Homework 1
AMS 517: Quantitative Risk Management
DUE: Monday, February 22, 6 PM EST, Online
(2) Homework must be submitted online
(3) Homework is due Monday, February 22, 6 PM EST, online.
(4) No late homework, under any circumstances, will be accepted.
(5) Your submitted solutions should contain all the derivations.
Problem 1. (12p) (VaR and ES for a distribution function with jumps)
Suppose that the loss L has distribution function
F (x) =

0 x < 0.5
1− 11+x x ∈ [0.5, 5)
1− 1
x2
x > 5
(a) (4p) Plot the graph of F and F←.
(b) (4p) Compute value-at-risk V aRα(L) at confidence levels α = 0.95 and α = 0.99
(c) (4p) Compute expected shortfall ESα(L) at confidence level α = 0.95.
Problem 2. (4p) (VaR for a binomial model of a stock price)
Consider a portfolio consisting of a single stock with current value St = 50. Each year, the
stock price either increases by 5% with probability 0.6 or decreases by 5% with probability
0.4.
Compute V aRα for α ∈ {0.7, 0.95, 0.96, 0.99} over a time horizon of two years.
Problem 3. (12p) (VaR and ES for bivariate normal risks)
Consider two stocks whose log-returns are bivariate normally distributed with annualized
volatilities σ1 = 0.2, σ2 = 0.25 and correlation ρ = 0.3. Assume that the expected returns
are equal to 0 and that one year consists of 250 trading days. Consider a portfolio with
current value Vt = 106 (in USD) and portfolio weights w1 = 0.7 and w2 = 0.3. Furthermore,
denote by L∆t + 1 the linearized loss (using log-prices of the stocks as risk factors).
(a)(4p) Compute the daily V aR0.99(L

t+1) and ES0.99(L

t+1) for the portfolio.
(b)(4p) How does the answer change if ρ == 0.9? (Compute the number and answer if the
increase in correlation increase or reduce the risk of the portfolio?)
(c)(4p) Suppose you have three random variables. The correlation between X and Y is 0.6,
the correlation between X and Z is 0.8. What is the range of correlation between Y
and Z?
Problem 4 (10p) (Superadditivity of V aR for a heavy tailed distribution)
Let L1 and L2 be independent random variables with distribution function F (x) = 1−x−1/2,
x ∈ [1,∞). (This is a form of Pareto distribution with infinite mean.)
Show that V aRα(L1 + L2) > V aRα(L1) + V aRα(L2) for all α ∈ (0, 1).
Hint: Show that the distribution function FL1+L2 of L1 + L2 is given by FL1+L2(x) =
1− 2√x− 1/x, x ≥ 2.
Problem 5 (12p) (Shortfall-to-quantile ratio for t)
Let L ∼ tν(µ, σ2) for ν > 1, µ ∈ R, σ > 0.
(a)(4p) Compute V aRα(L) and ESα(L).
(b)(8p) Show that
lim
α→1−
ESα(L)
V aRα(L)
=
ν
ν − 1
and interpret this result from a risk management perspective.
Problem 6 (16p) (Manipulating logarithmic and relative returns)
For an asset price with daily values given by the time series (St) the h-day log-return is
defined by X
(h)
t+h = log(St+h/St) and the h-day relative return by Y
(h)
t+h = (St+h − St)/St for
all h ≥ 1.
(a)(4p) Derive a formula for the h-day log-return X
(h)
t+h in terms of the h-day relative return
Y
(h)
t+h.
(b)(4p) Derive a formula for the h-day relative return Y
(h)
t+h in terms of the h-day log-return
X
(h)
t+h.
(c)(4p) Given a series of h-day log-returns X
(h)
t+h, X
(h)
t+2h, . . . and an initial stock price St, explain
how you would reconstruct the series of prices St+h, St+2h, . . . at h-day intervals.
(d)(4p) Given a series of h-day relative returns Y (h)t+h, Y
(h)
t+2h, . . . and an initial stock price
St, explain how you would reconstruct the series of prices St+h, St+2h, . . .. at h-day
intervals.
Problem 7 (16p) (Programming Question - Changing correlation over time)
For the S&P500 and DAX log-returns from 3 January 2011 to 31 December 2015 (data
available in the R package qrmdata:
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(a)(4p) Estimate correlations using a rolling 25-day window and plot the time series of esti-
mated correlations.
(b)(4p) Experiment with different window sizes and investigate how the plot changes.
(c)(4p) Carry out a regression analysis to investigate whether correlation estimates calculated
for non-overlapping 25-day blocks of observations tend to be higher in periods with
higher volatility estimates.
(d)(4p) Repeat the analysis using rolling estimates of correlation and volatility.
Problem 8 (18p) (Geometric spacings between largest values in iid samples)
Let X1, X2, . . . be a sequence of iid random variables from a continuous distribution function
F and that p = P(X1 > u) = F¯ (u) ∈ (0, 1) for some threshold u. Let T0 = 0 and
Tj = min {i > Tj−1 : Xi > u}, j ∈ N.
(a)(9p) Show that the spacings Sj = Tj − Tj−1, j ∈ N, of exceedances of the Xi’s over the
threshold u form a series of iid random variables following a geometric distribution
with parameter p.
(b)(9p) Explain how the insight in (a) can be used to implement a graphical test for volatility
clustering?
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