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 97  WACC, no explanation

A company has:

• 10 million common shares outstanding, each trading at a price of $90.
• 1 million preferred shares which have a face (or par) value of $100 and pay a constant dividend of 9% of par. They currently trade at a price of $120 each.
• Debentures that have a total face value of $60,000,000 and a yield to maturity of 6% per annum. They are publicly traded and their market price is equal to 90% of their face value.
• The risk-free rate is 5% and the market return is 10%.
• Market analysts estimate that the company's common stock has a beta of 1.2. The corporate tax rate is 30%.

What is the company's after-tax Weighted Average Cost of Capital (WACC)? Assume a classical tax system.

(a) 10.27%

(b) 10.39%

(c) 10.43%

(d) 11.47%

(e) 17.22%

Question 122  option

You have just sold an 'in the money' 6 month European put option on the mining company BHP at an exercise price of $40 for a premium of $3.

Which of the following statements best describes your situation?

(a) You have bought the right to buy a BHP share for $40 in 6 months.

(b) You have bought the right to sell a BHP share for $40 in 6 months.

(c) You have bought the obligation to sell a BHP share for $40 in 6 months.

(d) You have sold the right that allows your counterparty to buy a BHP share from you for $40 in 6 months. This means that you have the obligation to sell a BHP share to your counterparty for $40 in 6 months if he wants to buy it.

(e) You have sold the right that allows your counterparty to sell a BHP share to you for $40 in 6 months. This means that you have the obligation to buy a BHP share from your counterparty for $40 in 6 months if he wants to sell it.

Question 175  pay back period, no explanation

A project has the following cash flows. Normally cash flows are assumed to happen at the given time. But here, assume that the cash flows are received smoothly over the year. So the $250 at time 2 is actually earned smoothly from t=1 to t=2:

Project Cash Flows
Time (yrs)	Cash flow ($)
0	-400
1	200
2	250

What is the payback period of the project in years?

(a) 0.20

(b) 1.20

(c) 1.80

(d) 2.25

(e) 2.80

Question 543  price gains and returns over time, IRR, NPV, income and capital returns, effective return

For an asset price to triple every 5 years, what must be the expected future capital return, given as an effective annual rate?

(a) 0.6

(b) 0.319508

(c) 0.269705

(d) 0.245731

(e) 0.219755

Question 561  covariance, correlation

The covariance and correlation of two stocks X and Y's annual returns are calculated over a number of years. The units of the returns are in percent per annum ##(\% pa)##.

What are the units of the covariance ##(\sigma_{X,Y})## and correlation ##(\rho_{X,Y})## of returns respectively?

(a) Percentage points per annum ##(\text{pp pa})## and percentage points per annum ##(\text{pp pa})##.

(b) Percentage points per annum ##(\text{pp pa})## and percentage points per annum all squared ##\left( (\text{pp pa})^2 \right)##.

(c) Percentage points per annum all squared ##\left( (\text{pp pa})^2 \right)## and percentage points per annum ##(\text{pp pa})##.

(d) Percentage points per annum all squared ##\left( (\text{pp pa})^2 \right)## and percentage points per annum all squared ##\left( (\text{pp pa})^2 \right)##.

(e) Percentage points per annum all squared ##\left( (\text{pp pa})^2 \right)## and a pure number with no units.

Hint: Visit Wikipedia to understand the difference between percentage points ##(\text{pp})## and percent ##(\%)##.

Question 807  market efficiency, expected and historical returns, CAPM, beta, systematic risk, no explanation

You work in Asia and just woke up. It looked like a nice day but then you read the news and found out that last night the American share market fell by 10% while you were asleep due to surprisingly poor macro-economic world news. You own a portfolio of liquid stocks listed in Asia with a beta of 1.6. When the Asian equity markets open, what do you expect to happen to your share portfolio? Assume that the capital asset pricing model (CAPM) is correct and that the market portfolio contains all shares in the world, of which American shares are a big part. Your portfolio beta is measured against this world market portfolio.

When the Asian equity market opens for trade, you would expect your portfolio value to:

(a) Remain unchanged.

(b) Instantaneously fall by 10%.

(c) Slowly fall by 10% over the day.

(d) Instantaneously fall by 16%.

(e) Slowly fall by 16% over the day.

Question 858  indirect security, intermediated finance, no explanation

Which of the following transactions involves an ‘indirect security’ using a ‘financial intermediary’?

(a) Alice buys Alan’s shares using their share brokers and the securities exchange.

(b) Bell buys Bob’s house using his real estate agent.

(c) Carla lends Carlos money at an interest rate of 10% pa for 3 years.

(d) Dora lends money to her bank at 2% pa and the bank lends the funds to David at 5% pa.

(e) Eve buys Ernie’s government bond using their brokers and the securities exchange.

Question 918  duration, Macaulay duration, modified duration, bond convexity

A fixed coupon bond’s modified duration is 10 years, and yields are currently 5% pa compounded annually. Which of the following statements about the bond is NOT correct?

(a) The Macaulay duration of the bond is 10.5 years.

(b) If yields-to-maturity suddenly rose by 1 percentage point then bond prices would suddenly fall by 10% or less.

(c) If yields-to-maturity suddenly fell by 2 percentage points then bond prices would suddenly rise by 20% or less.

(d) If the bond is a zero-coupon bond then its maturity is 10.5 years.

(e) The duration-price-change formula assumes parallel shifts in the yield curve.

Question 927  mean and median returns, mode return, return distribution, arithmetic and geometric averages, continuously compounding rate

The arithmetic average continuously compounded or log gross discrete return (AALGDR) on the ASX200 accumulation index over the 24 years from 31 Dec 1992 to 31 Dec 2016 is 9.49% pa.

The arithmetic standard deviation (SDLGDR) is 16.92 percentage points pa.

Assume that the log gross discrete returns are normally distributed and that the above estimates are true population statistics, not sample statistics, so there is no standard error in the sample mean or standard deviation estimates. Also assume that the standardised normal Z-statistic corresponding to a one-tail probability of 2.5% is exactly -1.96.

If you had a $1 million fund that replicated the ASX200 accumulation index, in how many years would the mean dollar value of your fund first be expected to lie outside the 95% confidence interval forecast?

(a) 23.17 years.

(b) 41.31 years.

(c) 134.19 years.

(d) 536.75 years.

(e) Never.