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 128  debt terminology, needs refinement

An 'interest payment' is the same thing as a 'coupon payment'. or ?

Question 306  risk, standard deviation

Let the standard deviation of returns for a share per month be ##\sigma_\text{monthly}##.

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

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} = \sigma_\text{monthly}##

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

(c) ##\sigma_\text{yearly} = \sigma_\text{monthly} \times 144##

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

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

Question 418  capital budgeting, NPV, interest tax shield, WACC, CFFA, CAPM

Project Data
Project life	1 year
Initial investment in equipment	$8m
Depreciation of equipment per year	$8m
Expected sale price of equipment at end of project	0
Unit sales per year	4m
Sale price per unit	$10
Variable cost per unit	$5
Fixed costs per year, paid at the end of each year	$2m
Interest expense in first year (at t=1)	$0.562m
Corporate tax rate	30%
Government treasury bond yield	5%
Bank loan debt yield	9%
Market portfolio return	10%
Covariance of levered equity returns with market	0.32
Variance of market portfolio returns	0.16
Firm's and project's debt-to-equity ratio	50%

Notes

1. Due to the project, current assets will increase by $6m now (t=0) and fall by $6m at the end (t=1). Current liabilities will not be affected.

Assumptions

• The debt-to-equity ratio will be kept constant throughout the life of the project. The amount of interest expense at the end of each period has been correctly calculated to maintain this constant debt-to-equity ratio.
• Millions are represented by 'm'.
• All cash flows occur at the start or end of the year as appropriate, not in the middle or throughout the year.
• All rates and cash flows are real. The inflation rate is 2% pa. All rates are given as effective annual rates.
• The project is undertaken by a firm, not an individual.

What is the net present value (NPV) of the project?

(a) $5.772m

(b) $4.979m

(c) $4.959m

(d) $4.733m

(e) $4.584m

Question 433  Merton model of corporate debt, real option, option, no explanation

A risky firm will last for one period only (t=0 to 1), then it will be liquidated. So it's assets will be sold and the debt holders and equity holders will be paid out in that order. The firm has the following quantities:

##V## = Market value of assets.

##E## = Market value of (levered) equity.

##D## = Market value of zero coupon bonds.

##F_1## = Total face value of zero coupon bonds which is promised to be paid in one year.

What is the payoff to equity holders at maturity, assuming that they keep their shares until maturity?

(a) ##max(V_1 - F_1, 0)##.

(b) ##max(V_1 - D_1, 0)##.

(c) ##max(F_1 - V_1, 0)##.

(d) ##max(D_1 - V_1, 0)##.

(e) ##-min(F_1 - V_1, 0)##.

Question 482  market capitalisation of equity

The below screenshot of Microsoft's (MSFT) details were taken from the Google Finance website on 28 Nov 2014. Some information has been deliberately blanked out.

What was MSFT's market capitalisation of equity?

(a) $395.11 million

(b) $21.01 billion

(c) $121.92 billion

(d) $393.95 billion

(e) $1.02935 trillion

Question 495  risk, accounting ratio, no explanation

High risk firms in danger of bankruptcy tend to have:

(a) Low debt to assets ratios.

(b) Low quick ratios.

(c) High interest coverage ratios.

(d) Positive amounts of net working capital.

(e) High net profit margins.

Question 540  APR, effective rate, no explanation

Which one of the below statements about effective rates and annualised percentage rates (APR's) is NOT correct?

(a) An effective semi-annual rate could be called: "a six month rate compounding per six months".

(b) An APR compounding semi-annually could be called: "a yearly rate compounding per six months".

(c) An effective monthly rate could be called: "a monthly rate compounding every month".

(d) An APR compounding monthly could be called: "a monthly rate compounding per year".

(e) An effective daily rate could be called: "a daily rate compounding every day".

Question 793  option, hedging, delta hedging, gamma hedging, gamma, Black-Scholes-Merton option pricing

A bank buys 1000 European put options on a $10 non-dividend paying stock at a strike of $12. The bank wishes to hedge this exposure. The bank can trade the underlying stocks and European call options with a strike price of 7 on the same stock with the same maturity. Details of the call and put options are given in the table below. Each call and put option is on a single stock.

European Options on a Non-dividend Paying Stock
Description	Symbol	Put Values	Call Values
Spot price ($)	##S_0##	10	10
Strike price ($)	##K_T##	12	7
Risk free cont. comp. rate (pa)	##r##	0.05	0.05
Standard deviation of the stock's cont. comp. returns (pa)	##\sigma##	0.4	0.4
Option maturity (years)	##T##	1	1
Option price ($)	##p_0## or ##c_0##	2.495350486	3.601466138
##N[d_1]##	##\partial c/\partial S##		0.888138405
##N[d_2]##	##N[d_2]##		0.792946442
##-N[-d_1]##	##\partial p/\partial S##	-0.552034778
##N[-d_2]##	##N[-d_2]##	0.207053558
Gamma	##\Gamma = \partial^2 c/\partial S^2## or ##\partial^2 p/\partial S^2##	0.098885989	0.047577422
Theta	##\Theta = \partial c/\partial T## or ##\partial p/\partial T##	0.348152078	0.672379961

 

Which of the following statements is NOT correct?

(a) If the underlying stock price suddenly rose by $0.01, the put option price would be expected to fall by less than $0.00552034777953178.

(b) If the underlying stock price suddenly fell by $0.01, the put option price would be expected to rise by more than $0.00552034777953178.

(c) If the underlying stock price suddenly rose by $0.01, the call option price would be expected to rise by more than $0.00888138404936379.

(d) To Delta hedge the 1000 long put options, long 552.0348 stocks.

(e) To Delta and Gamma hedge the 1000 long put options, long 2078.4226 call options and short 2397.9617 stocks.

Question 892  foreign exchange reserve, foreign exchange rate, no explanation

The Chinese central bank has the largest amount of foreign currency reserves.

What could the large amounts of foreign exchange reserves held by the Chinese government be used for in a currency crisis? China's currency is called the Renminbi (RMB) or Yuan (CNY). In a Chinese currency crisis the Chinese government is likely to use its FX reserves to:

(a) Buy RMB.

(b) Buy USD.

(c) Buy EUR.

(d) Buy JPY.

(e) Sell RMB.