xuebaunion@vip.163.com

3551 Trousdale Rkwy, University Park, Los Angeles, CA

留学生论文指导和课程辅导

无忧GPA：https://www.essaygpa.com

工作时间：全年无休-早上8点到凌晨3点

微信客服：xiaoxionga100

微信客服：ITCS521

程序代写案例-AMS 517

时间：2021-02-21

Homework 1

AMS 517: Quantitative Risk Management

DUE: Monday, February 22, 6 PM EST, Online

(1) Please sign your home work with your name and Stony Brook ID number.

(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:

2

(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?

3

学霸联盟

AMS 517: Quantitative Risk Management

DUE: Monday, February 22, 6 PM EST, Online

(1) Please sign your home work with your name and Stony Brook ID number.

(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:

2

(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?

3

学霸联盟