B1. [3+3=6 marks] You have $100k to invest. Among risky assets, there are two funds that you can choose from. However, you are allowed to choose only one of the two funds. After you select a fund, you are allowed to divide your $100k in any way you would like across risk- free bonds and that fund.

The expected return and standard deviation of each fund is given in the following table:

Zero-coupon Commonwealth Government securities maturing in one year are currently priced at $97.561 per $100 par value. Your utility function is given by

U = E[r] – 2σ2

a) Assuming you can costlessly short sell both the risk-free asset and whatever fund you choose, what is your optimal portfolio? That is, which fund do you select, how much of your $100k do you invest in that fund, and how much money do you invest in the risk-free asset? Show your calculations. [3 Marks]

b) Now suppose that you cannot short sell either fund or the risk-free asset. What is your optimal portfolio in this case? [3 Marks]

B2.  [4 marks] Suppose you believe markets are efficient and that the single-index model properly describes security returns. You estimate that

•    On average, individual securities have idiosyncratic risk of 20% (measured as standard deviation)

•    A broad-based, equally-weighted market index has an expected return of 8% and a standard deviation of 25%

If the risk-free rate is 2%, what is your prediction of the Sharpe Ratio of a portfolio equally weighted in 10 randomly selected stocks? Clearly specify any assumptions you make.

B3. [4 Marks] According to the CAPM, expected return increases as the beta of the investor’s portfolio increases. Given that, evaluate the following statement as true or false:

“If investors have standard mean-variance preferences (U = E[r] - ½Aσ2) with A>0, CAPM investors will maximise utility by borrowing as much as possible at the risk-

free rate and investing the money in the market portfolio.”

You must explain the rationale for your answer for full credit.

B4. [1+3=4 marks] You are using a two-factor APT model to find the “fair” expected return on a well-diversified portfolio P that promises an “actual” expected return of 18%. The relevant factor portfolios, their betas in portfolio P, and their factor risk premia are shown in the table below. The risk-free rate is 2.5%.

a) According to the APT, what is the “fair” expected return on the portfolio P? [1 mark]

b) Suppose there exists an additional portfolio Q, which has a beta of 1 on each factor and an expected return of 15.875%. Assuming the risk premium on Factor A is estimated correctly, what must the risk premium on Factor B be to exclude arbitrage opportunities between P and Q? [3 marks]

B5. [5 marks] For this question, use the trading calendar on page 7 to assist with dates.

At 11 am in Sydney on 20 April 2022, you agreed to purchase $10,000 of face value of an Australian government bond paying coupons of 2% p.a. and maturing on 21 October 2023. The bond pays coupons semi-annually.

If the yield to maturity is 1.5%, what is the settlement price?

B6. [4 marks] A coupon bond can be thought of as a portfolio of zero-coupon bonds, so one way to price a coupon bond is to use the zero-rate curve. Suppose, for example, you wanted to price a 3-year bond paying 5% coupons annually. You could break up the bond into pieces – the cash flows paid in year 1, year 2, and year 3 – then calculate each piece’s present value and sum them to determine the price of the bond. You could then use this price to back out the bond’s yield to maturity.

Given the zero-rate curve below, what is yield to maturity on a 3-year bond paying 5% coupons annually? Report your answer in percent to two decimal places (e.g., “1.23%”). Assume all bonds have no default risk, and you may ignore technical issues like settlement dates, etc.

B7. [5 marks] You observe the following rates:

Supposing you could borrow (short sell) up to $100 million, derive the most profitable arbitrage that generates positive cash flows today and zero net cash flows in the future. Be concise in your answer. E.g., “At time t, Buy $x of Bond X, sell $y of Bond Y, etc..”

B8. [6 Marks] An investment manager wishes to establish a fund that will be worth $80 million in 4 years’ time. She plans to follow a duration-matching approach using a 3-year zero-coupon bond and a 5-year bond paying a coupon rate of 12% pa. The 5-year bond pays coupons annually. Both bonds are currently priced to yield 6% pa.

How much should she invest in the 3-year bond? How much should she invest in the 5-year bond?

B9. [2+2=4 Marks] Consider the following cash flows for Forest Financial’s Large Cap B mutual fund (all figures in million AUD):

For simplicity, assume all cash flows occur at the end of the year.

a) In % pa to two decimal places, what was the time-weighted return (TWR)?

b) In % pa to two decimal places, what was the dollar-weighted return (DWR)?