This question asks you to show howmuch better off you are with this particular risk-free asset for a particular risk choice.

(a) In Formula 8.12 on page 237, we showed that this no-risk-free minimum-variance portfolio with an investment weight of 76.2% in H and 24.8% in I has a risk of about 8.90%.

(b) The reward of this no-risk-free-asset-available, minimum-variance portfolio is E(˜r) = 76.2% . 6% +

24.8% . 9% ≈ 6.8%.

(c) With a weight of 30% in H and 70% in I, the rates of return in the four scenarios for the tangency portfolio T are as follows:

In Scenario ♣: 0.3. (−6%) + 0.7 . (−12%) = −10.2%

In Scenario ♦: 0.3. (12%) + 0.7 . (18%) = +16.2%

In Scenario ♥: 0.3.

(0%) + 0.7 . (24%) = +16.8%

In Scenario ♠: 0.3. (18%) + 0.7 . (6%) = +9.6%

(These calculations will reappear later in Table 9.2 on page 290.)

(d) The reward of the tangency portfolio is E(˜rT) = (−10.2% + 16.2% + 16.8% + 9.6%)/4 = 8.1%.

(e) Its risk is Sdv(˜rT) = 
[(−18.3%)2 + (8.1%)2 + (8.7%)2 + (1.5%)2]/4 ≈ 10.94%.

(f) You want the expected rate of return of a portfolio that uses the risk-free asset and that has a risk of 10.94% (i.e., the same that the no-risk minimum-variance portfolio had). Solve 8.9% = wT . 10.94%
Sdv(˜r) = wT . Sdv(˜rT)
Therefore, wT ≈ 81.35%. In words, a portfolio of 81.35% in the tangency portfolio T and 18.65% in the risk-free asset F has the same risk of 10.94%.
(g) You now want to know the expected rate of return on the portfolio (wT, wF) = (81.35%, 18.65%):
E(˜r) ≈ 81.35% . 8.1% + 18.65% . 4% ≈ 7.33%
E(˜r) = wT . E(˜rT) + wF . rF You therefore would expect to receive a 7.33% − 6.71% ≈ 62 basis points higher expected rate of return if you have access to this risk-free rate.