CIMA P2 Syllabus D. Risk and Control - Value at Risk - Notes ^{12 / 13}

VaR is measured by using normal distribution theory.

Confidence levels are often set at either:

95% (The VaR here shows the potential loss that has only a 5% chance of decline) or
99% (The VaR considers a 1% chance of loss of value).

Illustration

Cow plc estimates the expected NPV of a project to be £100 million, with a standard deviation of £9.7 million.

Required:

Establish the value at risk using both a 95% and also a 99% confidence level.

Solution

• Using Z = (X - μ) / σ 

  where

  X = result we are considering
  μ = mean
  σ = standard deviation

• Establishing Z from the normal distribution tables 

  ie at a 95% (0.95) confidence level, 1.65 is the value for a one tailed 5% probability of decline (i.e. 0.95 - 0.50 = 0.45 = 0.4505 from the normal distribution table) 

  and at a 99% (0.99) confidence level, 2.33 is the value for a one tailed 1% probability of loss of NPV (i.e. 0.99 - 0.50 = 0.49 = 0.4901 from the normal distribution table).

• At 95% confidence level, Z = (X-100) / 9.7 = –1.65; 

  therefore X = (9.7x–1.65)+100 = 84

• At 99% confidence level,  Z = (X-100) / 9.7 = –2.33;

  therefore X = (9.7x–2.33)+100 = 77.4

• There is a 5% chance of the expected NPV falling to £84 million or less and a 1% probability of it falling to £77.4 million or below.
  

Value at risk can be quantified for a project using simulation to calculate the project’s standard deviation.

In this context, the standard deviation needs to be adjusted by multiplying by the square root of the time period ie

• 95% value at risk = 1.645 x standard deviation of project x √time period of the project

Illustration

A four-year project has an NPV of $2m and a standard deviation of $1m per annum.

Required

Analyse the project’s value at risk at a 95% confidence level.

• The VAR at 95% is 1.645 x 1,000,000 x √4 = $3,290,000 

  ie worst case NPV (only 5% chance of being worse) = $2m – $3.29m = – $1.29m

Illustration

A simulation has been used to calculate the expected value of a project and is deemed to be normally distributed with the following results:

Mean = $40,000 (positive)
Standard deviation = $21,000

Calculate the following:

a) The probability that the NPV of the project will be greater than 0.

b) The probability that the NPV will be greater than $45,000.

• a) 

  Using Z = (X - μ) / σ 

  μ = $40,000
  σ = $21,000
  X = 0

  Z = (0 - 40,000) / 21,000
  Z = 1.90

• From normal distribution table 

  1.90 = 0.4713 + 0.50 = 0.9713 = 97% probability that NPV >0

• b) 

  Using Z = (X - μ) / σ 

  Z = (45,000 - 40,000) / 21,000
  Z = 0.24

• From normal distribution table 

  0.24 = 0.0948

  then

  0.50 - 0.0948 = 0.4052 = 41% probability that the project's NPV > $45,000