Fixed Income Markets Flashcards by Nicholas Lennon

What is the duration of a fixed income instrument

FIS 3

Duration D_P is the sensitivity of an FIS’s price P to a shift on the yield curve dr
dP = -(D_P) x P x dr

How well did you know this?

Not at all

Perfectly

What is the Macauley duration of an FIS

FIS 3

MacD is a time weighted average of PV cash flows
MacD = Σw_i(T_i-t)
w_i = CF(i)/P (1+y/2)^-2T_i

Modified Duration = MacD/(1+y/2)

How well did you know this?

Not at all

Perfectly

Duration approximation: Describe the error terms and how it comes about

FIS 3

FIS prices are not linear with interest rates. Using a linear approximation (based on duration) will create error terms that increase as the rate shock Δy widens

How well did you know this?

Not at all

Perfectly

Define the 95% VaR and what it measures

95% VaR is the 5% quantile (π_0.05). VaR is a measure of how bad the unfavorable tail events can be.

Not a coherent measure, not super great

How well did you know this?

Not at all

Perfectly

What are 5 issues with VaR (potentially under normal dist)

VaR depends on the distribution asumed for portfolio returns
VaR varies with whether we use normal dist assumption or historical data
Duration approximation of portfolio value is only close for small changes in rates, while VaR as an extreme range is concerned with large changes
VaR measures the maximum loss with 95% prob, but doesn’t say anything about the distribution of losses worse than this point
Under the normal dist assumption, VaR depends on the expected portfolio value μ_P which is subject to errors

How well did you know this?

Not at all

Perfectly

Give the VaR and TVaR approximations under normal distribution assumption

Let dP ~ N(μ_P, σ_P), dr ~ N(μ, σ²)
The 95% VaR for the portfolio = μ_P - 1.645σ_P
95% TVaR = μ_P - 2.062σ_P

How well did you know this?

Not at all

Perfectly

What is the convexity of a portfolio

FIS 4

The weighted average of components. Consider N securities with prices P_i, convexities C_i. A portfolio W is made up of N_i of each security, with total value W = ΣN_iP_i. Then the portfolio convexity is C_w = ΣN_iP_iC_i/W

How well did you know this?

Not at all

Perfectly

What is the duration-convexity approximation of portfolio value

dP/P = -(D_P) dr + 1/2 Convexity (dr²)

How well did you know this?

Not at all

Perfectly

Describe mechanics of Duration Convexity Hedging with 2 bonds and give the system of equations

Suppose we have two hedging instruments Z₁ and Z₂ to cover our bond P. The total portfolio V = P + k₁P₁ + k₂P₂ is such that
P D_P + k₁D₁P₁ + k₂D₂P₂ = 0 and
P C_P + k₁C₁P₁ + k₂C₂P₂ = 0

How well did you know this?

Not at all

Perfectly

Describe the dynamics under the factor model for the yield curve

Let (T₁,…,T_n) be points of the yield curve and let (r₁ = r(t,T₁),…,r_n = r(t,T_n)) be the corresponding ZC rates. A factor model for term structure dynamics assumed the change in points on the yield curve dr_i is due to a set of common factors (Φ₁,…,Φ_n)
dr₁ = β₁₁dΦ₁+…+β_1mdΦ_m
… dr_n = β_n1dΦ₁+…+β_nmdΦ_m
Each β determines the impact of one factor on one yield curve point

How well did you know this?

Not at all

Perfectly

What is the Factor Duration of a ZCB (D_j,ZCB) under the factor model of rates

Let P_Z(t,T_i) = 100e^{-(T_i - t) r(t,T_i)}
where dr_i = β_i1dΦ₁+β_i2dΦ₂+β_i3dΦ₃
Then dP_Z(t,T_i)/dΦ_j = -(T_i-t) P_Z(t,T_i) β_ij and
D_j,ZCB = (T_i-t) β_ij

How well did you know this?

Not at all

Perfectly

How does Duration based RM do vs Factor based RM, and why does this difference occur

When interest rates change undergo a parallel change, duration based hedging performs acceptable
When the slope and curvature of the term structure change significantly, duration based hedging fails to capture these dynamics and the hedge breaks down
Factor based risk mgmt is much more helpful in these circumstances
In reality, rates almost never move parallel as duration hedges assume

How well did you know this?

