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Question 999 duration, duration of a perpetuity with growth, CAPM, DDM

A stock has a beta of 0.5. Its next dividend is expected to be $3, paid one year from now. Dividends are expected to be paid annually and grow by 2% pa forever. Treasury bonds yield 5% pa and the market portfolio's expected return is 10% pa. All returns are effective annual rates.

What is the Macaulay duration of the stock now?

Question 434 Merton model of corporate debt, real option, option

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 debt holders at maturity, assuming that they keep their debt until maturity?

Question 383 Merton model of corporate debt, real option, option

In the Merton model of corporate debt, buying a levered company's debt is equivalent to buying the company's assets and:

Question 709 continuously compounding rate, APR

Which of the following interest rate quotes is NOT equivalent to a 10% effective annual rate of return? Assume that each year has 12 months, each month has 30 days, each day has 24 hours, each hour has 60 minutes and each minute has 60 seconds. APR stands for Annualised Percentage Rate.

Question 710 continuously compounding rate, continuously compounding rate conversion

A continuously compounded monthly return of 1% $(r_\text{cc monthly})$ is equivalent to a continuously compounded annual return $(r_\text{cc annual})$ of:

Question 712 effective rate conversion

An effective monthly return of 1% $(r_\text{eff monthly})$ is equivalent to an effective annual return $(r_\text{eff annual})$ of:

Question 722 mean and median returns, return distribution, arithmetic and geometric averages, continuously compounding rate

Here is a table of stock prices and returns. Which of the statements below the table is NOT correct?

Price and Return Population Statistics
Time	Prices	LGDR	GDR	NDR
0	100
1	50	-0.6931	0.5	-0.5
2	100	0.6931	2	1

Arithmetic average		0	1.25	0.25
Arithmetic standard deviation		0.9802	1.0607	1.0607

(a) The geometric average of the gross discrete returns (GAGDR) equals 1 which is 100%.

(b) $\text{GAGDR} = \exp \left( \text{AALGDR} \right)$ . The GAGDR is equal to the natural exponent of the arithmetic average of the logarithms of the gross discrete returns (AALGDR).

(c) $\text{GAGDR} = \left( P_T/P_0 \right)^{1/T}$ . The GAGDR equals the ratio of the last and first prices raised to the power of the inverse of the number of time periods between them.

(d) $\text{LGAGDR} = \text{AALGDR}$ . This is always true, regardless of the distribution of the prices or returns and the number of return observations. The logarithm of the geometric average of the gross discrete returns (LGAGDR) is always equal to the arithmetic average of the logarithms of the gross discrete returns (AALGDR).

(e) $\text{LAAGDR} = \text{AALGDR} + \text{SDLGDR}^2/2$ . This is always true, regardless of the distribution of the prices or returns and the number of return observations. The logarithm of the arithmetic average of the gross discrete returns (LAAGDR) equals the arithmetic average of the logarithms of the gross discrete returns (AALGDR) plus half the variance of the LGDR's.

Question 877 arithmetic and geometric averages, utility, utility function

Gross discrete returns in different states of the world are presented in the table below. A gross discrete return is defined as $P_1/P_0$ , where $P_0$ is the price now and $P_1$ is the expected price in the future. An investor can purchase only a single asset, A, B, C or D. Assume that a portfolio of assets is not possible.

Gross Discrete Returns
In Different States of the World
Investment	World states (probability)
asset	Good (50%)	Bad (50%)
A	2	0.5
B	1.1	0.9
C	1.1	0.95
D	1.01	1.01

Which of the following statements about the different assets is NOT correct? Asset:

Question 866 option, Black-Scholes-Merton option pricing

A one year European-style put option has a strike price of $4.

The option's underlying stock currently trades at $5, pays no dividends and its standard deviation of continuously compounded returns is 47% pa.

The risk-free interest rate is 10% pa continuously compounded.

Use the Black-Scholes-Merton formula to calculate the option price. The put option price now is:

Question 904 option, Black-Scholes-Merton option pricing, option on future on stock index

A six month European-style call option on six month S&P500 index futures has a strike price of 2800 points.

The six month futures price on the S&P500 index is currently at 2740.805274 points. The futures underlie the call option.

The S&P500 stock index currently trades at 2700 points. The stock index underlies the futures. The stock index's standard deviation of continuously compounded returns is 25% pa.

The risk-free interest rate is 5% pa continuously compounded.

Use the Black-Scholes-Merton formula to calculate the option price. The call option price now is:

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?