The Hillier Model provides a quantitative approach to assess project risk in capital budgeting by calculating the standard deviation of expected cash flows. Developed by F.S. Hillier, this method considers the correlation between cash flows across years and enables more precise evaluation of a project’s uncertainty. This document analyzes the model, its mathematical foundations, practical interpretation, and the relationship between risk-free and risk-adjusted discount rates, crucial for corporate finance decision-making.
1. Introduction
Capital budgeting decisions require not only expected profitability measurement but also a rigorous quantification of risk. The Hillier Model addresses this need by employing statistical measures—notably standard deviation—to evaluate uncertainty in future cash flows.
2. Hillier Model Overview
According to the Hillier Model, project risk is measured by the standard deviation of its expected cash flows. By examining deviations of cash flows from their mean, the model evaluates how uncertain the project’s outcome may be.
Key assumptions include:
• Cash flows can be either uncorrelated (independent) or perfectly correlated across years.
• The model provides specific formulas for both cases to compute expected Net Present Value (NPV) and the standard deviation of NPV.
3. Mathematical Formulation
3.1 Notations
• Ct: Expected cash flow in year t
• σt: Standard deviation of cash flow in year t
• i: Discount rate (usually risk-free rate)
• I: Initial investment
• n: Total number of years
3.2 Case 1: Uncorrelated Cash Flows
Here, cash flows in different years are statistically independent. The formulas are:
NPV = Σ [Ct / (1 + i)^t] − I
σ(NPV) = √ Σ [σt² / (1 + i)^(2t)]
3.3 Case 2: Perfectly Correlated Cash Flows
If cash flows move identically over all years, then:
NPV = Σ [Ct / (1 + i)^t] − I
σ(NPV) = Σ [σt / (1 + i)^t]
4. Role of Discount Rate
Typically, the discount rate i is considered as the risk-free rate while assessing the project. This approach isolates risk measurement via the standard deviation.
However, if a risk-adjusted discount rate is used (i.e., risk-free rate + risk premium), it impacts the valuation as follows:
• The NPV decreases as the risk-adjusted discount rate increases, due to the inverse relationship.
• This reflects higher demanded compensation for risky projects, often calculated via Capital Asset Pricing Model (CAPM):
Risk-adjusted rate = rf + β (rm − rf)
Where:
rf = Risk-free rate
rm = Expected market return
β = Project beta (risk measure)
5. Practical Interpretation and Use Cases
• The Hillier Model helps differentiate between projects with similar expected NPVs but different risk profiles.
• It is especially useful in industries with varying cash flow correlation patterns.
• Examples of uncorrelated cash flows: Start-ups with varying yearly outcomes independent of past performance; projects facing highly variable market-driven revenues annually.
6. Limitations and Considerations
• The assumption of independence or perfect correlation is a simplification; real cash flows may exhibit partial correlations.
• More sophisticated methods (e.g., Monte Carlo simulation or decision tree analysis) might be needed if cash flows have complex dependencies.
• The model requires reliable estimates of the variance (or standard deviation) of yearly cash flows—a challenge in uncertain environments.
7. Conclusion
Hillier’s Model enriches capital budgeting by quantifying risk through standard deviation of discounted cash flows, aiding more informed investment decisions. The flexibility to analyze correlated and uncorrelated cash flows makes it a versatile tool in project evaluation. Combining this risk measurement with appropriate discount rates supports achieving an optimal risk-return balance.
Related Posts:
