4/19/2017
Thefearlessmarketignoresperilsahead
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Risk and Portfolio Management
Spring 2011
PCA, Dynamic PCA
& Applications to Risk-Management
Basic investment equation
Et equity in a trading account at time t (liquidati on value)
Rit return on stock i from time t to time t t (includes dividend income)
Q
Probability Distributions for Financial Assets
and Risk Measures
Feb 3, 2016
Probability Distributions
To any portfolio, we associate a probability distribution for gain/losses over
certain time horizon.
This PDF will correspond to either
(i) the predic
What risk ?
The main risk for financial institutions is insolvency,
i.e. depletion of trading capital to the point that you have to stop operations
or are severely impaired.
Pension funds: not enough funds to cover retirement of pensioners
Banks: insolv
Portfolio Theory
Risk and Portfolio Management
Spring 2017
Theoretical set-up
N risky assets with returns 1 , , . These are assumed to be
random variables with
= , = 2
( , )=
Investor has $1 of capital to distribute among the N risky assets,
in amounts
Risk and Portfolio Management
Spring 2011
Principal Components Analysis and
Factors explaining stock returns
Correlation matrices in finance
ij
Corr Ri , R j
Ri
return of stock # i, i 1,., N
Estimation of correlation matrix from data requires selecting a
Term-structures of interest rates
Risk-management
Risk & Portfolio Management
Lecture 5
U.S. Treasury securities
U.S. Treasury issues securities on a periodic basis. This is done through auctions (weekly for short tenors, quarterly
for longer tenors).
Au
Homework #5
Submission date
1
2
3
4
5
6
Comparative Methodologies
Using the positions in question, calculate 99%, 95% and 90% and 80% VaR using V/CV
Using the positions in question, calculate 99%, 95% and 90% and 80% VaR using HS
Using the positions in qu
Using the positions in A6:F6, calculate 99%, 95% and 90% and 80% var using V/CV, HS, and MC.
Use an eigenvalue/eigenvector decomp
Show stand-alone 99% vars for each asset in C/CV, HS, and MC
Assume that the 10y has a 3.63 yield
Test your random normals fo
Using the positions in A6:F6, calculate 99%, 95% and 90% and 80% var using V/CV, HS, and MC.
Use an eigenvalue/eigenvector decomp
Show stand-alone 99% vars for each asset in C/CV, HS, and MC
Assume that the 10y has a 3.63 yield
Test your random normals fo
2017 HomeworMatrix Algebra in Excel
1 Compute the covariance and correlation matrices of the % changes in the dataset below, using Exc
2 Factor that covariance matrix using an eigensystem decomposition
3 Factor that covariance matrix using a Cholesky deco
Homework #3
Missing data as basis for Monte Carlo models in risk
1. In a spreadsheet, generate MC time series for the following:
Random walk
AR1
Mean reverting process
2. in a spreadsheet, generate 60,000 random normals
how many > 1std?
2std?
3std?
4std?
Homework #2
Pricing of simple bonds in Excel
Assuming that todays' date is 8/7/07, the bond has $100 notional - and using the Bloomberg screen below
1) for the following bonds, calculate
a) price at current yield
b) PV01 at current yield
c) duration at cu
FINC-UB.0044 PORTFOLIO MANAGEMENT
Fall 2016
Sample Quiz
For questions 1-6 use the following information on the historical annualized excess returns on two
funds (e.g., ETFs). Assume the current annual risk-free rate is 1%.
Average excess return
Volatility
FINC-UB.0044 Portfolio MANAGEMENT
Fall 2016
Quiz #1 Formula Sheet
Portfolio theory (statistics):
S
S
E[ri ] p(s)ri (s) var[ri ] i2 E[(ri (s) E[ri ]) 2 ] p(s)(ri ( s) E[ri ]) 2
s 1
s 1
S
cov[ri , r j ]
s 1
i j
cov[ri , r j ] E[(ri E[ri ])(r j E[r j ])] p(
FINC-UB.0044 PORTFOLIO MANAGEMENT
Fall 2016
Case #1: Innovating into Active ETFs
The assignment consists of two sets of questions: (1) quantitative, Excel-based questions, which
should be completed and submitted, and (2) more qualitative questions, which
FINC-UB.0044 PORTFOLIO MANAGEMENT
Fall 2016
Information for Quiz #1
The first quiz will be on Tuesday, October 11 in class. It will be closed book and closed
notes, except for the formula sheet on the reverse side (I will hand out a clean copy with
the ex
FINC-UB.0044 PORTFOLIO MANAGEMENT
Fall 2016
Case #2: ProShares Hedge Fund Replication ETF
The assignment consists of two sets of questions: (1) quantitative, Excel-based questions, which
should be completed and submitted, and (2) more qualitative question
Homework #1: [DUE (Date TBA) through NYU Classes]
5. Suppose that X is a discrete random variable taking the values -1, 0, 1 with Pr(X = -1) = Pr(X = 0) = Pr(X
= 1) = 1/3. Let Y = X2. Show that Cov(X,Y) = 0, but that X and Y are NOT independent.
6. Suppos