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MATH 423 Winter 2022

Homework 1

Instructions

• All results have to be justiﬁed by appropriate intermediate steps and/or argu- ments. Only complete and clear solutions will receive full credit.

Homework (50 Points)

Problem 1. (10 Points) Consider a market of two assets: the risk-free asset with price A(0) = 1 and A(1) = 1.2 and the stock with price S(0) = 100 and

S(1) =

Is this market arbitrage-free? If yes, explain why. If not, ﬁnd a arbitrage opportunity.

Problem 2. (10 Points) Consider a bond with prices A(0) = $100, A(T) = $115 and a stock with price S(0) = $100,

S(T) =

with probability p

with probability 1 − p .

(a) (2 point) Find the return of the bond.

(b) (2 points) Find the return of the stock.

V(T) =

when stock price goes up

when stock price goes down .

(d) (3 point) Let p = 0.6. Calculate the expected return and risk of the portfolio found in (c). (Write answer in terms of percentage and keep two decimal places.)

Problem 3. (10 Points) Let A(0) = 100, A(1) = 115 and S(0) = 35. Is it possible to ﬁnd an arbitrage opportunity if the forward price of stock is F = 38.60 with delivery time 1? Justify your answer.

Problem 4. (5 Points) How long will it take for $1000 attracting simple interest to become $1050 if the interest rate is 2%? Also, compute the return of this investment.

Problem 5. (5 Points) Find the present value of $100, 000 to be received after 10 years if the interest rate is assumed to be 5% and

(a) (2 points) semi-annual compounding applies (Round to the nearest cents); (b) (2 points) quarterly compounding applies (Round to the nearest cents);

(c) (1 points) what is the return of the investment in (a)? (Write answer in terms of percentage and keep two decimal places.)

Problem 6. (10 Points) Suppose that you took a mortgage of $600, 000 on a house to be paid back in full by 30 equal annual instalments, each consisting of the interest due on the outstanding balance plus a repayment of a part of the amount borrowed. If you decide to clear the mortgage after 20 years, how much would you need to pay in addition to the twentieth instalment, assuming that an annual compounding interest rate of 7% applies through out the period of the mortgage? (Round to the nearest cents)

MATH 423 Winter 2022

Homework 2

Instructions

• All results have to be justiﬁed by appropriate intermediate steps and/or argu- ments. Only complete and clear solutions will receive full credit.

Homework (50 Points)

Problem 1. (10 Points) A bond with annual coupons C = $5, maturing after 10 years, is trading at par. Let the implied continuous compounding rate be 6%. For the following questions, round your answer to the nearest hundredths:

(a) (4 Points) Find the face value of the bond;

(b) (1 Points) Find the price of the bond (at t = 0);

(d) (2 Points) After 15 months, what is the price of the bond?

Problem 2. (5 Points) One zero-coupon bond is purchased at the price of B(0, 1) = 0.88.

(a) (6 Points) After how many days will the bond produce a 5% return? (round your answer to the nearest hundredths)

(b) (4 Points) At the end of each year, the proceeds are reinvested in new bonds of the

same kind. How many bonds will be purchased at the end of year ﬁve? (round your answer to the nearest hundredths)

Problem 3. (10 Points) Suppose that in the ﬁnancial market there are two risky stocks S1 and S2 with current prices S1 (0) = $50 and S2 (0) = $50. You want to invest $1000 in a portfolio V of the two stocks. You have the following four scenarios about the returns of the stocks after one year:

Scenario	Probability	K1	K2
1	0.2	-10 %	10 %
2	0.3	-20 %	20 %
3	0.3	-5 %	-5 %
4	0.2	10 %	5 %

(a) (4 Points) Find the data of the market: µ1,µ2 , σ , σ and c12 := Cov(K1,K2). (Do not round your answer)

(b) (3 Points) You buy 12 shares of stock S1 and with the remainder of the money you buy shares in stock S2. Find µV and σ. (You may round your answer to 4 signiﬁcant digits)

(c) (3 Points) How many shares of each stock you should buy in order for your portfolio to have the minimum possible variance? (Round to the nearest thousandth)

Problem 4. (10 Points) Consider a market that consists of 2 independent stocks with the following data for the associated returns:

µ1 = 0.2, µ2 = 0.4, σ1 = 0.2, σ2 = 0.3.

An investor wishes to construct a portfolio V in this market with high expected return and low risk. For this, he decides to minimize σ − 2µV . (Note: the independence of the two stocks indicates that c12 = 0. Round your answer to 4 signiﬁcant digits)

(a) (4 Points) Find a portfolio that achieves this goal. (ﬁnd the weights in this portfolio)

(b) (3 Points) Find the mean and the variance of (the return of) the portfolio you found in part (a).

(c) (3 Points) Suppose the investor has $1000 to invest. Explain what he should do with his money and what is the expected amount that he will obtain after one year (note that by convention the return is calculated for one year).

Problem 5. (15 Points) Consider a market consisting of three securities with expected returns µ1 = 0.10, µ2 = 0.20, µ3 = 0.30, standard deviations of returns σ1 = 0.25, σ2 = 0.28, σ3 = 0.20, and the correlations between the returns ⇢12 = 0.30, ⇢23 = 0.00, ⇢13 = 0.15.

(a) (4 Points) Find the weights of the minimum variance portfolio; also calculate the ex- pected return and risk of the MVP. (Round to the nearest thousandth)

(b) (6 Points) An investor wants to lock the expected return of his/her portfolio to be at least 30%. Please ﬁnd the one with minimum risk among all such portfolios. (Round to the nearest thousandth)