编程辅导 ECON3206/5206 Financial Econometrics

Microsoft Word – Tutorial 1_2020.docx

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ECON3206/5206 Financial Econometrics

Tutorial 1
1. Let !! be the price of BHP share at the end of day !, adjusted for dividends. The daily

return may be calculated either as the simple !! = (!! − !!!!)/!!!! or the log return
!! = ln!(!!/!!!!). Show that !! ≈ !! when |(!! − !!!!)/!!!!| is small. Hint: use
Taylor expansion.

2. In the same setting as in Question 1, suppose !! = $30 at the end of day 1, !! = 5%
at the end of day 2, and !! = −3% at the end of day 3. What is the price at the end of
day 3? What is the return from the end of day 1 to the end of day 3? More generally,

based on the daily returns, how do you calculate the weekly return of the BHP stock?

Assume that a week consists of 5 trading days.

The above figure shows the probability mass function (pmf) on the left and the

probability density function (pdf) on the right for discrete and continuous random

variables, respectively.

The discrete random variable represents the number of times we observe heads when

we toss a fair coin 2 times.

In the future I will use asterisk *[ ] and parenthesis for materials or questions which

are not essential for performing well in the course, but which may be of interest.

*[Abraham de Moivre was a French mathematician who first noticed that as you

increase the number of tosses, the distribution of the total number of heads

(appropriately rescaled because the mean is increasing with the number of tosses)

takes a specific shape which can be approximated by a continuous bell shaped

distribution (he called it an ultimate curve). This is first special case of at the central

limit theorem. Later the ultimate curve, which is now known as Normal or Gaussian

distribution was formalised by , a German mathematician, the greatest

contributor the mathematics and science since antiquity.

For more insights on de Moivre reasoning you may read this

http://www.mathpages.com/home/kmath642/kmath642.htm ]

(a) What are all possible outcomes of the experiment: toss a fair coin 2 times and

count heads?

(b) What is the probability to observe outcome 0?

The continuous random variable corresponds to a height of a person in some

population.

(d) What is the probability to observe outcome 180?

(e) Which proportion of the population is expected to have height below 180?

(f) In your own words, explain the notions of the probability density function and the

cumulative probability distribution for a continuous random variable. What about

their counterparts for a discrete random variable?

(g) In your own words, explain what the mean and the variance of a random variable

(h) In your own words, explain the central limit theorem.

4. Use an example with two (random) variables to explain the notion of the conditional

distribution of one variable given the other.

5. Suppose {!!,!!,… ,!!} is a set of random variables that (i) are uncorrelated with one

another; (ii) have common mean ! and variance !!. Let ! = !!
! be the sample

mean. Find (a) ! ! , (b) Var ! , (c) ! ! ! , where ! ! is a function. Hint: think
carefully about (c), you may need to impose additional restriction on !(!).

(d) Can you do (b) without assuming that the random variables are uncorrelated?

(e) What happens to !”# ! as n gets larger (! →∝)? What happens to !?
ECON5206: How does this relate to the law of large numbers?

(f) ECON5206: Apply the CLT theorem in the context of !

6. Recently Chinese stock market received a lot of attention. We are going to use a

recent data set on the SHANGHAI SE A SHARE – PRICE INDEX and S&P/ASX200

– PRICE INDEX. The data was downloaded from Data-steam, but you may get these

data also from finance.yahoo.com. The data is in the file ASX200-SE-indexes.xlsx on

Moodle. Using excel1 try to:

(a) Plot the indices;

(b) Generate the log return series of the indices and plot the log return series;

(d) Compute the statistics !!”, !!” and !”;
(e) Compute the correlations of log return series

(f) Summarise the features of the log return series and compare with the lecture

examples .

1 You may ask why Excel. This is coming from a recent interaction from an employer who claimed that
Economics graduate do not know how to use Excel and that is the only software they use. Sounds fishy, but let’s
do this one in Excel.

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