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 80  CAPM, risk, diversification

Diversification is achieved by investing in a large amount of stocks. What type of risk is reduced by diversification?

(a) Idiosyncratic risk.

(b) Systematic risk.

(c) Both idiosyncratic and systematic risk.

(d) Market risk.

(e) Beta risk.

Question 125  option, speculation, market efficiency

Suppose that the US government recently announced that subsidies for fresh milk producers will be gradually phased out over the next year. Newspapers say that there are expectations of a 40% increase in the spot price of fresh milk over the next year.

Option prices on fresh milk trading on the Chicago Mercantile Exchange (CME) reflect expectations of this 40% increase in spot prices over the next year. Similarly to the rest of the market, you believe that prices will rise by 40% over the next year.

What option trades are likely to be profitable, or to be more specific, result in a positive Net Present Value (NPV)?

Assume that:

• Only the spot price is expected to increase and there is no change in expected volatility or other variables that affect option prices.
• No taxes, transaction costs, information asymmetry, bid-ask spreads or other market frictions.

(a) Buy one year call options on fresh milk.

(b) Buy one year put options on fresh milk.

(c) Sell one year call options on fresh milk.

(d) All of the above option trades are likely to have a positive NPV.

(e) None of the above option trades are likely to have a positive NPV.

Question 167  NPV, IRR

A project's net present value (NPV) is negative. Select the most correct statement.

(a) The project should be accepted.

(b) The project's IRR is less than its required return.

(c) The project's IRR is more than its required return.

(d) The project's IRR is equal to its required return.

(e) The project is under-priced.

Question 307  risk, variance

Let the variance of returns for a share per month be ##\sigma_\text{monthly}^2##.

What is the formula for the variance of the share's returns per year ##(\sigma_\text{yearly}^2)##?

Assume that returns are independently and identically distributed (iid) so they have zero auto correlation, meaning that if the return was higher than average today, it does not indicate that the return tomorrow will be higher or lower than average.

(a) ##\sigma_\text{yearly}^2 = \sigma_\text{monthly}^2##

(b) ##\sigma_\text{yearly}^2 = \sigma_\text{monthly}^2 \times 12##

(c) ##\sigma_\text{yearly}^2 = \sigma_\text{monthly}^2 \times 12^2##

(d) ##\sigma_\text{yearly}^2 = \sigma_\text{monthly}^2 \times \sqrt{12}##

(e) ##\sigma_\text{yearly}^2 = \sigma_\text{monthly}^2 \times {12}^{1/3}##

Question 560  standard deviation, variance, needs refinement

The standard deviation and variance of a stock's annual returns are calculated over a number of years. The units of the returns are percent per annum ##(\% pa)##.

What are the units of the standard deviation ##(\sigma)## and variance ##(\sigma^2)## 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) Percent per annum ##(\% pa)## and percent per annum ##(\% pa)##.

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

Question 581  APR, effective rate, effective rate conversion

A home loan company advertises an interest rate of 6% pa, payable monthly. Which of the following statements about the interest rate is NOT correct? All rates are given to four decimal places.

(a) The APR compounding monthly is 6.0000% per annum.

(b) The effective monthly rate is 0.5000% per month.

(c) The effective annual rate is 6.1678% per annum.

(d) The effective 6 month rate is 3.0000% per six months.

(e) The APR compounding semi-annually is 6.0755% per annum.

Question 595  future, continuously compounding rate

A 2-year futures contract on a stock paying a continuous dividend yield of 3% pa was bought when the underlying stock price was $10 and the risk free rate was 10% per annum with continuous compounding. Assume that investors are risk-neutral, so the stock's total required return is the risk free rate.

Find the forward price ##(F_2)## and value of the contract ##(V_0)## initially. Also find the value of the contract in 6 months ##(V_{0.5})## if the stock price rose to $12.

(a) ##F_2=11.5027, V_0=0, V_{0.5}=1.5715##

(b) ##F_2=11.5027, V_0=0, V_{0.5}=1.8258##

(c) ##F_2=12.9693, V_0=0, V_{0.5}=1.3218##

(d) ##F_2=12.9693, V_0=0, V_{0.5}=1.3896##

(e) ##F_2=12.9693, V_0=0, V_{0.5}=1.6144##

Question 599  bond pricing

On 22-Mar-2013 the Australian Government issued series TB139 treasury bonds with a combined face value $23.4m, listed on the ASX with ticker code GSBG25.

The bonds mature on 21-Apr-2025, the fixed coupon rate is 3.25% pa and coupons are paid semi-annually on the 21st of April and October of each year. Each bond's face value is $1,000.

At market close on Friday 11-Sep-2015 the bonds' yield was 2.736% pa.

At market close on Monday 14-Sep-2015 the bonds' yield was 2.701% pa. Both yields are given as annualised percentage rates (APR's) compounding every 6 months. For convenience, assume 183 days in 6 months and 366 days in a year.

What was the historical total return over those 3 calendar days between Friday 11-Sep-2015 and Monday 14-Sep-2015?

There are 183 calendar days from market close on the last coupon 21-Apr-2015 to the market close of the next coupon date on 21-Oct-2015.

Between the market close times from 21-Apr-2015 to 11-Sep-2015 there are 143 calendar days. From 21-Apr-2015 to 14-Sep-2015 there are 146 calendar days.

From 14-Sep-2015 there were 20 coupons remaining to be paid including the next one on 21-Oct-2015.

All of the below answers are given as effective 3 day rates.

(a) -0.035%

(b) 0.0267%

(c) 0.035%

(d) 0.2667%

(e) 0.3078%

Question 857  DuPont formula, accounting ratio

The DuPont formula is:

###\dfrac{\text{Net Profit}}{\text{Sales}} \times \dfrac{\text{Sales}}{\text{Total Assets}} \times \dfrac{\text{Total Assets}}{\text{Owners' Equity}}###

Which of the following statements about the DuPont formula is NOT correct?

(a) The DuPont formula gives the book Return on Equity (ROE).

(b) The net profit to sales ratio is called the ‘gross profit margin’.

(c) The sales to total assets ratio is called the ‘total asset turnover’ ratio.

(d) The total assets to owners' equity ratio is called the ‘equity multiplier’.

(e) The equity multiplier and ROE will rise if the firm uses more leverage by replacing equity with debt.