FINC5001 Module C – Mathematics of Risk
Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit
FINC5001
Module 3 – Financial Markets, Institutions, and Funds Management
Module C – Mathematics of Risk
Tutorial Solutions
Question 1
What role do financial intermediaries play in the financial system?
Financial intermediaries facilitate the optimal allocation of capital in the financial system. These intermediaries help create efficient markets by reducing trading frictions and increasing information dispersion. Key benefits that financial intermediaries, like investment funds, provide include creating market liquidity, pooling investments and risk, reducing costs, and creating economies of scale. Pooled investments also mean that investors can access larger scale investments that they would not be able to access by themselves, like large-scale commercial real estate.
Question 2
What are the differences between commercial banks and investment banks?
The key differences between commercial banks and investment banks can be seen from two factors: their clients and their operating activities.
Commercial banks:
- Clients: consumers, small-medium businesses, retail investors.
- Main activities: accept deposits, make loans, offer basic financial products.
Investment banks:
- Clients: large businesses, institutional investors, high net worth individuals, governments.
- Main activities: underwrite new debt and equity issues (IPOs), help with selling shares,
mergers and acquisitions, reorganisations, offer more complex financial products. It is important to note that some banks have both commercial and investment banking arms.
Question 3
What is risk and how can it be quantified in finance?
In finance, risk refers to the possibility that actual investment returns will be less than expected, including the possibility of negative returns.
Investors prefer a narrow distribution of possible returns. While it is great to receive more than you expect, it is painful to receive less than you expect . This is the idea of diminishing marginal utility of wealth. Because the pain of a loss (negative utility) is more than the positive utility from a gain of the same size, investors do not like variability of possible returns.
One way of measuring risk is to calculate the variance of investment returns and related to this is the standard deviation, which is the square root of the variance. Both measures give us an idea of the width of the distribution of possible outcomes — the wider the distribution, the more risk there is for this asset.
An alternative risk measure is to compute the beta coefficient of an investment, which we will look at in the next topic. A beta coefficient is a measure of the systematic risk of an individual asset. The beta tells us the percentage increase or decrease we would expect in an asset’s return as a result of a percentage change in the market return.
Question 4
The following historical month-end price data was found for Rio Tinto Ltd:
a) Calculate the arithmetic and geometric average rates of return over this period given that the company paid a dividend of $7.60 in August.
b) Explain which average rate of return is most appropriate for an investment in Rio Tinto Ltd.
Geometric returns are the compounded rate of return earned over the period and so describe the increase in the value of an investment over time. The return Rio Tinto has achieved over this multi period investment is better described by the geometric rate of 1.8390% p.a. However, in a typical year Rio Tinto earned the arithmetic average of 1.9596% p.a.
Arithmetic returns are widely used in investments because they provide a better forecast of the investment’s expected return in the near-term future. The arithmetic rate of return of 1.9596% p.a. is a better predictor of how much Rio Tinto can expect to earn in the coming investment period.
c) Calculate the standard deviation of returns using the arithmetic average return .
d) Assuming the returns in these months follow a normal distribution, provide an interpretation of the risk from investing in Rio Tinto.
If returns follow a normal distribution, then about 68.27% of the values are within 1 standard deviation of the mean, about 95.46% of the values are within two standard deviations and about 99.73% lie within 3 standard deviations. Applied to the current stock, returns of RIO would lie within 1.9596% ± 1 standard deviation approximately 68.27% of the time. This gives a range of -3.3497% to 7.2689% per month.
Question 5
What is the relationship between compound interest and geometric average rate of return?
The arithmetic average is the simple average of returns, which does not consider compounding. The geometric average is different as it takes compounding into account. The geometric average rate of return answers the question, ‘What was the growth rate of your investment?’ while the arithmetic average rate of return answers the question, ‘What was the average of the yearly rates of return?’ If you are trying to determine what annual rate of return you can expect over a multi-year horizon, then the geometric mean is better.
Question 6
The following table details an analyst’s prediction of the probabilities of different states of the market over the next year, along with the forecast returns on two shares in each different market state:
a) Calculate the expected return and standard deviation of returns on the shares of both Softbank Group and Toyota Motor Corporation.
b) What is the covariance of returns between these two stocks?
c) What is the correlation coefficient for these two stocks?
2022-02-04