Question

A portfolio of $ 100,000 is composed of two assets: A stock whose expected annual return is 10% with an annual standard deviation of 20%; A bond whose expected annual return is 5% with an annual standard deviation of 12%. The coefficient of correlation between their returns is 0.3. An investor puts 60% in the stock and 40% in bonds.

What is the expected annual return, standard deviation of the portfolio

What is the 1-year 95% VaR? Explain in non-technical terms the meaning of the number you calculated.

What is the 1-year 99% VaR? Explain in non-technical terms the meaning of the number you calculated

Discuss the weaknesses of Value-at-Risk as a measure of risk.

Answer 1

A) Expected return of portfolio= weight of stock × return of stock + weight of bond × return of bond

= 0.6 × 10% + 0.4 × 5%

= 6% + 2%

= 8%

Standard deviation of portfolio

= √ (weight of stock )^2 (std deviation of stock)^2 + (weight of bond)^2 (std deviation of bond)^2 + 2 × weight of stock × weight of bond × std deviation of stock × std deviation of bond × correlation

= √ (0.6)^2 (0.20)^2 + (0.4)^2 (0.12)^2 + 2 × 0.6 × 0.4 × 0.20 × 0.12 × 0.3

= √ (0.36) ( 0.04) + (0.16) (0.0144) + 0.003456

= √ 0.0144 + 0.002304 + 0.003456

= √ 0.02016

= 14.20%

B) 95% VAR

VAR= standard deviation × z score

= 14.20% × -1.645

= -23.359%

Over a year, the portfolio could lose 23.359% of 100,000 or $23,359. This means that the portfolio has 95% chance that more than this amount is lost and 5% chance that less than this will be lost.

C) 99% Var

VAR= standard deviation × z score

= 14.20% × -2.33

= -33.086%

Over a year, the portfolio could lose 33.086% of 100,000 or $33,086 . This means that portfolio has 99% chance that more than this amount is lost and 1% chance that less than this will be lost.

D) The weakness of Value -at-risk

i) different methods of value of risk gives different result.

ii) It does not provide with worst case loss

iii) VAR is only as goods as its assumptions. If the assumptions go wrong, then the VAR estimate will also be wrong

iv) Value at risk is not additive.