Measurement of Risk

Analysis of Risk & Uncertainty¶

#Description & Measurement of Risk
#Risk Evaluation Approaches
#Risk & Real Options

Description & Measurement of Risk¶

Risk¶

Variability in actual returns wrt \(E(r)\), in terms of cashflows¹.

Sensitivity Analysis (Absolute measure)¶

How sensitive estimated project parameters are to estimation errors
- Parameters: cash flows, cost of capital, economic life
- Estimates made under three assumptions: pessimistic, most likely, optimistic
Example 1: Calculate NPV of Project \(X,Y\) for each possible cash flow (worst, most-likely, best) using sensitivity analysis
- NPV determination
- conservative and risk-taking attitude towards risk

Assign probabilities¶

Drawback of sensitivity analysis: Doesn't disclose the chances of occurrence of these variations
Example 2: Find expected return on the project. PV of expected monetary values.

Scenario Analysis (Absolute measure)¶

Evaluate the impact on the project's profitability of simultaneous changes in more than one variable at a time (inflows, outflows, cost of capital)
- Ask operating manager (production, sales, personnel)
  - worst-case (high fixed costs, high VC, low Selling price, low sales volume, higher cost of capital) etc.
  - best-case - vice versa
- NPV guides the decision and helps assess the risk
- Highly unlikely (both the scenarios), but useful \(\implies\) If NPV positive in worst case, then project is worth accepting.
- Limited Usefulness: Considers few discrete outcomes, infinite number of possibilities exist

Simulation¶

get a feel of the risk (statistics based)
Apply predetermined probability distributions + random numbers \(\implies\) estimate risky outcomes

Precise Measures of Risk: Standard Deviation and CoV¶

Standard Deviation (Absolute measure)¶

\[\sigma = \sqrt{\sum_{i}^n P_{i} (CF_{i}- \bar{CF})^2 }\]

CoV (Relative measure)¶

\[V = \dfrac{\sigma}{\bar{CF}}\]

Example 4: Compare the risk of projects on the basis of SD and CoV

Risk Evaluation Approaches¶

1. Risk-adjusted Discount Rate Approach (RAD)¶

relatively risky \(\implies\) relatively high discount rates
Accept-reject Decision
- NPV and IRR approach
- -ve NPV \(\implies\) Reject
- internal rate of return, \(r \gt\) risk-adjusted rate \(\implies\) Accept
Example 5: Evaluate project (accept/reject)
Evaluation (+1 -3)
- + Simple to calculate
- - How to determine risk-adjusted discount rate?
- - Doesn't use direct info available distribution of expected future CF, it adjust \(r\) instead of CF
- - Risks compounds overtime, not theoretically desirable into practice. Discounting process should only consider time value and not risk.

2. Certainty-Equivalent Approach¶

We adjust the expected cash flow directly.
Risk-adjustment factor is expressed in terms of a certainty-equivalent coefficient \(a_{t}\)
- Based on firm's utility preference: \((12000) \sim (20000,0.6)\)²
- This coefficient \(a = 0.6\) when multiplied to a risky CF gives us a riskless CF
- \(a\propto \dfrac{1}{\text{Risk}}\)

Accept-Reject Rule

\[NPV = \sum_{i=1}^n \dfrac{a_{i}CFAT_{i}}{(1+i)^i} - CO\]

Illustration using Example 5
Evaluation (+2 -1)
- + simple to calculate
- + modifies CF that is subject to risk

3. Probability Distribution Approach¶

Dependent CF
Independent CF

\[NPV = \sum_{t=1}^n \dfrac{\bar{CF}_{t}}{(1+i)^t} - CO\]

\[\sigma(NPV) = \sqrt{ \sum_{t=1}^n \dfrac{\sigma_{t}^2}{(1+i)^{2t}}}\]

You are essentially finding \(Var\left(\dfrac{CF}{(1+i)^t}\right)\)
Where \(\sigma_{t} = \sqrt{ \sum (CF_{jt} - \bar{CF_{t}})^2 \cdot P_{jt}}\)

Problem Checklist

Example 6: Expected Cashflow and standard deviation
Normal distribution: (\(\leq 0, \gt 0, a \lt CF \lt b\))
Example 7: Find probabilities (i) \(\leq 0\); (ii) \(\gt 0\); (iii) \((25,45)\); (iv) \((15,30)\)
Example 8:
- (i) \(\mu = E(NPV)\);
- (ii) \(\sigma_{NPV}\) ;
- (iii) \(P(NPV\dots)\) :
  - (a) \(\leq 0\); (b) \(\gt 0\); (c) \(\geq \mu\)
- (iv) Profitability Index of \(\mu\);
- (v) \(P(PI < 1)\)

4. Decision-tree Approach¶

Problem Checklist¶

Example 9: Investment outlay with 9 distinct possibilities (shown by decision tree)

Notation¶

Decision is squared, chance is circled
Index like so \(D_{11}, D_{22}\) for Decision \(D_{1}\)

At \(D_{1}\), choose \(D_{11}\) and wait for outcome at \(C_{1}\)

Risk & Real Options¶

Types of Option¶

Growth¶

expand (demand > expectations)
open new doors, if successful \(\implies\) invest cash
embedded in capital budgeting projects

Abandonment¶

abandon/terminate/shutdown prior to its expected economic useful life
minimize firm's losses
projects with abandonment value, lower the project's risk by limiting downside losses and improve NPV
sell off some capacity and put to other use
variant: suspend (temporarily)
- mineral extractions
- extraction costs > selling price

Timing¶

optimal time postpone
accelerate or slow process of implementation
negative NPV today doesn't mean forever
traditional contrast, now or never

Flexibility¶

accept multiple inputs and…