Financial Economics May 2012 Solutions

FINANCIAL ECONOMICS MAY 2012 – SOLUTIONS

SECTION A(60 MARKS)

Answer ALL questions from this section.

QUESTION 1

1. State and define the three components of interest rate that are considered in investment analysis. (9 marks)

1. Risk-free rate – This is a rate that investors could invest fund at no or little risk. Normally this is a ten-year government bond as a proxy to a risk-free rate. (3 marks)
2. Risk premium – This is a rate above the risk-free rate that investors demand as compensation for losing the use of the money today and potentially investing in more risky assets than government bonds at the risk-free rate. (3 marks)
3. Inflation premium – inflation eats away at real returns and again investors need compensation for the loss of purchasing power. This would however lead to different results due to the intervals between payments.

1. Differentiate between effective and nominal interest rate. Provide clear examples for each. (6 marks).

Nominal Interest rate

This is the interest rate that makes no allowance for inflation and this is the rate that many banks advertise also known as annual percentage yield (APY). It is the return on the principal amount over an entire year. For example, a 5% rate compounded monthly would have an approximate APY of 5.12%.

Example: Let's assume a nominal interest rate of 6% per annum, which is credited as of = 0.5% every month.

After one year, the initial investment is increased by the factor (1+0.005)12 ≈ 1.0616. As a result, this nominal interest rate of 6% is equivalent to an effective interest rate of 6.16%.

Effective Interest rate

This is the interest rate that makes allowance for inflation and is the actual rate paid (or received) after accounting for compounding that occurs during the year. If you want to compare two alternative investments with different compounding periods you need to compute the effective interest rate also known as effective annual rate (EAR) and use that for comparison.

An interest rate is called nominal if the period of time after that the interest is credited (e.g. a month) is not identical to the basic time unit (normally a year).

QUESTION 2

1. What do you understand by the term annuity and under what circumstance could it be used? (4 marks)

The term annuity is used in finance theory to refer to any terminating stream of fixed payments over a specified period of time.

It is mostly used in connection with the valuation of the stream of payments, taking into account time value of money concepts such as interest rate and future value.

1. Give two types of annuities with clear examples for each. (6 marks

1. Ordinary annuity – payments are required at the end of each period. For example, straight bonds usually pay coupon payments at the end of every six months until the bond's maturity date. (3 marks)
2. Annuity due – payments are required at the beginning of each period. Rent is an example of the annuity due. (3 marks)

1. Calculate the payment at the beginning of a contract for a warehouse at 10% per annum whose payment is K5,000 per annum. (5 marks)

QUESTION 3

a.Compare and contrast current yield and adjusted current yield. Please give examples for each. (10 marks)

Current yield

A simple yield calculation that is often used to calculate the yield on both bonds and the dividend yield for stocks is the current yield. The current yield calculates the percentage return that the annual coupon payment provides the investor. In other words, this yield calculates what percentage the actual dollar coupon payment is of the price the investor pays for the bond. The multiplication by 100 in the formulas below converts the decimal into a percentage, allowing us to see the percentage return:

Current Yield = (Annual Kwacha Interest Paid/ Market price) * 100. So, if you purchased a bond with a par value of Mk100 for Mk95.92 and it paid a coupon rate of 5%, the current yield will be = ((0.05 * Mk 100)/Mk 95.92) * 100% = 5.21%

Adjusted yield

The modified current yield formula then takes into account the discount or premium at which the investor bought the bond. This is the full calculation:

Adjusted Current yield = (Annual coupon/Market price) * 100 + ((100- Market price)/years of maturity)

To re-calculate the yield of the bond in our first example, which matures in 30 months and has a coupon payment of Mk 5

Adjusted current yield = (Mk 5/Mk 95.92) * 100 + ((100- 95.92)/2.5) = 6.84%

b.In the absence of a financial calculator and a programme, define how you would use an approximation method to define yield to maturity. (5 marks)

Review the relationship between a bond's price and its yield. In general, as a bond's price increases, yield decreases. This relationship is measured using the price value of a basis point (PVBP). By taking into account factors such as the bond's coupon rate and credit rating, the PVBP measures the degree to which a bond's price will change when there is a 0.01% change in interest rates.

The charted relationship between bond price and required yield appears as a negative curve below.

Figure 1: Price Yield Curve

This is due to the fact that a bond's price will be higher when it pays a coupon that is higher than prevailing interest rates. As market interest rates increase, bond prices decrease.

QUESTION 4

Regression analysis is the statistical technique that identifies the relationship between two or more quantitative variables. The analysis has however, has strengths and limitations.

