Exercises

Question 1

As a pension fund manager, you have promised to make perpetual payments of €1mn per year to your beneficiaries. You can invest in the 3-year coupon bond mentioned above (assume duration 2 years) and in a zero-coupon bond (par value = €1000) with a time to maturity of 40 years. Yield to maturity is 4%. How much of each bond will you hold in your portfolio?

Answer 1

Perpetual payments of $1 million

Duration of Perpetuity = (1 + y)/y = 1.04 / 0.04 = 26 years

Equalise the Duration = 26 = (w * 2) + [(1 – w) * 40]

w = 0.3684 | (1-w) = 0.6316
36.84% in the 3-year coupon bond and 63.16% zero-coupon bond

In millions:
0.3684 * 26 million = 9.58 million in the 3-year coupon bond
0.6316 * 26 million = 16.42 million in the zero-coupon bond

Question 2

In the absence of arbitrage opportunities, expected excess returns of portfolios that are:

a. proportional to its standard deviation.
b. proportional to its beta coefficient.
c. proportional to its weight in the market portfolio.
d. inversely proportional to its standard deviation.

Answer 2

proportional to its beta coefficient.

Question 3

The term ‘Active share’ was introduced by Cremers and Petajisto. Please explain (in words) the concept of active sharing. In addition, answer the following two questions:

• Did funds with a high active share outperform or underperform the market?
• Which type of mutual funds contributed to the average underperformance of mutual funds as a group, according to Cremers and Petajisto?

Answer 3

Funds with a high active share make ‘active’ choices, meaning they pick just a handful of stocks within an industry. Funds with a high active share outperformed others. Underperformance was driven by index huggers and market timers

Question 4

Assume that the single index model holds and that you held the (well-diversified) market portfolio with a very large number of securities. If the standard deviation of your portfolio was 0.20 and σM was 0.10, the β of the portfolio would be approximately:

a. 1.5
b. 2
c. 1
d. 0.5

Answer 4

2

Question 5

XYZ just issued a 5-year bond on which the YTM is 8%. CDS are also traded on this stock. The spread for 5-year protection is currently 500bps. CDS contracts are settled on the basis of cash settlements. Risk-free investments are also possible, with a yield of 2.50% with a flat yield curve. Is an arbitrage strategy possible here? If yes, how do you set it up and what is the profit of the strategy?

Answer 5

Yes, there is an arbitrage opportunity.

Buy the 5-year bond and receive 8%, the CDS (insurance) costs 5%.

The bond with CDS provides a return of 8% - 5% = 3%

Borrow funds by shorting (selling) treasury bill => 3% – 2.5% = 0.5%

Profit from Arbitrage is 0.5%