basis points higher than when the contract was agreed?
$250 per contract.
49
What is the main difference between forwards and futures contracts?
Forwards often have no cash flow until maturity, while futures involve daily marking-to-market.
50
What happens when interest rates are constant in terms of forward and futures prices?
The values of the cashflows are equal and the prices must also be equal.
51
Why do forward and futures prices differ when interest rates vary unpredictably?
Due to daily cashflows from settlement and interest earned on cash received.
52
What is the effect of a long futures contract being more attractive than a similar long forward contract when asset prices are positively correlated with interest rates?
Futures prices will tend to be higher than forward prices.
53
What typically happens to the gains from a long futures position when interest rates are high?
The gains tend to be invested at higher than average interest rates.
54
What is the convexity adjustment formula applied to convert futures to forward interest rates?
Forward rate = Futures rate - ½σ²t₁t₂.
55
What is the typical value for σ in the convexity adjustment formula?
1.2%.
56
Why are forward rates on un-margined interest rate forwards lower than the corresponding futures rates?
Because of the lack of daily cash flows and interest earned on cash.
57
How is an interest rate swap valued?
As a long position in one bond compared to a short position in another bond.
58
What is typically used to discount cashflows in an interest rate swap?
LIBOR zero-coupon interest rates.
59
What does Bfix represent in the context of valuing an interest rate swap?
The value of the fixed-rate bond underlying the swap.
60
What is the formula for valuing the fixed-rate bond in an interest rate swap?
Bfix = Σ (k * e^(-r*t)) where cashflows are k at time t.
61
What is the formula for valuing a fixed-rate bond?
nBfix = L ke-r,t + Le-r,t, i=1
## Footnote
Where k represents cash flows at time t, and L is the principal amount.
62
What is LIBOR in the context of floating-rate bonds?
LIBOR is the continuously-compounded zero rate used for estimating future coupon payments and discounting them.
63
How does the value of a floating-rate bond behave immediately after a payment date?
The value of the floating-rate bond will be L immediately after a payment date.
64
What is the relationship between market price and redemption value for floating-rate bonds?
There is no discrepancy between the market price and the redemption value.
65
What is the value of an interest-only variable-rate mortgage on Day I?
The value is equal to the amount of the loan (the principal).
66
In a floating-rate bond, what does the coupon paid at the end of each year depend on?
The coupon depends on the value of the floating interest rate at the start of that year.
67
What is the formula for the price of a bond with zero rates?
The price can be expressed using the formula: (1 + R2t) = (1 + R1)(1 + RF2) and (1 + R3t) = (1 + R1)(1 + RF2)(1 + RF3).
68
What is the significance of coupon payments in floating-rate bonds?
The argument applies immediately after any coupon payment during the lifetime of the bond.
69
What is the value of a bond today, just before the next payment date?
Its value will be L + k*, where k* is the floating-rate payment due at time t1.
70
What are the terms of the interest rate swap in Question 11.15?
Company X receives 6% pa fixed and pays 1-year LIBOR on a notional principal of $50 million.
71
What is the first step in valuing a swap as a series of forward rate agreements?
Calculate forward rates for each of the LIBOR rates that will determine swap cashflows.
72
How is the swap value determined?
The swap value is equal to the present value of the cashflows, discounted using the appropriate LIBOR zero rates.
73
What is the net present value of the cashflows payable in 6 months for the swap?
The net present value is equal to 50 x (0.06 - 0.0572)e^(-0.05 x 7.5) = 0.13607.
74
What is the purpose of calculating forward rates in the context of the swap?
To determine the amount of the floating-rate payment at a future time.
75
What is the annually-compounded forward rate for the period starting in 18 months' time?
The forward rate is equal to 6.1837% pa.
76
What does the no-arbitrage principle state about two assets providing identical payoffs?
They must have the same price.
77
What can an investor do if F0 < S0erT?
Short the asset at the current spot price, invest the sale proceeds risk-free, and enter into a forward contract to buy the asset.
78
What is the risk-free profit generated by the strategy if F0 < S0erT?
A risk-free profit of S0erT - F0.
79
What happens if F0 > S0erT?
Unlimited profit can be made by buying the asset now and entering into a forward contract to sell it at time T.
80
What is the relationship between the spot price and the forward contract price in arbitrage?
If the prices do not hold, arbitrage opportunities will exist.
81
What is the profit from a strategy of buying an asset and entering into a forward contract to sell it?
A risk-free profit of S0er' - F0
## Footnote
Where S0 is the current spot price and F0 is the forward price.
82
What does the forward price F0 eliminate?
Arbitrage opportunities
## Footnote
Arbitrage refers to the practice of taking advantage of price differences in different markets.
