STAD70 Statistics & Finance II Final Exam 2022
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STAD70 Statistics & Finance II
Final Exam
2022
Academic integrity statement. Academic integrity is a fundamental value of learning and schol- arship at the UofT. Participating honestly, respectfully, responsibly, and fairly in this academic community ensures that your UofT degree is valued and respected as a true signifies of your individual academic achievement. By submitting this midterm you agree with the following: You will not commit academic misconduct, and are aware of the penalties that may be imposed if you commit an academic offence.
There are 2 parts in the midterm. This exam is open notes and you need to use R in the second part. There are 110 total points, but you will be graded out of 100.
Part I. Theoretical.
Write your answers on the provided exam booklet. If needed, you may justify your answer numerically using R.
1. (15 points) Consider n assets whose nominal simple returns over a holding period [t0 , t1]
are R1 , . . . , Rn . So
Xi(t1 )
Xi(t0 )
where Xi is the price of asset i. (We say this is nominal because the unit of price is the underlying currency, e.g. $.) Let X0(t) > 0 be the price of another asset (e.g. a market index), and define the relative price of asset i by
i(t) = , t e {t0 , t1 }.
Let i be the simple return of the relative price; we call this the relative return. Consider a portfolio with weights w1 , . . . , wn, where wi > 0 and wi = 1. Show that the excess growth rate of the portfolio is the same if we use either the normal or relative (log) returns. [Financially, this result says that the excess growth rate does not depend on the choice of the numeraire.]
Hint: Write down the algebraic expressions of both excess growth rates. Simplify and show that they are equal.
Let
Xi(t1 )
ri = log(1 + Ri) = log
and
i = log(1 + i) = log = log = ri + c,
where c = log Let
does not depend on i.
γ L = log ╱ wi eri\ -
n
z
be the “usual excess growth rate”, and let
L = log ╱ z(n)wi ei \ -z(n)wii
be the one using relative returns.
Writing i = ri + c, we have
L = log ╱ wi eri+c\ - wi(ri + c)
= log ╱ z(n)wi eri\ + c -z(n)wiri - c
L
[Note: This result was proved in Lemma 3.2 of the paper
❼ Pal, S., & Wong, T. K. L. (2013). Energy, entropy, and arbitrage. arXiv preprint
arXiv:1308.5376.]
2. (15 points) Consider an ARCH(1) model where yt = σtεt , εt i. . N (0, 1), and σt(2) = ω + γyt(2)↓1 .
Assume ω > 0 and γ > 0.
(a) (10 points) Find 屯[y2(2)|y0 = 1].
(b) (5 points) Is the conditional distribution of y2 given y0 = 1 normal? Explain your
answer.
(a) We have
y2(2) = σ2(2)ε2(2)
= (ω + γy 1(2))ε2(2)
= (ω + γσ1(2)ε1(2))ε2(2)
= (ω + γ(ω + γy0(2))ε1(2))ε2(2)
= ωε2(2) + γ(ω + γy0(2))ε1(2)ε2(2) .
Since ε 1 and ε2 are i.i.d. N (0, 1) and are independent of y0 , we have
屯[y2(2)|y0 = 1] = 屯 ┌ωε2(2) + γ(ω + γy0(2))ε1(2)ε2(2)|y0 = 1┐
= 屯 ┌ωε2(2) + γ(ω + γ)ε1(2)ε2(2)|y0 = 1┐
= 屯 ┌ωε2(2) + γ(ω + γ)ε1(2)ε2(2)┐
= ω屯[ε2(2)] + γ(ω + γ)屯[ε1(2)]屯[ε2(2)]
= ω + γ(ω + γ).
(b) No. Given y0 = 1, we have
y2 = ←ω + γ(ω + γ)ε1(2)ε2 ,
which is not normal.
3. (20 points) Let R = (R1 , . . . , Rn)基 represent the simple returns of n assets. Suppose the distribution of R is multivariate normal, i.e., R ~ N (µ, Σ) for some mean vector µ and (invertible) covariance matrix Σ . Let W0 > 0 be a constant representing the initial wealth. Given a vector w = (w1 , . . . , wn)基 of portfolio weights, the final wealth is W = W0(1 + w基 R).
(a) (10 points) Consider the exponential utility function
u(W) = -e ↓W .
Derive a formula for the expected utility 屯[u(W)] as a function of w .
Hint: You may use the fact that if X ~ N (m, σ2 ) then
屯[etX ] = etm+t2 σ 2 .
(b) (10 points) Let 1 = (1, . . . , 1)基 be a vector of ones. Consider the expected utility
maximization problem
max 屯[u(W)].
w:w十 1=1
Note that maximizing 屯[u(W)] is the same as minimizing 屯[-u(W)] which is the same as minimizing log 屯[-u(W)] (this is because log(.) is strictly increasing). Using this transformation, solve the problem using the method of Lagrange multiplier. (This includes finding the value of the multiplier in terms of the given parameters.)
Hint: If you formulate the problem properly, you will get a quadratic programming problem. Also make you that you have solved part (a) correctly before starting part (b).
We have
w基 R ~ N (w基 µ, w基 Σw)
Hence
W = W0(1 + w基 R) ~ N ╱ W0(1 + w基 µ), W w0(2) 基 Σw、.
It follows from the mgf of normal distribution that
屯[-e ↓W] = -e ↓W0(1+w十 µ)+W0(2)w十 Σw .