1. Give four limitations of regression analysis (8 marks).
2. The technique is demanding because it requires quantitative data relating to several thousand individuals.
3. Implementing the data collection can be time-consuming and expensive.
4. Regression analysis is likely to reach the conclusion that there is a strong link between two variables, whereas the influence of other, more important, variables may not have been estimated (this error is called "data snooping"). The tool should therefore be used with care.
5. Relations between the different explained and explanatory variables are often circular (X explains Y and Y explains X). In this case, the method is inapplicable.
6. The observations must present sufficiently contrasted evolutions to allow for adjustment. For example, if all the observations concern the 30-40 age group, it will not be possible to estimate the influence of age on employment.
7. Explain and define two factors that analysts must consider when making their estimates for investment evaluation. (7 marks)

SECTION B

Answer ANY TWO questions from this section.

QUESTION 5

1. Explain the sinking funds and give two common uses of sinking funds. (5 marks)

A sinking fund is defined as an annuity invested in order to meet a known commitment at some future date. (1 mark)

Common uses of Sinking funds –

(i)Repayment of debts (2 marks)

(ii)(ii) Providing funds to purchase a new asset when the existing one is fully depreciated. (2 marks)

1. Mr Jones has acquired a loan of K 650, 000 from one of the local banks at an interest rate of 7.75%. The loan is to be paid back over a period on 20 years. Calculate the annual payment necessary to amortize a debt for Mr. Jones. (15 marks)

P = K 650,000, n = 20 and i = 7.75/100 = 0.0775. Working through the formula, the annual payment necessary to amortise the debt is K64, 977.08

QUESTION 6

1. What is dispersion in financial economics? (2 marks)

It is how spread or variability the data is in statistics. It describes how spread out or scattered a set or distribution of numeric data is.

1. Compare and contrast with examples the range and mean deviation. (8 marks)

Both the range and the mean deviation are the particular measures of dispersion. However, the range is defined as the numerical difference between the smallest and largest values of the items in a set or distribution. Thus, it can be calculated as largest value minus smallest value.

An example for the range: two industrial machines over fourteen days were:

machine 1: 4,7,1,2,2,6,2,3,0,4,5,3,7,4

machine 2: 3,2,2,3,3,2,4,1,1,3,2,4,2,2

The range of values for machine 1 is 7-0=7 and for machine 2 is 4-1=3. Thus the daily production of rejects is more variable for machine 1.

On the other hand, mean deviation is a measure of dispersion that gives the average absolute difference between each item and the mean. It is a much more representative measure than the range since all item values are taken into account in its calculation.

An example of the mean deviation: Suppose an assembly line produced 3,10,5 and 2 defective products on four successive runs. the mean number of defectives is

The absolute differences between each value and the mean (5) are respectively:

3-5=-2(or 2 ignoring the minus sign)

10-5=5

5-5=0

2-5=-3 (or 3 ignoring the minus sign)

The mean deviation can now be calculated as the average of the above absolute differences. that is:

Mean deviation

1. Outline the main steps involved in rejecting a hypothesis and what does this mean?. (10 marks)

The hypothesis is tested basing on the assumptions about the population, assumptions which may or may not be true. This is always given in the form of a statement about the statistical nature of the population. Then if, on the evidence from a single sample taken from the population, it is found that the results obtained from the sample would have been unlikely to occur if the hypothesis were true, we would be inclined to reject the hypothesis.

Steps involved in rejecting the hypothesis.

STEP 1: State the null hypothesis H0 and the alternative hypothesis Ha.

To do a significance test, you need 2 hypotheses: a). Null Hypothesis (H0): the statement being tested and b), Alternative Hypothesis (Ha): the statement we hope or suspect is true instead of H0.

Hypotheses can be one-sided or two-sided.

One-sided hypothesis: covers just part of the range for your parameter

OR

Two-sided hypothesis: covers the whole possible range for your parameter

Even though Ha is what we hope or believe to be true, our test gives evidence for or against H0 only.

STEP 2: Calculate the value of the test statistic.

A test statistic measures compatibility between the H0 and the data. The formula for the test statistic will vary between different types of problems.

STEP 3: Draw a picture of what Ha looks like, and find the P-value.

P-value: the probability, computed assuming that H0 is true, that the test statistic would take a value as extreme or more extreme than that actually observed due to random fluctuation. It is a measure of how unusual your sample results are.

The smaller the P-value, the stronger the evidence against H0 provided by the data.

Calculate the P-value by using the sampling distribution of the test statistic

STEP 4: Compare your P-value to a significance level. State your conclusion about the data in a sentence.Compare P-value to a significance level, .