83
What is a hedge?
A trade to reduce market risk
## Footnote
Market risk is the risk of losses due to unpredictable changes in market values.
84
What is a short hedge?
Taking a short futures position to eliminate the effect of price movement
## Footnote
This is done by agreeing to sell the asset at a fixed price at a future date.
85
What happens if the price of the asset goes down in a short hedge?
The investor gains on the short futures position
## Footnote
This offsets the loss on the actual sale of the asset.
86
What is a long hedge?
Taking a long futures position to hedge against future price increases
## Footnote
This protects against the risk of higher costs when buying the asset.
87
What is basis risk?
The risk that the asset to be hedged is not the same as the asset underlying the futures contract
## Footnote
It can also arise from uncertainty in timing of the asset transaction.
88
What is the optimal hedge ratio (h)?
The ratio of the size of the position taken in futures contracts to the size of the exposure
## Footnote
It is calculated using the standard deviations of changes in spot and futures prices.
89
What is the formula for the optimal hedge ratio?
h = (σS / σF) * ρ
## Footnote
Where σS is the standard deviation of the change in spot prices, σF is the standard deviation of the change in futures prices, and ρ is the correlation coefficient.
90
What is meant by 'leptokurtic' in the context of return distributions?
Return distributions are peaked at the mean and fat-tailed compared with the normal distribution
## Footnote
This indicates a higher probability of extreme values.
91
What does the independence of increments assumption imply?
Price changes are expected to be independent and exhibit a random walk
## Footnote
This means that past price movements do not influence future price movements.
92
What is a heteroscedastic model?
A model where the variance of the error terms is not constant over time
## Footnote
This contrasts with a homoscedastic model, where the variance is constant.
93
What is the opposite of a heteroscedastic model?
A homoscedastic model
## Footnote
This model assumes constant variance across observations.
94
What is the relationship between share prices at time u and time t according to the continuous-time lognormal model?
Share prices follow a geometric Brownian motion
## Footnote
This relationship is described mathematically in financial theory.
95
What does slower-than-linear rate in equity values suggest?
Equity values exhibit long-run mean reversion
## Footnote
This contradicts the assumption that increments in equity prices are independent.
96
What have studies found regarding the dependence of equity returns?
Significant non-linear dependence with autocorrelated absolute values and squared returns
## Footnote
These studies generally do not find evidence of serial correlation in the prices themselves.
97
What does non-linear dependence of squared returns suggest about returns series?
Returns series could be heteroscedastic
## Footnote
This contradicts the homoscedasticity assumption of the random walk model.
98
How does financial market volatility behave during different market conditions?
Increases in recessions and financial crises; lower in bull markets than in bear markets
## Footnote
This is usually explained by the effect of leverage.
99
What is volatility clustering in asset returns?
Large and small shocks tend to be clustered together
## Footnote
This feature indicates non-linear dependence in the returns series.
100
What is a contentious issue regarding equity values, dividend growth rates, and inflation?
The relationship between these variables is unclear; there is little direct correlation
## Footnote
Some evidence suggests equity returns are negatively related to inflation.
101
What can the relationship between equity values and inflation indices indicate for ALM?
It is fundamental for strategic asset allocation
## Footnote
If equities do not provide a reasonable match for real liabilities, investing in them as an inflation hedge makes less sense.
102
What is the zero-coupon spot yield?
The continuously-compounded rate of return on a zero-coupon bond
## Footnote
It reflects the return earned on a bond that does not make periodic interest payments.
103
What is the bond yield?
The single interest rate that equals the discounted present value of a bond's payments to its market value
## Footnote
It represents the effective return on a bond.
104
What is the par yield?
The coupon rate required for the theoretical value of a bond to equal its nominal value
## Footnote
This is under the prevailing pattern of zero-coupon interest rates.
105
How can the zero-coupon yield curve be constructed?
From the observed prices of coupon-bearing bonds using bootstrapping
## Footnote
Bootstrapping is a method for deriving zero-coupon yields from coupon-bearing bond prices.
106
What is a forward interest rate?
Interest rate implied by current zero-coupon rates for a specified future time period
## Footnote
It can be calculated using the instantaneous forward rate formula.
107
What defines the value of a Forward Rate Agreement (FRA)?
The present value of the difference between interest payments
## Footnote
This helps in evaluating the FRA's market position.
108
What are three-month interest rate futures contracts?
Contracts available in a wide range of currencies
## Footnote
They provide a way to hedge against interest rate fluctuations.
109
What is the relationship between forward and futures rates?
Forward rate = futures rate - ½o-2tit2
## Footnote
A convexity adjustment is applied for accurate conversion.
110
What is a hedge?
