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)𝑇𝑖−0/ −1+𝑖>0 –
+- These two conditions make sure ) (*,+
higher than perpetual bond’s duration.
The required condition does not depend on 𝜃.
* ‘.( ,- ‘.( -*’ .(
> 0. So in this case the corresponding D is
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 0 ( * =<
decreasing function of 𝑖. On the other hand, when 𝐻 is large, 0; becomes relatively less significant, and the horizon rate of return becomes an increasing function of 𝑖.
(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 𝜎∗C = 𝜎C − 𝜌C 𝜎C, if coefficient is zero, no variance reduction at all (no A D DFD
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 + 𝜎C 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 + 𝜎C or 𝜎' + 𝜎C is bigger. After taking square on both expressions, we know 𝜎'C +𝜎C ≤ 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)