a
To compute: The proceeds of stock short sales needed to earn arbitrage profits. If the current interest rate is 2.5%,
Introduction:
Arbitrage Profit: Arbitrage is an act of buying an asset in the market and at the same time, the selling part is handled in another market. Such a calculated act of buying and selling is done to earn a profit when there is an imbalance of prices is called arbitrage profit.
a
Answer to Problem 26PS
The fraction of the proceeds which helps in earning the arbitrage profit is 0.76.
Explanation of Solution
S&P 500 index=1950
June maturity contract F0=1951
Current rate of interest=2.5%
Dividend=1.9%
Let us consider the S&P index value at S0.
Spot index S0=1950
Future index F0=1951
Time T=1
Rate of interest rf=2.5% when converted it becomes 0.025 (2.5/100=0.025)
Dividend=1.9% When converted in becomes 0.019(1.9/100=0.019)
To calculate arbitrage profits, proceeds of short sales is required. Let us assume the fraction of proceeds of short sales to be ‘p.’
We have to use the formula of spot-future parity here.
Where,
F0= Future index
S0= Current index
Rf= Interest rate
P=Proceeds
T=Time
D= Dividend payment
To proceed further, we have to first calculate the dividend payment.
By substituting the values in the formula, we get
After simplifying the equation, we get
By interchanging the values, we get
Therefore, the fraction of the proceeds which helps in earning the arbitrage profit is 0.76.
b.
To compute: The lower bound on the future prices which rules the arbitrage opportunities. Having 90% of the sales proceeds.
Introduction:
Arbitrage opportunity: It is an opportunity which can be availed to make a risk-free profit even in market fluctuations. The process of arbitrage involves buying of an asset in one market with a lesser price and sell it another market with a higher price.
b.
Answer to Problem 26PS
The lower bound on the future prices which rules the arbitrage opportunities is $1956.83
Explanation of Solution
S&P 500 index=1950
June maturity contract F0=1951
Current rate of interest=2.5%
Dividend=1.9%
We are told that the proceeds from short sales is 90%. So, p=90%
Therefore p=0.9
By substituting the values in the formula, we get
or 1956.83 (when rounded off)
Therefore, the lower bound on the future prices which rules the arbitrage opportunities is $1956.83
c.
To evaluate: The value of actual future price which falls below the no-arbitrage bound.
Introduction:
Arbitrage Profit: Arbitrage is an act of buying an asset in the market and at the same time, the selling part is handled in another market. Such a calculated act of buying and selling is done to earn a profit when there is an imbalance of prices is called arbitrage profit.
c.
Answer to Problem 26PS
The fall of actual price fall below the no-arbitrage opportunities will be by 5.83.
Explanation of Solution
S&P 500 index=1950
June maturity contract F0=1951
Current rate of interest=2.5%
Dividend=1.9%
From the above calculations, the values of lower bound on future price is $1956.83.
The calculations have to be done by using the formula:
So, the value of $5.83 reflects the potential profit by using arbitrage strategy.
Therefore, the fall of actual price fall below the no-arbitrage opportunities will be by 5.83.
d.
To determine: The arbitrage strategy and profits earned by using it.
Introduction:
Arbitrage Profit: Arbitrage is an act of buying an asset in the market and at the same time, the selling part is handled in another market. Such a calculated act of buying and selling is done to earn a profit when there is an imbalance of prices is called arbitrage profit.
d.
Answer to Problem 26PS
The profit per contract will be $1720 when the multiplier is $250.
Explanation of Solution
S&P 500 index=1950
June maturity contract F0=1951
Current rate of interest=2.5%
Dividend=1.9%
There are many strategies used by the investor. When the investor wants to earn risk free profits, the choice of arbitrage strategy proves to be good. The fundamental in this strategy is very simple. Arbitrage is the act of buying an asset in the market and at the same time, the selling part is handled in another market. Expecting two markets to deal with the same price is impossible. When there is a mismatch of prices between the two markets, arbitrage takes place. Like all other strategies even arbitrage strategy works on some rules namely,
- When the actual prices are greater than the future price, then the investor should purchase the spot and fell the futures.
- When the actual price is found to be lower than the future price, the investor should purchase futures and sell spot.
Her in this situation, we find that actual price is less than the future price, so investor should short the stock. The profit earned if the investor sells at 90% of the sales as proceeds can be calculated as below:
Selling price of the stock=1950
Proceeds = 90% of the sale
So, now we have to calculate the remaining proceeds.
So, 195 has to be kept in the margin account till the short position gets covered within 1 year. Therefore, the investor purchases future and lends 1755.
Particulars | Current cash flow | Cash flow after 1 year |
Purchase futures | 0 | |
Sale of shares | 1950-195 | |
Lend | -1755 | |
Total payoff | 0 | 6.875 |
Let us now consider the multiplier of S&P 500 contract to be $250.
The profit from arbitrage is 6.875 or 6.88 (when rounded off)
So, profit per contract will be calculated as follows:
The profit per contract will be $1720 when the multiplier is $250.
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Chapter 23 Solutions
INVESTMENTS (LOOSELEAF) W/CONNECT
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