# Empirical Finance: Problem

Question 1

An investment fund has generated the following yearly simple net returns over a 5 year period:

R1 = 0.15, R2 = 0.08, R3 = −0.10, R4 = 0.12, R5 = −0.05.
If you had invested £10000 into the scheme, with how much would you have ended up with at the end of the 5 year period? (assume no further deposits or withdrawals are made).

Compute the APR of this investment.

Obtain the 5 year continuously compounded growth rate associated with the investment.

You deposit £1000 in an account that earns 6% annual interest compounded continuously. Assume no further deposits or withdrawals are made.

How long will it take for the balance to double?

If you want to move your account to one that compounds interest monthly what nominal annual rate would give the same APR as the original account?

Suppose that monthly simple net returns were 1% per month in year 1 and -0.5% per month in year

What is the annualised rate?

Suppose that log stock prices evolve according to pt = µ + pt−1 + et with et NID(0, σe2) (normally and independently distributed with mean 0 and variance σe2).
Assume µ = 0.10 and σe2 = 0.025. What is the probability that pT +1, the price at time T + 1, exceeds pT the price at time T ? Show your calculations. Hint: P (Z > −0.63) = 0.7357 for
Z N(0, 1).

Suppose µ > 0 and σe > 0. Consider P [pT +k > pT ]. How does this probability evolve as k is allowed to increase and where does it converge to?

Question 2

Explain the diﬀerence between stock returns that are serially uncorrelated and stock returns that are independent.

In the context of testing the Eﬃcient Markets Hypothesis discuss why the distinction between independence and uncorrelatedness of stock returns is important.

Using a suitable model describing the dynamics of stock prices define and briefly explain the concepts of a stochastic trend and deterministic trend.

A trader claims that there is often positive momentum in stock returns. You are provided with the following sample autocorrelations of a broad market return series based on T = 500 monthly observations: ρˆ1 = 0.32, ρˆ2 = 0.15, ρˆ3 = 0.10, ρˆ4 = 0.09.

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Implement a suitable test for verifying whether the trader’s claim is supported by the data and briefly discuss your findings (note: some useful cut-oﬀs of the standard normal distribution include P (Z > 1.28) = 0.10, P (Z > 1.65) = 0.05 and P (Z > 1.96) = 0.025 for Z N(0, 1)).

Discuss the shortcomings of the test you implemented in question (i).

Suppose that for a large number of stocks with returns rt, Corr[rt, rt−1] > 0 in the population. Explain how you could take advantage of this information to design a profitable investment strategy.

Question 3

Suppose that returns evolve according to rt+1 = µ + et+1 with E[et+1|It] = 0 (It denotes all available information up to and including time t). For any xt It show that Cov[rt+1, f(xt)] = 0, for any function f(.). Argue whether xt could be useful for designing a profitable investment strategy in such an environment.

Suppose that returns evolve according to rt+1 = µ + et+1 with et IID(0, σe2). Can such a model capture volatility clustering phenomena?

Suppose that returns evolve according to rt+1 = µ+et+1 with E[et+1|It] = 0 and E[e2t+1|It] = δ0 +δ1e2t with δ0 > 0 and 0 < δ1 < 1.

Are returns serially uncorrelated? Formally justify your answer.

Are returns independent? Formally justify your answer.

Is it worth investing in the stock market in such an environment?

Question 4

You wish to launch an investment fund based on a value investment philosophy. As part of your marketing brochure you wish to implement a historical backtest using yearly data on a universe of UK stocks (say N = 1000).

Briefly discuss the data you will need to collect in order to implement your backtest.

Provide a step by step explanation of the implementation of your backtest.

During a client presentation a potential investor is concerned about survivorship bias that may be aﬀecting your backtest. How would you adress this concern?

Given the historical profile of portfolio returns generated by your backtest explain how you would test whether there is a value anomaly in your universe.

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