Fixed Income Markets Flashcards
What is the duration of a fixed income instrument
FIS 3
Duration DP is the sensitivity of an FIS’s price P to a shift on the yield curve dr
dP = -(DP) x P x dr
What is the Macauley duration of an FIS
FIS 3
MacD is a time weighted average of PV cash flows
MacD = Σwi(Ti-t)
wi = CF(i)/P (1+y/2)-2Ti
Modified Duration = MacD/(1+y/2)
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
Define the 95% VaR and what it measures
95% VaR is the 5% quantile (π0.05). VaR is a measure of how bad the unfavorable tail events can be.
Not a coherent measure, not super great
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 μP which is subject to errors
Give the VaR and TVaR approximations under normal distribution assumption
Let dP ~ N(μP, σP), dr ~ N(μ, σ2)
The 95% VaR for the portfolio = μP - 1.645σP
95% TVaR = μP - 2.062σP
What is the convexity of a portfolio
FIS 4
The weighted average of components. Consider N securities with prices Pi, convexities Ci. A portfolio W is made up of Ni of each security, with total value W = ΣNiPi. Then the portfolio convexity is Cw = ΣNiPiCi/W
What is the duration-convexity approximation of portfolio value
dP/P = -(DP) dr + 1/2 Convexity (dr2)
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 DP + k1D1P1 + k2D2P2 = 0 and
P CP + k1C1P1 + k2C2P2 = 0
Describe the dynamics under the factor model for the yield curve
Let (T1,…,Tn) be points of the yield curve and let (r1 = r(t,T1),…,rn = r(t,Tn)) be the corresponding ZC rates. A factor model for term structure dynamics assumed the change in points on the yield curve dri is due to a set of common factors (Φ1,…,Φn)
dr1 = β11dΦ1+…+β1mdΦm
… drn = βn1dΦ1+…+βnmdΦm
Each β determines the impact of one factor on one yield curve point
What is the Factor Duration of a ZCB (Dj,ZCB) under the factor model of rates
Let PZ(t,Ti) = 100e-(Ti - t) r(t,Ti)
where dri = βi1dΦ1+βi2dΦ2+βi3dΦ3
Then dPZ(t,Ti)/dΦj = -(Ti-t) PZ(t,Ti) βij and
Dj,ZCB = (Ti-t) βij
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
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
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
Define the ctsly compounded spot rate r(0,Tn) as the avg of forward rates
At time t=0 we know the spot rate r(0,Tn) and all the forward rates for n future periods of length Δ. We then have that
r(0,Tn) = 1/Tn (r(0,T1)T1 + f(0,T1,T2)Δ + … + f(0,Tn-1,Tn)Δ
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 VFRA(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[rn(T1,T2)-fn(0,T1,T2)]
The time t value is VFRA(t) = NΔZ(t,T2(fn(0,T1,T2) - fn(t,T1,T2))
where fn(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
PCFwd(0,T,Tn) = 100c/2 ΣF(0,T,Ti) + 100F(0,T,Tn)
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 = PC(T,Tm) - 100
PC(0,T,Tm) = 100c/2 ΣZ(T,Ti) + 100Z(T,Tm)
How do we find the forward swap rate
The forward swap rate fS is the annual coupon rate that make the initial cost of the contract 0. I.e., it makes PCFwd(0,T,Tm) = 100
f2S(0,T,Tm) = 2 (1-F(0,T,Tm))/ΣF(0,T,Tj)
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 Ti = 0.25 N Max(r4(Ti-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 Ta can enter the swap that expires at Tb, with payment period τ, that swap has value of
Notional Σ(a+1, b) Z(Ta, Ti) τi (F(Ta; Ti-1, Ti)-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 Tf
- 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 FFut(T,TF) 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 TB
- 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 KP
Use the premium to buy calls on T bills with strike KC
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
KPPut(KP) = KCCall(KC)
What is the payoff or profit from a zero cost collar
Negative under KP, 0 between KP and KC, positive above KC.
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,…vk), the prices of instruments are also a function of those variables. In fact, we can represent it as a vector
Price = vec(P1…PN) = vec(P1(v1,…vk)…PN(v1,…vk))
The matrix of partial derivatives is the Partial DV01 wrt curve parameters
PDV01 = matrix((∂P1/∂v1…∂PN/∂v1)…(∂P1/∂vk…∂PN/∂vk))
What is the derivative of prices wrt rates, the DpDr
Let (r1,…,rk) = the k curve input rates,
Pi = price of ith instrument, i = 1…n
The DpDr is expressed as a matrix of partial derivatives
DpDr = matrix((∂P1/∂r1…∂PN/∂r1)…(∂P1/∂rk…∂PN/∂rk))
What is the derivative of prices wrt alternate variables, the DpDx
Assume some alternate variable set (x1,…,xk) and assume we can calculate these from the forward curve parameters (v1,…vk). The DpDx is a matrix of partial dervatives
DpDx = matrix((∂P1/∂x1…∂PN/∂x1)…(∂P1/∂xk…∂PN/∂xk))
How can we express DpDx as a product of 2 matrixes
The DpDx = DpDv * (DxDv)-1
DpDv = matrix((∂P1/∂v1…∂PN/∂v1)…(∂P1/∂vk…∂PN/∂vk))
and DxDv = The Jacobian of the alternate variables = matrix((∂x1/∂v1…∂xk/∂v1)…(∂x1/∂vk…∂xk/∂vk))
How can we express the DpDy
- Choose a set of traded instruments (x1,…,xk) that match the curve variables (v1,…,vk) 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
