RMBI 4210 – Quantitative Methods for Risk Management Mid-term Test, 2021
Time allowed: 80 minutes
Instructor: Prof. Y. K. Kwok
1. We consider the duration of a bond with T years of maturity and coupons of dollar amount c are paid annually at t = 1,2,…,T. The par value BT is paid at time T. Suppose the current time is θ, where 0 ≤ θ < 1, which is before the first coupon payment date (t = 1). Recall that the duration D at t = 0 is given by
1 T(i−Bc )−(1+i)
D=1+ + T , (A)
[points]
i Bc [(1+i)T −1]+i T
where i is the prevailing interest rate.
(a) Modify formula (A) to obtain the new formula for D at time θ, 0 ≤ θ < 1. Explain
your answer. [4]
Hint: Before any coupon payment, duration is reduced by the same amount with the passage of the calendar time. Is there any change of the relative weights of the cashflows at t = 0 and t = θ? The first term “1” in formula (A) refers to the special case of θ = 0.
(b) People have the myth that duration is always an increasing function of T, that is,
bond with longer maturity has duration larger than the shorter maturity counterpart,
both bonds have the same coupon rate. Deduce from the new formula for D obtained
in part (a) that under certain condition, bond with finite maturity may have duration exceeds that of the perpetual bond. Find explicitly such required condition in terms
of θ, c, BT , i and T . Is the required condition dependent on θ? If not, why? Give justification to your answer. [6]
2. Recall the horizon rate of return rH as defined by
[B(i)]1/H
rH = B (1+i)−1
0
for horizon H and interest rate i. Here, B0 is the bond price at the initial interest rate i0
and B(i) is the new bond price when the interest rate moves to the level i.
(a) Using mathematical argument to explain why rH is monotonically decreasing func-
tion of i when H is small and becomes monotonically increasing in i when H increases beyond certain threshold. [3]
(b) Give the financial interpretation of
lim rH = i. [1]
H→∞
1
3.
(c) Explainusingfinancialintuitionwhyimmunizationofthebondinvestmentisachieved
when the duration of the bond D matches with the target horizon H. [2]
(d) Explain why duration D and horizon H do not change by the same amount with
the passage of calendar time. How does this feature of unequal changes impact on
the immunization procedure? [2]
(a) Can we achieve perfect hedging using the delta hedging procedure in a call option?
What are the potential risks faced by the writer of the call option even with the implementation of delta hedging? [2]
(b) When the call option is very deep-in-the-money on two days before expiry; that is,
the stock price is significantly above the strike price at time close to expiry. Explain
why the delta is close to one. Suppose the stock price increases by 10% on the next
day (one day before expiry), does the writer need to purchase roughly 10% more shares? Explain your answer. [2]
(c) In the use of minimum variance hedge of jet fuel using heating oil, explain in finan-
cial intuition why no hedging can be achieved if the correlation coefficient between
the prices of the two oils is zero. How about the hedging performance when the correlation coefficient tends to one? [2]
Suppose the loss random variables of two risky investments, denoted by L1 and L2, are normally distributed with mean μi and variance σi2, i = 1, 2. Assume that their random losses are independent so that the sum of random losses, Lsum = L1 + L2, remains to be normal with mean μ1 + μ2 and variance σ12 + σ2.
(a) Show that
VaRα(Li)=μi +σiN−1(α), i=1,2.
where α is the confidence level. [3]
(b) Find VaR(Lsum). [2] (c) For this portfolio of two investments, check whether subadditivity is violated in VaR. [2]
(a) The Expected Shortfall ES is generally recognized as the better risk measure when compared with the VaR. Give two reasons why ES is preferred over VaR by risk managers? [2]
(b) How can one compute VaRα and ESα at a given confidence level α using one mini- mization calculation? Is VaRα always less than ESα? Explain your answer. [2]
Let V be the random loss on a portfolio over a certain time horizon. We use the Pareto
4.
5.
6.
distribution
( y )−1/ξ Gξ,β(y)=1− 1+ξβ
2
to model the random loss beyond u, where the probability that V > u + y conditional on V > u is 1 − Gξ,β (y). Let F (v) be the cumulative distribution function for the random loss variable V . The unconditional probability that V > x is given by
P[V > x] = [1−F(u)][1−Gξ,β(x−u)]. Suppose 1 − F (u) is estimated by empirical data to be nu/n so that
n[ξ ]−1/ξ P [ V > x ] = nu 1 + β ( x − u ) ,
where n is the total number of scenarios and nu is the number of scenarios that the loss exceeds u. Note that x = u + y. Recall that VaR with a confidence level of q is given by
F(VaR) = q,
whereF(u)=1−P[V >u].
(a) VaR determined from the above formula may not agree with the estimation of VaR
from empirical data. Which estimate of VaR is more trustworthy? Explain why. [1]
(b) Note that VaR can be expressed in terms of q, u, n and other parameters of the u
Pareto distribution. Recall the formula for Expected Shortfall ES, where
1∫1
ES=1−α
use this formula to find ES under this extreme value theory model. Show the details of the mathematical derivation. No direct citation of the formula from the lecture note.
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3
α
VaRq dq.
[4]