AF5343: Quantitative Methods for Finance – Individual Assignment 2 Duncan Wong

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Q1 – (10 points)

In futures markets, profits or losses on contracts are settled at the end of each trading day. This procedure is called
marking to market or daily resettlement. By preventing a trader’s losses from accumulating over many days,
marking to market reduces the risk that traders will default on their obligations. A futures markets trader needs a
liquidity pool to meet the daily mark to market. If liquidity is exhausted, the trader may be forced to unwind his
position at an unfavorable time.

Suppose you are using financial futures contracts to hedge a risk in your portfolio. You have a liquidity pool
(cash and cash equivalents) of λ dollars per contract and a time horizon of T trading days. For a given size
liquidity pool, λ, Kolb, Gay, and Hunter (1985) developed an expression for the probability stating that you
will exhaust your liquidity pool within a T-day horizon as a result of the daily mark to market. Kolb et al.
assumed that the expected change in futures price is 0 and that futures price changes are normally distributed.
With σ representing the standard deviation of daily futures price changes, the standard deviation of price
changes over a time horizon to day T is √, given continuous compounding. With that background, the Kolb
et al. expression is
Probability of exhausting liquidity pool = 2[1 − N(x)]

where x = λ/( √). Here x is a standardized value of λ. N(x) is the standard normal cumulative distribution
function. For some intuition about 1 − N(x) in the expression, note that the liquidity pool is exhausted if losses
exceed the size of the liquidity pool at any time up to and including T; the probability of that event happening
can be shown to be proportional to an area in the right tail of a standard normal distribution, 1 − N(x).

Using the Kolb et al. expression, answer the following questions:

A. Your hedging horizon is five days, and your liquidity pool is \$2,500 per contract. You estimate that the
standard deviation of daily price changes for the contract is \$500. What is the probability that you will
exhaust your liquidity pool in the five-day period?

B. Suppose your hedging horizon is 20 days, but all the other facts given in Part A remain the same. What is the
probability that you will exhaust your liquidity pool in the 20-day period?

AF5343: Quantitative Methods for Finance – Individual Assignment 2 Duncan Wong

Q2 – (25 points)

Please use the data in the "Data-Part2" tab of homework data file for this exercise.

1. Assume the Hang Seng Index returns follow an i.i.d. normal distribution. Given the index data in the "Data-
Part2" tab, construct 95% and 99% confidence intervals for the population mean of the Hang Seng Index
returns. Explain what distribution you should use to construct the confidence interval (Student-t or normal
distribution)? Explain how to interpret the confidence intervals.

2. Following the assumptions in Q1, construct a hypothesis testing to evaluate whether the variance of HSI
returns is less than 1% per month. Choose 10% level of significance.

Hint: You can use excel function CHISQ.INV to calculate the critical point for hypothesis testing.

Q3 – (15 points)

Assume that the equity risk premium is normally distributed with a population mean of 8 percent and a population
standard deviation of 20 percent. Over the last five years, equity returns (relative to the risk-free rate) have
averaged −2.0 percent. You have a large client who is very upset and claims that results this poor should never

A. Construct a 95 percent confidence interval for the population mean for a sample of five-year returns.

B. What is the probability of the <= −2.0 percent returns over a five-year period?

Q4 – (15 points)

During a 15-year period, the standard deviation of annual returns on a portfolio you are analyzing was 20
percent a year. You want to see whether this record is sufficient evidence to support the conclusion that the
portfolio’s underlying variance of return was less than 900%, the return variance of the portfolio’s benchmark.

A. Formulate null and alternative hypotheses consistent with the verbal description of your objective.

B. Identify the test statistic for conducting a test of the hypotheses in Part A.

C. Identify the rejection point or points at the 0.05 significance level for the hypothesis tested in Part A.

D. Determine whether the null hypothesis is rejected or not rejected at the 0.05 level of significance.

AF5343: Quantitative Methods for Finance – Individual Assignment 2 Duncan Wong

Q5 – (35 points)

Please use the data in the "Data-Part5" tab of homework data file for this exercise. Read the following background
information first.

Background: The most important economic release in the US is the job report. At 8:30 pm of every first Friday
of every month, the US Labor Department reports the previous month's unemployment rate and number of jobs
(payroll) gained or lost. About one week before the release, economists from dozens of big financial institutions
submitted their forecasts to the media (i.e., Bloomberg). The median of these forecast payroll numbers form the
so called market consensus. In the Friday morning, if actual payroll number released is significantly different
from the consensus, the stock market, bond market, and foreign exchange market could move significantly. In
this exercise we are going to see how this payroll surprise may impact the bond market through 10 year treasury
rate. We will organize data to run the following regression:

Delta (10 year treasury rate) = alpha + beta*payroll surprise + error term

where

Delta (10 year treasury rate)= 10 year treasury rate of the release date - 10 year treasury rate of the date prior the
release date;

Payroll surprise = actual payroll release – consensus

Intuitively, payroll surprise > 0 implies that job market and the general economy is stronger than economists have
expected, and so market would expect the Fed to have more reasons to raise interest rate forward, this implies
that beta>0.

The excel file “part3_data.xls” includes four variables: “date”, “treasury_10y_dif”, “payroll_surprise”, and
"payroll_revision_previous_month". “treasury_10y_dif” represents delta (10 year treasury rate) which is
Delta_treasury_10y below.

Now please work out the following questions (please use 1% level of significance as reference):

1. Run the regression model and discuss briefly your results based on the above background information:

Delta_treasury_10y = b0 + b1*payroll_surprise + error term.

2. Run another regression by adding payroll_revision_previous_month as an independent variable and discuss

Delta_treasury_10y = b0 + b1*payroll_surprise + b2*payroll_revision_previous_month + error term.