Not at all

Perfectly

Define the forward curve

FIS 5

Let t=0, T be the first future date, and T+Δ be the second. Keeping Δ fixed, we get the forward curve f(0,T,T+Δ) = -ln(F(t,T,T+Δ))/Δ
The forward curve and spot curve r(0,T) move in tandem

How well did you know this?

Not at all

Perfectly

Formula connecting the spot rate and forward rate

f(0,T,T+Δ) = r(0,T) + (T+Δ)(r(0,T+Δ)-r(0,T))/Δ
This is positive if r(0,T) increases with T and negative if r decreases with T

How well did you know this?

Not at all

Perfectly

Define the ctsly compounded spot rate r(0,T_n) as the avg of forward rates

At time t=0 we know the spot rate r(0,T_n) and all the forward rates for n future periods of length Δ. We then have that
r(0,T_n) = 1/T_n (r(0,T₁)T₁ + f(0,T₁,T₂)Δ + … + f(0,T_n-1,T_n)Δ

How well did you know this?

Not at all

Perfectly

Valuation of an FRA based on ZCB Prices

Under an FRA, the net payment is made at contract maturity T₂
The initial value of the contract V(0) = 0
We estimate the ratio of ZCB(0,T₁)/ZCB(0,T₂) = M at t=0
At valuation time t, we recompute the ZCB prices
The FRA value is V^FRA(t) = N(Z(t,T₂)M-Z(t,T₁))

Valuation of FRA based on difference of forward rates

The net payment between counterparties at maturity T₂ is NΔ[r_n(T₁,T₂)-f_n(0,T₁,T₂)]
The time t value is V^FRA(t) = NΔZ(t,T₂(f_n(0,T₁,T₂) - f_n(t,T₁,T₂))
where f_n(t,T₁,T₂) = n[1/F(t,T₁,T₂)/^{1/[nx(T₂-T₁))-1]}

Valuation of a forward contract written on a semi-annual coupon bond

The no arbitrage forward price is
P_C^Fwd(0,T,T_n) = 100c/2 ΣF(0,T,T_i) + 100F(0,T,T_n)

Payoff of a forward swap contract

The payoff from entering the forward swap is the same as entering into a forward for purchasing a fixed rate bond with semiannual coupons c for par value of 100
Forward Swap Payoff = P_C(T,T_m) - 100
P_C(0,T,T_m) = 100c/2 ΣZ(T,T_i) + 100Z(T,T_m)

How do we find the forward swap rate

The forward swap rate f^S is the annual coupon rate that make the initial cost of the contract 0. I.e., it makes P_C^Fwd(0,T,T_m) = 100
f₂^S(0,T,T_m) = 2 (1-F(0,T,T_m))/ΣF(0,T,T_j)

Payoffs of an interest rate cap

FIS 6

A cap is a portfolio of caplets, each of which has payoff based on the excess of the qtrly compounded rate set at the prior quarter and the strike rate K
Payoff of caplet at T_i = 0.25 N Max(r₄(T_i-1)-K,0)

What is a Swaption, and when would it be exercised

A swaption is an interest rate contract where the buyer has the right to enter an interest rate swap at maturity T
At T, the buyer compares the strike rate of the swaption to the market quote of the swap rate of the underlying floating-for-fixed swap
The long position exercises the swaption if the market swap rate > strike swap rate
A holder of a call on this underlying swap is a payer swaption
a put option allows the buyer (the receiver swaption) to enter the swap, which they will if market rate < strike rate

What is the PV of cash flows for a swaption

If the option owner at T_a can enter the swap that expires at T_b, with payment period τ, that swap has value of
Notional Σ(a+1, b) Z(T_a, T_i) τ_i (F(T_a; T_i-1, T_i)-K)

How do options on a futures contract work

The underlying instrument is a futures contract
The option maturity is T is lower than the futures maturity T_f
The futures is written on some FIS (bond, ZCB, interest rate)
The option strike price K is called the stike futures price
At T, we compare the actual futures price F^Fut(T,T_F) to the strike K
The opition can be a call or put, and includes both European and American

How do options written on bonds work

* The underlying instrument is a ZCB or coupon bond * The option maturity is T is lower than the bond maturity T_B * The opition can be a call or put, and includes both European and American * The derivative mechanics work the same way as in Neftci 1, except the underlying is now a bond price observed at T

Which interest rate derivatives have 0 inception value and why?

