# Hillier Model

**Definition:** According to the **Hillier model**, the risk associated with the project can be assessed through the standard deviation of expected cash flows. In other words, determining the viability of the project through calculating the deviations in the cash flows from the mean of expected cash flows.

Thus, Hillier model asserts that the computation of standard deviations of several ranges of cash flows enables a firm to determine the uncertainty involved in the future projects.

This model was proposed by F.S. Hillier and according to him, the expected Net Present Value and the standard deviation of the Net present value of the project can be determined through analytical derivations. Under this model, there are two cases of analysis:

- When there is no correlation among the cash flows
- When there is a perfect correlation among the cash flows.

When the cash flows of different years are uncorrelated, then the cash flow in the year “t” is independent of the cash flow in the year “t-n”. Whereas, if the cash flows of different years are perfectly correlated, then the cash flows in each period will be alike.

The formula to compute the Net present Value and the standard deviation under both the cases is given below:

*Uncorrelated Cash Flows*

**NPV = ^{n}∑_{t=1 }[C_{t} / (1+i)^{t}] – I**

**∂ (NPV) = ^{n}∑_{t=1 }[∂_{t}^{2}/ (1+i)^{2t}]^{1/2}**

*Correlated Cash Flows*

**NPV = ^{n}∑_{t=1 }[C_{t} / (1+i)^{t}] – I**

**∂(NPV) = ^{n}∑_{t=1}[ ∂_{t}/ (1+t)^{t}]**

Where, **C _{t }**

_{= }Expected cash flow of the year “t”

**∂**= standard deviation of cash flow for the year “t”

_{t}**i**= risk free rate

**I**= initial investment

## 4 Comments

Can you tell me what will be the effect if we use i = risk adjusted rate ?

We normally use the risk-free rate to discount the future cash flows principally to quantify the project risk, evaluate it and then decide on a risk-adjusted discount rate. The risk-adjusted rate is used to further discount the cash flows in case the profile of the investment project is highly risky.

The same formula, as mentioned in the content, will be used to compute the value of NPV, just in the place of a risk-free rate the value of risk-adjusted discount rate will be used. The risk-adjusted discount rate is the sum of the risk-free rate and the risk premium. Symbolically,

Risk-adjusted discount rate = risk-free rate + risk premium

The risk premium can be calculated by using the CAPM method:

Risk premium = β (rm – rf)

rm= market risk

rf=risk free rate

β= risk of the project

The amount of risk premium depends on the investor’s level of risk aversion and his perception of the risk associated with the investment. when the investment risk is high, a high risk-adjusted discount rate is used and vice-versa.

Effect: There is an inverse relationship between the value of NPV and the risk adjusted discount rate, which means, if the adjusted rate increases the value of NPV decreases,thereby making the project a riskier one.

can you give me the examples of real life project having uncorrelated cash flow??

In the independent formula why t is multiply with 2?