Exam in TIØ4317 – Empiriske og kvantitative metoder i finans, 2009

Norges teknisk-naturvitenskapelige universitet
Institutt for industriell økonomi og teknologiledelse
Contacts during exam:
1. Alexei Gaivoronski, tlf: 48243742
2. Sjur Westgaard, tlf. 97122019
Sensureringsfrist: 05.06.2009
Exam in TIØ4317 – Empiriske og kvantitative
metoder i finans, 2009
All types of calculators are admitted to exam
Exercise 1
Portfolio manager of pension fund Trygg Havn have to make decision about
distribution of funds wealth between two classes of assets. One is governmental bonds
which give 6% yearly return and are considered to be risk free. Another is index fund
which follows the overall index of Rosenborg Stock Exchange. During the last 200
trading days this index has shown the following daily returns:
return -5% -4% -3% -2% -1% 0% 1% 2% 3%
Number of 3 7 9 15 22 38 72 29 5
days
The pension fund manager uses these data as the empirical distribution of future
returns of the index. In order to serve his liabilities the manager has to construct
portfolio with at least 8% yearly return.
At the same time in order to fulfill requirements of regulatory bodies the daily 95%
VaR of his portfolio should not exceed 1%.
1. Is it possible for manager to reach both return and risk targets?
2. What will be the answer if 1% risk bound is expressed in CVaR instead of VaR?
Please give detailed answer.
Exercise 2
Portfolio revision when the assets can be bought and sold in lots.
There are many types of assets that can be bought and sold only in fixed amounts
called lots. You are invited to formulate portfolio rebalancing problem similar to
those described in the book of Zenios, chapter 3, when portfolio is composed from
such assets. Let us denote:
yi – current holding of asset i, i=1:n
p0i – current price of one unit of asset i
pi - random future price of one unit of asset i
These assets can be bought and sold in fixed lots of two sizes: smaller lot of asset i
consists of k1i units and larger lot consists of k2i units. Buying or selling of one smaller
lot entails transaction cost c1i. Buying or selling of one larger lot entails transaction
cost c2i.
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1. Please formulate portfolio revision model with these assets
2. Discuss its properties and possibilities for implementation.
Exercise 3
In this analysis we study credit spreads in the UK bond market. That is the difference
between long term yields on corporate AAA rated bonds and government bonds. We
have run an ADF test and confirmed that the spread is stationary. Then we model the
credit spread with 2 ARMA models:
• ARMA(2,2)
• ARMA(2,0)
The results from these 2 models together with residual diagnostics are given below:.
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a) Write down the two models with their parameters. Give comments to the
parameter values and their significance. Test formally if the MA(1) term and
the MA(2) term in the ARMA(2,2) are significant. Choose a significance
level at 5%.
b) Describe and perform a test for normality and serial-correlation of the
residuals for the two models. How are the results compare the assumptions we
usually apply to linear regression models and ARMA models?
c) Describe a criterion for model selection that can be applied. According to this
criterion, which model will you choose?
Exercise 4
In this exercise we analyze volatility in the FX market. We are looking at daily returns
for USDJPY in the period 7/11 1984 to 31/8 2007. We estimate a GARCH(1,1)
model, a GJR(1,1) model and an EGARCH(1,1) model for the volatility equation. For
the mean equation we just assume a constant. The results for the models are given
below:
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a) Write down the different models with their parameter estimates and intepret
the models. Give comments to the parameter significance. What type of
restrictions should be apply to the parameters?
b) Calculate the unconditional variance and standard-deviation for the
GARCH(1,1) model.
c) Discuss if asymmetry in volatility should be modeled for USDJPY.
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