(b) We have
log 屯[-u(W)] = -W0(1 + w基 µ) + W0(2)w基 Σw,
which we want to minimize subject to the constraint w基 1 = 1. (Note that W0 > 0 is a constant which can be removed. We keep it here.)
The Lagrangian is
2 = -W0(1 + w基 µ) + W0(2)w基 Σw - λ(w基 1 - 1).
Setting the derivative with respect to w to zero, we have
-W0µ + W0(2)Σw - λ1 = 0 ÷ w = Σ↓1(λ1 + W0µ).
The constraint w基 1 is satisfied if
λ 1基 Σ↓11 + 1基 Σ↓1µ = 1.
So
W0(2) + W01基 Σ↓1µ
1基 Σ↓1 1 .
4. (10 points) (True/False, 1 point each)
(a) Under the setting of the classical one-period CAPM, some investors may invest all
their capital in a single asset.
(b) The Ljung-Box test, when applied directly to the ACF of the return series, can detect
volatility clustering.
(c) The mean-variance portfolio optimization problem has an explicit, analytical solution even when short-selling is forbidden.
(d) If two distributions have the same VaR for all level α, then they are the same.
(e) In the i.i.d. setting, the growth optimal portfolio can be computed as long as the first
four moments of the return distribution is known.
(f) Even if stock returns follow exactly the single index model, the R2 (of the regression)
for individual stocks can still be close to zero.
(g) Suppose the returns of n stocks are i.i.d. Then the asymptotic growth rate of Cover’s
universal portfolio is the same as that of the growth optimal portfolio.
(h) The tail index of the normal distribution is 1.
(i) If the semi-strong form of efficient market hypothesis holds, then stock returns over different periods must be independent.
(j) It is an established stylized fact of stock returns that volatility is highest on Mondays
and lowest on Fridays.
(a) False. (Every agent invests in a combination of the risk-free asset and the market (tangency) portfolio. [While it is logically possible that an investor invests everything in the riskfree asset, under very mild condition it can be shown that each investor must invest something in the risky market portfolio.]
(b) False. (Volatility clustering is visible in the acf of e.g. the squared returns.)
(c) False.
(d) True.
(e) False. (The entire distribution is required.)
(f) True.
(g) True.
(h) False.
(i) False.
(j) False.
Part II. Data analysis .
Submit a single R file (containing all codes) and a separate text file (e.g. MS word, markdown, or plain text file) which contains your answers. You don’t have to export the graphs as images. Comment your codes. The R file must be self-contained. In particular, it must contain commands to load the necessary packages.
To begin, load the data in the file final data .RData by using the code load("final data .RData").
5. (10 points) Consider the ARCH(1) model in Problem 2. Assume ω = 0.001 and γ = 0.02. Suppose R is a random variable whose distribution is the conditional distribution of y2 given y0 = 1. Regard R as the simple return of a portfolio whose initial value is $100 , 000.
Simulate N = 50000 values of R and estimate the 5% value at risk of the portfolio value using the sample quantile (here α = 0.05, so that the confidence is 95%).
6. (20 points) The object r HSI contains the daily log returns of the Hang Seng Index from January 2017 to March 2022.
(a) (10 points) Consider fitting the series with a GARCH(p, q) model where the condi-
tional distribution is normal. You can use the fGarch package and garchFit(), and set include .mean = TRUE. [Our convention for the GARCH(p, q) model is
p q
σt(2) = ω + βiσt(2)↓i + γjyt(2)↓j ,
i=1 j=1
which may defer from that of the package.] Use AIC (which can be obtained by applying summary() to the output of the fit) to pick the model when 0 < p < 2 and 0 < q < 2. Examine the standardize residuals of your chosen model and discuss the fit of the model.
(b) (10 points) Convert the log returns to simple returns, and name the resulting series by
R HSI. [Do this directly, using the relationship between simple and log returns.] Using the package rugarch, fit a Riskmetrics model (again the conditional distribution is
normal) to R HSI and write down the fitted model, i.e.,
σt(2) = . . .
with the estimated coefficients. Compute the 1% expected shortfall (in terms of the return) for the next trading day (given the observed data). Use the sign convention such that this expected shortfall is negative.
7. (25 points) The object fundreturns contains the daily simple returns of 8 equity funds from 2002 to 2007.
(a) (5 points) Plot a heat map of the empirical covariance matrix.
(b) (5 points) Suppose each fund starts initially at $1. Compute the final value of all
portfolios. Which fund has the best performance (in terms of the final value)?
(c) (5 points) Consider also the equal-weighted fund of funds (rebalanced daily) [essen- tially, regard each fund as single asset and form the equal-weighted portfolio]. Does the equal-weighted fund of funds outperform the best fund you find in (b)?
(d) (5 points) Apply PCA to the returns. How much variation do the first two principal components explain?
(e) (5 points) Consider all constant-weighted fund of funds [again, you may regard each
fund as a stock]. What are the weights which maximize the final value of the portfolio? Hint: This part may be quite time consuming, and may be regarded as a bonus question. If we let bi be the weights, and ri(t) be the log return of fund i for period t, then the final log value of the fund of funds is
t log ╱ i bieri(t)\ .
You want to maximize this value over b e ∆8 . (See Assignment 4.)
Please refer to the associate R file.
2022-08-06