If the P-value ≤, we can reject H0.

In this case, we reject H0, meaning that the results significant.

QUESTION 7

1. What is risk diversification under modern portfolio theory? (6 marks)

The risk on an individual asset is derived from the probability distribution and it is usually assumed that the wealth owner cannot change that. But one of the most important ideas in financial economics is that the portfolio owners can reduce the average risk on the asset he/she owns by holding more than the same amount of wealth placed in one share. The notion that risk can be reduced by owning more than an asset is called risk diversification. In other words, the process of spreading an investment across assets (and thereby forming a portfolio) is called diversification. The principle of diversification indicates that spreading an investment across many assets will eliminate some of the risk.

1. Mention any three approaches to risk diversification. (4 marks)

1. Random or Naïve Diversification
2. Inter-industry diversification
3. Inter-quality rating diversification
4. Markowitz Risk diversification

1. Under Capital Asset Pricing Model (CAPM) describe systematic and unsystematic risk and how the risk can be reduced. (10 marks)

Systematic Risk (also known as non-diversifiable risk, non-specific, unavoidable or market risk) refers to that portion of risk of individual security’s returns caused by factors affecting market as a whole (macroeconomic variables) such as changes in interest rates, inflation, taxation etc. Indeed, systematic risks have market wide effects. Unsystematic Risk (diversifiable risk, specific or avoidable risk) refers to risk unique to a particular firm such a firm going bankrupt or staff of Post Office going on strike. Unsystematic risk accounts for approximately 70% of a firm’s total risk and can be reduced through diversification. Reducing unsystematic risk through holding diversified portfolios of share form the basis of Markwitz’s portfolio theory. The area labeled ‘unsystematic risk’ is the part that can be eliminated by diversification. CAPM uses the systematic risk of individual securities to determine the fir price.

Systematic and Unsystematic Risks

Risk

(Std. dev)Unsystematic risk

Total risk

Systematic risk

2030 No. of securities in a portfolio

CAPM uses the systematic risk of individual securities to determine the fair price.

QUESTION 8

1. According to Say's Law, when an economy produces a certain level of real GDP, it also generates the income needed to purchase that level of real GDP. It is believed that the economy is always capable of achieving the natural level of real GDP. Please illustrate with a diagram how an economy would move out of natural real GDP. (10 marks)

If aggregate demand falls below aggregate supply due to aggregate saving, suppliers will cut back on their production and reduce the number of resources that they employ. When employment of the economy's resources falls below the full employment level, the equilibrium level of real GDP also falls below its natural level. Consequently, the economy may not achieve the natural level of real GDP if there is aggregate saving. The classical theorists' response is that the funds from aggregate saving are eventually borrowed and turned into investment expenditures, which are a component of real GDP. Hence, aggregate saving need not lead to a reduction in real GDP.

Consider, however, what happens when the funds from aggregate saving exceed the needs of all borrowers in the economy. In this situation, real GDP will fall below its natural level because investment expenditures will be less than the level of aggregate saving. This situation is illustrated in the figure below.

Figure 2: Classical theory of interest rate adjustment in the money market

Aggregate saving, represented by the curve S, is an upward-sloping function of the interest rate; as the interest rate rises, the economy tends to save more. Aggregate investment, represented by the curve I, is a downward-sloping function of the interest rate; as the interest rate rises, the cost of borrowing increases and investment expenditures decline. Initially, aggregate saving and investment are equivalent at the interest rate, i. If aggregate saving were to increase, causing the S curve to shift to the right to S′, then at the same interest rate i, a gap emerges between investment and savings. Aggregate investment will be lower than aggregate saving, implying that equilibrium real GDP will be below its natural level.

1. How would you deal with the situation, explain with the aid of classical theory. (10 marks).

The figure below, considers a decrease in aggregate demand from AD1 to AD2.

The immediate, short-run effect is that the economy moves down along the SAS curve labeled SAS1, causing the equilibrium price level to fall from P1 to P2, and equilibrium real GDP to fall below its natural level of Y1 to Y2. If real GDP falls below its natural level, the economy's workers and resources are not being fully employed. When there are unemployed resources, the classical theory predicts that the wages paid to these resources will fall. With the fall in wages, suppliers will be able to supply more goods at lower cost, causing the SAS curve to shift to the right from SAS1 to SAS2. The end result is that the equilibrium price level falls to P3, but the economy returns to the natural level of real GDP.

END OF THE EXAMINATION PAPER

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A qualification examined by the Institute of Bankers in Malawi