Fight Finance

Question 65  annuity with growth, needs refinement

Which of the below formulas gives the present value of an annuity with growth?

(a) ##\dfrac{C_1}{r-g}\left(1-\dfrac{1}{(1+r)^T}\right)##

(b) ##\dfrac{C_1(1+g)^T}{r-g}\left(1-\dfrac{1}{(1+r)^T}\right)##

(c) ##\dfrac{C_1}{r-g}\left(1-\dfrac{1+g}{(1+r)^T}\right)##

(d) ##\dfrac{C_1}{r-g}\left(1-\left(\dfrac{1+g}{1+r}\right)^T \right)##

(e) ##\dfrac{C_1(1+g)}{r-g}\left(1-\dfrac{1}{(1+r)^T}\right)##

Hint: The equation of a perpetuity without growth is: ###V_\text{0, perp without growth} = \frac{C_\text{1}}{r}###

The formula for the present value of an annuity without growth is derived from the formula for a perpetuity without growth.

The idea is than an annuity with T payments from t=1 to T inclusive is equivalent to a perpetuity starting at t=1 with fixed positive cash flows, plus a perpetuity starting T periods later (t=T+1) with fixed negative cash flows. The positive and negative cash flows after time period T cancel each other out, leaving the positive cash flows between t=1 to T, which is the annuity.

###\begin{aligned} V_\text{0, annuity} &= V_\text{0, perp without growth from t=1} - V_\text{0, perp without growth from t=T+1} \\ &= \dfrac{C_\text{1}}{r} - \dfrac{ \left( \dfrac{C_\text{T+1}}{r} \right) }{(1+r)^T} \\ &= \dfrac{C_\text{1}}{r} - \dfrac{ \left( \dfrac{C_\text{1}}{r} \right) }{(1+r)^T} \\ &= \dfrac{C_\text{1}}{r}\left(1 - \dfrac{1}{(1+r)^T}\right) \\ \end{aligned}###

The equation of a perpetuity with growth is:

###V_\text{0, perp with growth} = \dfrac{C_\text{1}}{r-g}###

Question 90  CAPM, risk

According to the theory of the Capital Asset Pricing Model (CAPM), total variance can be broken into two components, systematic variance and idiosyncratic variance. Which of the following events would be considered the most diversifiable according to the theory of the CAPM?

(a) Global economic recession.

(b) A major terrorist attack, grounding all commercial aircraft in the US and Europe.

(c) An increase in corporate tax rates.

(d) The outbreak of world war.

(e) A company's poor earnings announcement.

Question 269  time calculation, APR

A student won $1m in a lottery. Currently the money is in a bank account which pays interest at 6% pa, given as an APR compounding per month.

She plans to spend $20,000 at the beginning of every month from now on (so the first withdrawal will be at t=0). After each withdrawal, she will check how much money is left in the account. When there is less than $500,000 left, she will donate that remaining amount to charity.

In how many months will she make her last withdrawal and donate the remainder to charity?

(a) In 31 months (t=31 months).

(b) In 30 months (t=30 months).

(c) In 28 months (t=28 months).

(d) In 27 months (t=27 months).

(e) In 26 months (t=26 months).

Question 313  foreign exchange rate, American and European terms

If the AUD appreciates against the USD, the American terms quote of the AUD will or ?

Question 575  inflation, real and nominal returns and cash flows

You expect a nominal payment of $100 in 5 years. The real discount rate is 10% pa and the inflation rate is 3% pa. Which of the following statements is NOT correct?

(a) The nominal cash flow of $100 in 5 years is equivalent to a real cash flow of $86.2609 in 5 years. This means that $86.2609 will buy the same amount of goods and services now as $100 will buy in 5 years.

(b) The real discount rate of 10% pa is equivalent to a nominal discount rate of 13.3333% pa.

(c) The nominal price of goods and services will increase by 3% every year.

(d) The real price of goods and services will increase by 3% every year.

(e) The present value of your payment will increase by the nominal discount rate every year.

Question 605  cross currency interest rate parity, foreign exchange rate

If the Reserve Bank of Australia is expected to keep its interbank overnight cash rate at 2% pa while the US Federal Reserve is expected to keep its federal funds rate at 0% pa over the next year, is the AUD is expected to , , or remain against the USD over the next year?