FRAs, Forwards, Swaps, and Interest rate futures They cost nothing to enter because either party may be called upon to make a payment at maturity

Two shortcomings of futures vs forwards

* Basis Risk: Using futures, a firm may retain some residual risk because the available instrument may not be exactly right to hedge the risk * Tailing the Hedge: The cash flows arising from a futures position accrue over time which implies the need to account for TVM between the time the CF is realized and the maturity of the hedge

How do we construct a Zero cost collar contract

Sell puts on Treasury bills with strike K_P Use the premium to buy calls on T bills with strike K_C Look for strikes so that the option premiums perfectly offset each other. Technically ensure the total cost for purchasing the calls matches the premium from selling the puts K_PPut(K_P) = K_CCall(K_C)

What is the payoff or profit from a zero cost collar

Negative under K_P, 0 between K_P and K_C, positive above K_C. By entering this strategy, we effectively trade some upside potential for some downside protection. We can also change strikes so that the contract is no longer zero cost

What is the DV01

DV01 = -d/dy PV(y) This is the dollar price change per change in yield. The DV01 (dollar duration) for a portfolio is the sum of dollar durations for each instrument in the portfolio

What is Modified Duration

ModD = -100/PV(y) d/dy PV(y) = 100 DV01/PV(y) This can be thought of as the DV01 expressed in percentage units. Choosing between DV01 and ModD is largely driven by convenience

Describe Macaulay Duration

* MacD applies to instruments only with fixed cash flows while ModD can work for more general cash flows * MacD is a time weighted average while ModD is a percentage rate of change of price wrt yield

How are the forward, zero, and discount curves related?

Let the forward curve = f(t), zero coupon yield curve = z(t), discount curve = d(t) Assume all rates are continuously compounded. We have z(t) = ∫(0,t) f(u)du/t d(t) = e^{-z(t) t} = e^{-∫(0,t) f(u)du}

What is the Partial DV01

If the forward curve is a function of variables (v₁,...v_k), the prices of instruments are also a function of those variables. In fact, we can represent it as a vector Price = vec(P₁...P_N) = vec(P₁(v₁,...v_k)...P_N(v₁,...v_k)) The matrix of partial derivatives is the Partial DV01 wrt curve parameters PDV01 = matrix((∂P₁/∂v₁...∂P_N/∂v₁)...(∂P₁/∂v_k...∂P_N/∂v_k))

What is the derivative of prices wrt rates, the DpDr

Let (r₁,...,r_k) = the k curve input rates, P_i = price of ith instrument, i = 1...n The DpDr is expressed as a matrix of partial derivatives DpDr = matrix((∂P₁/∂r₁...∂P_N/∂r₁)...(∂P₁/∂r_k...∂P_N/∂r_k))

What is the derivative of prices wrt alternate variables, the DpDx

Assume some alternate variable set (x₁,...,x_k) and assume we can calculate these from the forward curve parameters (v₁,...v_k). The DpDx is a matrix of partial dervatives DpDx = matrix((∂P₁/∂x₁...∂P_N/∂x₁)...(∂P₁/∂x_k...∂P_N/∂x_k))

How can we express DpDx as a product of 2 matrixes

The DpDx = DpDv * (DxDv)^-1 DpDv = matrix((∂P₁/∂v₁...∂P_N/∂v₁)...(∂P₁/∂v_k...∂P_N/∂v_k)) and DxDv = The Jacobian of the alternate variables = matrix((∂x₁/∂v₁...∂x_k/∂v₁)...(∂x₁/∂v_k...∂x_k/∂v_k))

How can we express the DpDy

* Choose a set of traded instruments (x₁,...,x_k) that match the curve variables (v₁,...,v_k) in terms of maturity/other characteristics to ensure the Jacobian is well behaved (invertible) * Calculate the risk of these instruments wrt the curve variables to get the DpDv * Calculate the inverse Jacobian DvDy * Transform the risk using the equation DpDy = DpDv * DvDy