A trade to reduce market risk
## Footnote
It can involve various financial instruments to mitigate potential losses.
111
What causes basis risk in hedging?
Arises when the asset to be hedged is not the same as the asset underlying the futures contract
## Footnote
Additional uncertainties regarding timing and closure of futures contracts can also contribute.
112
What are the empirical characteristics of asset prices identified by studies?
Departures from assumptions of normality, independence, and constancy of parameters
## Footnote
These assumptions include normality of increments in asset prices and constancy of drift and volatility.
113
What is a forward rate agreement (FRA)?
A forward contract where the parties agree that a specified interest rate will apply to a specified principal amount during a specified future time period.
## Footnote
This agreement is used in financial markets to lock in interest rates for future borrowing or lending.
114
What is the formula for continuously-compounded forward rate for Year 2?
0.065
## Footnote
This is derived from the formula: = 2x0.07-1x0.075 = 0.065.
115
What is the annually compounded forward rate for Year 2?
0.06716
## Footnote
This is calculated using the formula R = e^0.065 - 1.
116
What is the value of the forward rate agreement (FRA) to the lender?
$6,678
## Footnote
The value is calculated based on the difference between the fixed and floating rates and the principal amount.
117
How does a change of one basis point affect a Eurodollar futures quote?
$25
## Footnote
This is calculated using the formula: $10,000 X 0.25 X 0.01.
118
What is the equivalent annually-compounded interest rate given?
5.344% pa
## Footnote
This is known as the effective annual rate.
119
What is the cash payment on settlement for a futures contract priced at 100-5.24?
$325
## Footnote
The settlement cash payment is derived from the contract price differences.
120
What happens to an investor with a long futures position when the asset price increases?
Makes an immediate gain
## Footnote
This gain occurs due to the daily margining procedure of marking to market.
121
What is the impact of interest rate movements on a long forward position?
Not affected by interest rate movements
## Footnote
This contrasts with long futures positions, which are affected by interest rates.
122
What is the convexity adjustment in bond pricing?
The adjustment reflects the curvature of the price-yield relationship
## Footnote
It accounts for the non-linear behavior of bond prices as interest rates change.
123
What is the relationship between forward prices on un-margined forwards and futures prices?
Forward prices will be higher than corresponding futures prices
## Footnote
This is due to the lack of margining in forwards.
124
What is the formula for the value of a swap to Company X receiving fixed interest?
V_swap = B_fixed - B_float
## Footnote
This captures the difference in present values of fixed and floating cash flows.
125
How is the continuously-compounded 1-year forward rate calculated?
Using the formula R2T2 - R1T1 / (T2 - T1)
## Footnote
This measures the expected future interest rate based on current rates.
126
What is the basis of a futures contract?
The difference between the spot price of the asset (S0) and the futures price (F0)
## Footnote
It indicates how well a futures contract hedges against market risk.
127
Define basis risk.
The risk that the basis cannot be predicted with complete certainty
## Footnote
This risk is especially relevant at the settlement date.
128
Fill in the blank: A change of one basis point will change the contract price by _______.
$25
129
True or False: A long futures contract will always be more attractive than a long forward contract.
False
## Footnote
Futures contracts can be less attractive due to interest rate fluctuations.
130
What is the impact of negative correlation between asset prices and interest rates on long futures positions?
They tend to be less attractive than long forward contracts
## Footnote
This is due to the daily margining and interest rate effects.
131
What is basis risk?
The risk that the basis cannot be predicted in advance with complete certainty, except at the settlement date.
## Footnote
The basis should be zero if the asset to be hedged and the asset underlying the futures contract are identical, ignoring transaction costs.
132
What primarily causes basis risk for an investment asset?
Uncertainty regarding the level of the risk-free rate and the asset's future yield.
133
In the continuous-time lognormal model of security prices, what is the distribution of log(Su) - log(S1)?
N[µ(u-1), a^2(u-t)], where u > t.
134
What are the key parameters in the continuous-time lognormal model?
µ (drift) and σ (volatility).
135
True or False: In the continuous-time lognormal model, returns over non-overlapping intervals are assumed to be independent.
True.
136
What does leptokurtic distribution imply about market behavior?
Market crashes and jumps appear more often than expected from a normal distribution; days with little or no change also occur more frequently.
137
What characterizes a heteroscedastic model?
The volatility of the time series being modeled varies in some way.
138
What is the opposite of a heteroscedastic model?
A homoscedastic model, where volatility is modeled as a constant parameter.
139
List main departures in price and returns data from common asset model assumptions identified in empirical studies.
* Lack of normality of increments in (log) asset prices
* Degree of dependence between increments in asset prices
* Lack of constancy of parameters, e.g., drift (µ) and volatility (σ).