Question 625  dividend re-investment plan, capital raising

Which of the following statements about dividend re-investment plans (DRP's) is NOT correct?

(a) DRP's are voluntary, shareholders only participate if they choose.

(b) DRP's increase the number of shares.

(c) The number of shares issued to a shareholder participating in a DRP is usually calculated as their total dividends owed, divided by the allocation share price which is usually close to the current market share price.

(d) DRP's do not incur brokerage costs for the shareholder. This is unlike the case where the shareholder uses the cash dividend to buy more shares herself.

(e) If all shareholders participated in a company's DRP, the company would not pay any dividends and the firm's share price would not fall due to the cash dividend or the DRP.

Question 706  utility, risk aversion, utility function

Mr Blue, Miss Red and Mrs Green are people with different utility functions.

Note that a fair gamble is a bet that has an expected value of zero, such as paying $0.50 to win $1 in a coin flip with heads or nothing if it lands tails. Fairly priced insurance is when the expected present value of the insurance premiums is equal to the expected loss from the disaster that the insurance protects against, such as the cost of rebuilding a home after a catastrophic fire.

Which of the following statements is NOT correct?

(a) Mr Blue, Miss Red and Mrs Green all prefer more wealth to less. This is rational from an economist's point of view.

(b) Mr Blue is risk averse. He will not enjoy a fair gamble and would like to buy fairly priced insurance.

(c) Miss Red is risk-neutral. She will not enjoy a fair gamble but wouldn't oppose it either. Similarly with fairly priced insurance.

(d) Mrs Green is risk-loving. She would enjoy a fair gamble and would dislike fairly priced insurance.

(e) Mr Blue would like to buy insurance, but only if it is fairly or under priced.

Question 733  DDM, income and capital returns

A share’s current price is $60. It’s expected to pay a dividend of $1.50 in one year. The growth rate of the dividend is 0.5% pa and the stock’s required total return is 3% pa. The stock’s price can be modeled using the dividend discount model (DDM):

##P_0=\dfrac{C_1}{r-g}##

Which of the following methods is NOT equal to the stock’s expected price in one year and six months (t=1.5 years)? Note that the symbolic formulas shown in each line below do equal the formulas with numbers. The formula is just repeated with symbols and then numbers in case it helps you to identify the incorrect statement more quickly.

(a) ##P_{1.5}=P_0 (1+g)^1 (1+r)^{0.5}=60(1+0.005)^1 (1+0.03)^{0.5}##

(b) ##P_{1.5}=(P_0 (1+r)^1-C_1 ) (1+r)^{0.5}=(60(1+0.03)^1-1.5) (1+0.03)^{0.5}##

(c) ##P_{1.5}=\dfrac{C_1}{r-g} (1+r)^1 (1+g)^{0.5}=\dfrac{1.5}{0.03-0.005} (1+0.03)^1 (1+0.005)^{0.5}##

(d) ##P_{1.5}=\dfrac{C_1 (1+g)^1}{r-g} (1+r)^{0.5}=\dfrac{1.5(1+0.005)^1}{0.03-0.005} (1+0.03)^{0.5}##

(e) ##P_{1.5}=\dfrac{C_1 (1+g)^2}{r-g}/(1+r)^{0.5} +C_1 (1+g)^1/(1+r)^{0.5} =\dfrac{1.5(1+0.005)^2}{0.03-0.005}/(1+0.03)^{0.5} +1.5(1+0.005)^1/(1+0.03)^{0.5} ##

Question 941  negative gearing, leverage, capital structure, interest tax shield, real estate

Last year, two friends Lev and Nolev each bought similar investment properties for $1 million. Both earned net rents of $30,000 pa over the past year. They funded their purchases in different ways:

• Lev used $200,000 of his own money and borrowed $800,000 from the bank in the form of an interest-only loan with an interest rate of 5% pa.
• Nolev used $1,000,000 of his own money, he has no mortgage loan on his property.

Both Lev and Nolev also work in high-paying jobs and are subject personal marginal tax rates of 45%.

Which of the below statements about the past year is NOT correct?

(a) Lev’s annual interest expense would have been $40,000.

(b) Lev’s annual loss before tax on the investment property was $10,000

(c) Lev’s annual loss after tax on the investment property was $5,500

(d) Lev’s personal tax saving due to the investment property was $5,500.

(e) Nolev’s personal tax payable due to the investment property was $13,500.