Question

The table below shows the no-arbitrage prices of securities A and B. The economy is equally likely to strengthen or weaken in one year.

		Cash Flow in One Year
Security	Market Price Today	Weak economy	Strong Economy
A	231	0	600
B	346	600	0

1. What are the payoffs of a portfolio of one share of security A and one share of security B?
2. What is the market price of this portfolio? What expected return will you earn from holding this portfolio?
3. What is the risk-free interest rate?

2You are the beneficiary of a trust fund that will start paying you cash flows in five years. The cash flows will be $25,000 per year and will continue for 40 years. If the interest rate is 4% per year, what is the value needed in the trust fund now to fund these cash flows?

Answer 1

In case of weak economy (50% probability)

Expected Payoff of a portfolio of one share of security A and one share of security B

= 0+600

=600

In case of strong economy (50% probability)

Expected Payoff of a portfolio of one share of security A and one share of security B

= 600+0

=600

So, no matter what the state of economy is, one gets a payoff of 600 in one year

Expected payoff = 0.5*600+0.5*600 = 600

market price of the portfolio

= no arbitrage price of one share of Security A +no arbitrage price of one share of Security B

=231+346

=577

Expected return from holding the portfolio = 600/577 -1 = 0.03986 or 3.986%

Since the portfolio of one share of A and one share of B always gives a payoff of 600 after one year. there is no variance in the cashflows and hence it is a riskless portfolio . Thus, the expected return from holding this portfolio 3.986% is the same as the risk free interest rate in the economy.

So,   risk free interest rate is 3.986%

2.

value needed in the trust fund now to fund these cash flows

=present value of future cashflows

=25000/1.04^5+25000/1.04^6+........+25000/1.04^44

= 1/1.04^4* (25000/1.04+25000/1.04^2+........+25000/1.04^40)

=1/1.04^4* 25000/0.04*(1-1/1.04^40)

=422973.65

So, to fund the cash flows, amount required is $422973.65