.
An investment that requires $2000 initial investment will return $800 at the end of first year, $850 at the end of second year, and $900 at the end of third year. Assume the discount rate is continuously compounded at 10%. What is the Net Present Value of the investment?
NPV=PV-I = $800*e-.10*1+$850*e-.10*2+$900*e-.10*3 -$2,000=723.87+695.92+666.74-2,000=86.53
.
The current price of a stock is $40, and two-month European call options with a strike price of $39 currently sell for $5. An investor who feels that the price of the stock will increase is trying to decide between buying 100 shares and buying 800 call options (8 contracts). Both strategies involve an investment of $4,000.
a. Which strategy will earn more profits if the stock increases to $42?
b. How high does the stock price have to rise for the option strategy to be more profitable?
A:
a. If the stock closes at $42, buying100 shares will have a profit 0f $200 and buying 800 calls will have a loss of $1600. So buying stocks will be better than buying options.
Shares Price Position Cost Profit
stock investing 100 42.00 $4200 4000 $200.00
options only investing 800 $3(ITM) $2400 4,000 -$1600
Stock investing makes a profit of $200 and a loss of $1600 for options investing.
b. The strategies are equally profitable if the stock price rises to a level, S
The profit to buy 100 shares: 100 x S – 4,000
The profit to buy 1000 calls: 800(S – 39) – 4000
100 x S – 4,000 = 800(S – 39) – 4000
Find S:
100 x S – 4,000 = 800x S – 31200 – 4,000
700S=31200
S=31200/700=44.57
The option strategy is therefore more profitable if the stock price rises above $44.57
. (protective put)
The stock price of BAC is currently $120 and a put option with strike price of $120 is $5. A trader goes long 200 shares of BAC stock and long 2 contracts of the put options with strike price of $120.
a. What is the maximum potential loss for the trader?
b. When the stock price is $140, what is the trader’s net profit?
Solution:
a. What is the maximum potential loss for the trader?
S= $120 K= $120 P= $5
long 200 long 200 P
The costs to establish this protective put position: 200*$120+200*$5=$24,000+$1,000=$25,000
When the stock goes below $120, you will exercise the protective put option to limit the losses.
The sale proceeds from exercising put option (to sell the stock at $120) =200*$120=$24,000
The profit= sale proceeds – initial costs=$24,000-$25,000=-$1000
So the maximum loss=$1,000
b. When the stock price is $140, what is the trader’s net profit?
S= $140 K= $120 P= $5
The stock value at $140 per share =200*$140=$28,000
initial costs=$25,000
Net profit=28000-25000=$3000
. (Compare Options versus stock investments)
Suppose you think Tesla stock is going to appreciate substantially in value in the next year. Say the stock’s current price, S, is $500, and a call option expiring in one year has an exercise price, X, of $510 and is selling at a price, C, of $50. With $50,000 to invest, you are considering three investment alternatives.
Alternative A: Invest all $50,000 in the stock, buying 100 shares.
Alternative B: Invest all $50,000 in 1,000 options (10 contracts).
Alternative C: Buy 100 options (one contract) for $5,000, and invest the remaining $45,000 in a money market fund paying 6% annual interest.
a. What is total value of the investment for alternative A if Tesla stock goes up to $700 one year later?
b. What is your rate of return for alternative A if Tesla stock goes up to $700 one year later?
c. What is total value of the investment for alternative C if Tesla stock goes up to $700 one year later?
d. What is your rate of return for alternative C if Tesla stock goes up to $700 one year later?
Dollar Value Rate of Returns
Stock price= 700 700
Alternative A All stocks(100 shares) 70000 40.00%
Alternative B All Options(1000 shares) 190000 280.00%
Alternative C Bills+100 Options 66782.64 33.57%
Please make sure you feel comfortable with the answers in the table. Refer to the teaching note for the detailed steps in the calculations.
Answer a: Alternative C will buy 100 options (one contract) for $5000, and invest the remaining money ($45000) in a money market fund paying 6% annual interest:
the investment in the money market fund will becomes: 45000*ert= 45000*e0.06*1= 45000*1.061837 =47782.64 (t is the number of years)
When the stock price is $700, each call option will make a gain of $190 ($700-510=190), 100 call options will make 100*$190=19,000
The total dollar value will be 47782.64+19000=66782.64
.
Suppose that zero and forward rates are as follows:
Zero rate for an n-year
investment (% per annum)
Year(n) (% per annum)
1 3.0
2 4.0 (rate for two years)
3 4.6 (rate for three years)
4 5.0 (rate for four years)
5 5.3 (rate for five years)
Please calculate
Solution:
Zero rate for an n-year Forward Rate
investment (% per annum) For nth Year
Year(n) (% per annum) (% per annum)
1 3.0
2 4.0 (rate for two years) 5.0 (2nd yr rate)
3 4.6 (rate for three years) 5.8 (3rd yr rate)
4 5.0 (rate for four years) 6.2 (4th yr rate)
5 5.3 (rate for five years) 6.5 (5th yr rate)
In this example, future LIBOR rate (RM) is not given, we will use the forward rate for (RM).
Determine the forward rate for the second year: (please review the teaching notes)
The forward rate for the period between times TT and TT+1 is
(RT+1TT+1 -RTTT)/(TT+1-TT)
Forward rate for the second year: (R2T2 -R1T1)/(T2-T1)=(4*2-3*1)/(2-1)=5/1=5% (Please verify the forward rates for the other periods)
So the forward rate in the second year is 5% with continuous compounding or 5.127% (er-1=e0.05-1=1.05127-1=0.05127) with annual compounding.
the cash flow to the lender at T2: L(Rk-RF)(T2-T1)
=100m(6%-5.127%)(2-1)=$873,000
What is the value today? VFRA=PV=873,000*e^(-R2T2)= 873,000*e^(-R2T2)= 873,000*e^(-4%*2)=$805,880.6
Or use the equation (4.9) from the book, VFRA=L(Rk-RF)(T2-T1)*e-R2T2
VFRA= 100,000,000(0.06-0.05127)(2-1)*e-0.04*2=$805,880.6