RMBI4210 Midterm Answer (2021)

Q1(a) 𝐷 = 1 − 𝜃 + '(+

) (* +,-

* '.(+,-

'.( -*' .(

Explain it: before any coupon payment, there is no cash flow incurred. So duration is reduced

by the same amount with passage of time.

(b) Perpetual bond has duration 1 − 𝜃 + '( or 1 + '

( (depends which formula you use), and

explain /0-

1 + 𝑖 ) − 1 + 𝑖 > 0

1) 𝑐 ≠ 0

If 𝑐 = 0, 𝐷 = 𝑇, there is no way a finite maturity bond has a higher 𝐷 than a perpetual bond.

2) 𝑖 > /0-

3) 𝑇 𝑖 − /0-

− 1 + 𝑖 > 0

These two conditions make sure ) (* +

,-* '.(

+,-

'.( -*' .(> 0. So in this case the corresponding D is

higher than perpetual bond’s duration.

The required condition does not depend on 𝜃.

Again, as before any coupon payment, there is no cash flow incurred. So duration is reduced

by the same amount with passage of time. 𝜃 has the same negative linear relationship with

duration of either a perpetual bond or a finite bond (we can observe it through the formula).

The required conditions only affect the last term in the formula as a result it has nothing to do

with 𝜃.

Q2 (a) (Refer to Topic 1 P70 & 79)

When 𝐻 is small, ':

is large and the decreasing factor term is more significant. So it is a

decreasing function of 𝑖. On the other hand, when 𝐻 is large, 0 (0;

*<= becomes relatively less

significant, and the horizon rate of return becomes an increasing function of𝑖.

(b) Financial interpretation: With infinite time of horizon, the immediate change of bond price

is immaterial in the long term, so the horizon rate of return 𝑟: is dominated by the prevailing

interest rate 𝑖.

(c) (refer to P75 and P78)

Price risk and reinvestment risk are offsetting

(d) D is depending on time and interest rate while H only depends on calendar time. So they

do not change the same. As a result, we have to construct the investment such that matching

horizon with duration to achieve bond immunization dynamically.

Q3

(a) Theoretically, yes; but practical No, such as transaction cost; we may not be able to find the

underlying or highly correlated products to hedge in reality; option price depends on volatility,

but the implied volatility for hedging is different from the actual volatility which makes perfect

hedging impossible; other than keeping delta neutral, there are other factors like gamma,

liquidity risk, credit risk and so on.

(b) (Topic 1 P28) Deep-in-the-money, highly likely (almost 100%) to exercise the option, delta

is close to 1.

No, we don’t have to purchase 10% more shares as we already purchased the full amount of

shares due to delta is 1.

(c) Topic 1 P40 as 𝜎A∗C = 𝜎DC − 𝜌DFC 𝜎DC, if coefficient is zero, no variance reduction at all (no

hedging can be achieved); if coefficient tends to 1, variance is reduced to almost 0.

Q4 (a) Proof:

(b) V𝑎𝑅 𝐿JKL = V𝑎𝑅 𝐿' + 𝐿C = 𝜇' + 𝜇C + 𝜎'C + 𝜎CC 𝑁*'(𝛼)

(c) V𝑎𝑅 𝐿' + V𝑎𝑅 𝐿C = 𝜇' + 𝜇C + (𝜎' + 𝜎C)𝑁*'(𝛼), compare with part (b), we just need

to compare whether 𝜎'C + 𝜎CC or 𝜎' + 𝜎Cis bigger. After taking square on both expressions,

we know 𝜎'C + 𝜎CC ≤ 𝜎' + 𝜎C C , so V𝑎𝑅 𝐿' + V𝑎𝑅 𝐿C ≥ V𝑎𝑅 𝐿JKL , subadditivity is

satisfied.

Q5 (a) ES satisfies subadditivity while VaR sometimes violates it.

Expected shortfall reduces credit concentration because it takes into account losses beyond the

VaR level as a conditional expectation.

(b) One minimization calculation can compute both VaR and ES based on the following

function:

Yes, based on the formula on Topic 1 P140, ES is at least as large as VaR.

Q6 (a) EVT is

more trustworthy, as it can be used to improve VaR and ES estimates with a very high

confidence level. It involves smoothing and extrapolating the tails of an

empirical distribution.

(b)