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week3_calculation_practice

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IS4226 Week 3 Calculation Practice

Use decimal form in working unless stated otherwise. Try each question without looking at its topic label or solution. The formulas follow L3A and L3B; standard deviation is population SD when the question supplies the full population.

Questions

  1. A stock closes at 54. Find its simple return and log return.
  2. A stock has a log return of −0.12. Find its simple return. If it began at $75, find its ending price.
  3. Prices are 110, 108.90. Find each simple return, the arithmetic mean simple return, and the total holding-period return.
  4. An asset returns −8%, 5%, or 18% with probabilities 0.20, 0.50, and 0.30. Find expected return.
  5. A portfolio allocates 25% to A, 45% to B, and 30% to C. Their returns are 16%, 4%, and −6%. Find portfolio return.
  6. A portfolio allocates 35% to A returning 12% and the rest to B. Overall return is 5.5%. Find B’s return.
  7. The complete population of returns is −2%, 0%, 4%, and 6%. Find mean, variance, and population SD.
  8. Daily returns are normal with mean 0.15% and SD 1.1%. Give approximate 68%, 95%, and 99.7% daily-return bands.
  9. A $250 asset has an expected 10-day return of 0.5% and a 10-day SD of 3%. Give its approximate 95% return and price bands using the lecture’s simple price conversion.
  10. Daily mean is 0.03% and daily SD is 1.4%. Find 20-day mean and SD.
  11. An equity has daily mean 0.05% and daily SD 1.25%. Annualise both using 252 days.
  12. Crypto annualised volatility is 76.45%. Infer daily volatility using 365 days.
  13. Fund A returns 11%, Fund B returns 15%, and the risk-free rate is 3%. Their volatilities are 16% and 30%. Calculate both Sharpe ratios and select the better risk-adjusted performance.
  14. A portfolio has Sharpe ratio 0.6, return 11%, and risk-free rate 2%. Find its volatility.
  15. Covariance between an asset and market is 0.024; market variance is 0.016. Find beta. If market rises 2%, state the beta-only predicted asset move.
  16. A portfolio is 20% in beta 1.8, 50% in beta 0.9, and 30% in beta 0.2. Find portfolio beta.
  17. A portfolio targets beta 1.0. It holds 40% in beta 1.4 and 60% in asset B. Find B’s required beta.
  18. Risk-free rate is 2.5%, expected market return is 8.5%, and beta is 1.3. Find CAPM expected return.
  19. An asset actually returns 12.5%; risk-free rate is 2%; market return is 9%; beta is 1.2. Find alpha using full CAPM.
  20. Alpha is −1%, actual return is 6%, risk-free rate is 2%, and market return is 10%. Find beta.
  21. Two assets have covariance −0.0036 and SDs 6% and 10%. Find correlation and interpret it.
  22. Correlation is 0.25 and SDs are 12% and 20%. Find covariance.
  23. A stock goes from 52 over three days. Its first two log returns are 0.08 and 0.12. Find the third log return.
  24. A 5-day 68% return band is −2% to 4%. Infer the 5-day mean and SD, then infer daily mean and SD.

Worked solutions

  1. Simple return: (54/48-1=0.125=12.5%). Log return: (\ln(54/48)=\ln(1.125)=0.1173), or 11.73% in log-return units.

  2. (R=e^{-0.12}-1=-0.11308=-11.31%). Ending price: (75e^{-0.12}=$66.52) (equivalently (75(1-0.11308))).

  3. Simple returns: (110/100-1=10%); (99/110-1=-10%); (108.9/99-1=10%). Arithmetic mean = ((10-10+10)/3=3.333%). Total return = (108.9/100-1=8.9%). Trap: (3\times3.333%=10%) is not the compound return.

  4. (E[R]=0.2(-0.08)+0.5(0.05)+0.3(0.18)=-0.016+0.025+0.054=0.063=6.3%).

  5. (R_p=0.25(0.16)+0.45(0.04)+0.30(-0.06)=0.04+0.018-0.018=0.04=4%).

  6. (0.055=0.35(0.12)+0.65R_B). Thus (R_B=(0.055-0.042)/0.65=0.02=2%).

