Fixed Income Markets

Question 1

What is the duration of a fixed income instrument

FIS 3

Answer 1

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

Question 2

What is the Macauley duration of an FIS

FIS 3

Answer 2

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)

Question 3

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

FIS 3

Answer 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

Question 4

Define the 95% VaR and what it measures

Answer 4

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

Question 5

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

Answer 5

• 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

Question 6

Give the VaR and TVaR approximations under normal distribution assumption

Answer 6

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

Question 7

What is the convexity of a portfolio

FIS 4

Answer 7

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

Question 8

What is the duration-convexity approximation of portfolio value

Answer 8

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

Question 9

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

Answer 9

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

Question 10

Describe the dynamics under the factor model for the yield curve

Answer 10

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

Question 11

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

Answer 11

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

Question 12

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

Answer 12

• 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

Question 13

Define the forward curve

FIS 5

Answer 13

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

Question 14

Formula connecting the spot rate and forward rate

Answer 14

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

Question 15

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

Answer 15

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

Question 16

Valuation of an FRA based on ZCB Prices

Answer 16

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))

Question 17

Valuation of FRA based on difference of forward rates

Answer 17

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]}

Question 18

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

Answer 18

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)

Question 19

Payoff of a forward swap contract

Answer 19

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)

Question 20

How do we find the forward swap rate

Answer 20

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)

Question 21

Payoffs of an interest rate cap

FIS 6

Answer 21

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)

Question 22

What is a Swaption, and when would it be exercised

Answer 22

• 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

Question 23

What is the PV of cash flows for a swaption

Answer 23

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)

Question 24

How do options on a futures contract work

Answer 24

• 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

Question 25

How do options written on bonds work

Answer 25

• 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

Question 26

Which interest rate derivatives have 0 inception value and why?

Answer 26

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

Question 27

Two shortcomings of futures vs forwards

Answer 27

• 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

Question 28

How do we construct a Zero cost collar contract

Answer 28

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)

Question 29

What is the payoff or profit from a zero cost collar

Answer 29

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

Question 30

What is the DV01

Answer 30

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

Question 31

What is Modified Duration

Answer 31

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

Question 32

Describe Macaulay Duration

Answer 32

• 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

Question 33

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

Answer 33

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}

Question 34

What is the Partial DV01

Answer 34

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))

Question 35

What is the derivative of prices wrt rates, the DpDr

Answer 35

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))

Question 36

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

Answer 36

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))

Question 37

How can we express DpDx as a product of 2 matrixes

Answer 37

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))

Question 38

How can we express the DpDy

Answer 38

• 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