  7. Mean: ((-2+0+4+6)/4=2%). Squared deviations in percentage points: 16, 4, 4, 16; variance = (40/4=10) percentage-points², or 0.001 in decimal-return units. SD = (\sqrt{10}=3.162%).

  8. 68%: (0.15%\pm1.1%=[-0.95%,1.25%]). 95%: (0.15%\pm2.2%=[-2.05%,2.35%]). 99.7%: (0.15%\pm3.3%=[-3.15%,3.45%]).

  9. 95% return band: (0.5%\pm2(3%)=[-5.5%,6.5%]). Prices: (250(0.945)=236.25\) to \(250(1.065)=266.25).

  10. (\mu_{20}=20(0.03%)=0.6%). (\sigma_{20}=\sqrt{20}(1.4%)=6.261%).

  11. Annual mean: (252(0.05%)=12.6%). Annual SD: (\sqrt{252}(1.25%)=19.84%).

  12. (\sigma_d=76.45%/\sqrt{365}=4.00%) (approximately).

  13. A: ((11-3)/16=0.50). B: ((15-3)/30=0.40). Fund A has better risk-adjusted performance despite the lower raw return.

  14. (0.6=(0.11-0.02)/\sigma), so (\sigma=0.09/0.6=0.15=15%).

  15. (\beta=0.024/0.016=1.5). A beta-only prediction for a 2% market rise is (1.5(2%)=3%). This is a sensitivity estimate, not a guaranteed realised return.

  16. (\beta_p=0.2(1.8)+0.5(0.9)+0.3(0.2)=0.36+0.45+0.06=0.87).

  17. (1.0=0.4(1.4)+0.6\beta_B). Therefore (\beta_B=(1-0.56)/0.6=0.7333).

  18. Market risk premium = (8.5%-2.5%=6%). CAPM return = (2.5%+1.3(6%)=10.3%).

  19. CAPM benchmark = (2%+1.2(9%-2%)=10.4%). Alpha = (12.5%-10.4%=2.1%).

  20. (-1%=6%-[2%+\beta(10%-2%)]). Hence (7%=2%+8%\beta), so (\beta=5/8=0.625).

  21. (\rho=-0.0036/(0.06\times0.10)=-0.6). The assets have a moderately strong negative linear relationship, which may provide diversification benefit.

  22. (\operatorname{Cov}=\rho\sigma_x\sigma_y=0.25(0.12)(0.20)=0.006).

  23. Total log return = (\ln(52/40)=\ln(1.3)=0.262364). Third log return = (0.262364-0.08-0.12=0.062364). Its corresponding simple return is (e^{0.062364}-1\approx6.44%).

  24. The midpoint gives (\mu_5=1%); half-width gives (\sigma_5=3%). Daily mean: (\mu_d=1%/5=0.2%). Daily SD: (\sigma_d=3%/\sqrt5=1.342%).

Formula selection checklist

  • Prices at two times → simple or log return.
  • Mutually exclusive states with probabilities → expected return.
  • Portfolio allocations → weighted return or weighted beta.
  • Raw observations → mean, squared deviations, variance, SD.
  • Time-horizon conversion → mean × (N), volatility × (\sqrt N).
  • Excess return per unit of total volatility → Sharpe ratio.
  • Covariance with the market divided by market variance → beta.
  • Required/benchmark return from market risk → CAPM.
  • Actual return minus CAPM benchmark → alpha.
  • Standardised co-movement → correlation.

Assumption and error checklist

  • Convert percentages to decimals consistently.
  • Do not add simple returns across time; compound them or add log returns.
  • Do not call arithmetic mean return the compound growth rate.
  • Use (N), not (N-1), when the question follows the lecture’s population-SD formula.
  • Scale variance by (N), so SD scales by (\sqrt N), under the independence assumption.
  • Use 252 for equities and 365 for crypto unless the question specifies otherwise.
  • Subtract (R_f) in Sharpe and full CAPM alpha unless it is explicitly zero.
  • Beta measures benchmark sensitivity/systematic risk, not total volatility.
  • Correlation is unitless and bounded by −1 and +1; covariance is not.
  • Correlation does not establish causation.

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