## Monte Carlo simulation for real estate and infrastructure investments

Financial return on investment, which allows trade-offs between costs and benefits, always plays a role in Open Design problems. As mentioned in Chapter 4 of Open Design, a Collaborative Approach to Architecture, we recommend using discounted cash flow analysis with a preference for using the Internal Rate of Return (IRR) criterion over the Net Present Value (NPV) criterion. The net present value of a project is:

where:

Ci = cash flow (positive or negative) in year i; r = (yearly) cost of capital (as a fraction of that capital); m = life time of the project in years.

In general, the cost of capital r is closely linked with inflation. The internal rate of return is the discount rate which would give NPV = 0:

The internal rate of return can be considered to be made up of two parts:

1. Cost of capital r without risk allowance,

2. Profit p representing the reward for accepting the risk of the investment. In real estate financing, the cash flows can be characterised by:

• A large investment I, i.e. negative cash flow, at the start of the project;

• A yearly net exploitation result E, i.e. the difference between the yearly exploitation revenues and costs;

• A rest value V at the end of the project. In real estate investments V constitutes the selling price at the end of the lifetime.

The statistics of stored results | |

> u |
r |

1 3 |
/ |

ü E |
/ |

O | |

NPV or IRR |

Interpretation and assessment of total risk

Figure 2.1 Monte Carlo simulation for a real estate investment

In the real world, the variables I, r, E and V will never have the same values as assumed in the investment calculation. We can, however, estimate risk profiles for these variables, i.e. probability distributions for their occurrence. The probability distributions of the variables determining the financial return on the investment are the basis of Monte Carlo simulations as shown in Figure 2.1.

Instead of making one estimate for each variable that affects the return on investment, three estimates are made:

1. A pessimistic estimate, defined as having a probability of 10% that reality will be worse than that;

2. A best guess; in general, this value represents the outcome of a cost-benefits calculation;

3. An optimistic estimate, defined as having a probability of 10% that reality will be better than that.

These three points determine the probability distribution for the variable concerned. With these distribution curves, Monte Carlo simulation finally gives the probability distribution for the financial return.

The arithmetic of the Monte Carlo simulation is: whenever a risk variable enters into the calculation, a random number generated by the computer is corrected with the (skewed) distribution of the variable concerned. The calculation is done, say, 2 000 times. The resulting 2 000 different outcomes provide the probability distribution of the financial return on the investment.

This approach has two important advantages compared to conventional investment analysis based on single values:

1. It allows moderate return-low risk investments to be traded off against high return-high risk investments. The decision support information provided by the two different risk profiles is extremely relevant for an investor;

2. By asking experts a range instead of a single estimate, they tend to be genuine. When people are asked to give only one estimate, they tend to give their pessimistic guess without saying so.

An underlying assumption of Monte Carlo simulation is that the variables involved are stochastically independent.

The trade-off of a moderate return-low risk against a high return-high risk alternative often boils down to choosing between an inexpensive option 'X' and an expensive option 'Y'. For instance, should we spend extra money for a prestigious entrance and a large parking lot or should we keep the investment as low as possible? The former - the expensive option - is more risky in the sense that users may not be prepared to pay extra rent for the nice entrance and the parking area. There is also a reasonable chance, however, that they will be prepared to pay more for it than its (discounted) cost. In that case the financial return will be higher than for the inexpensive option (no prestigious entrance and only a limited parking lot).

The probability curves for the financial return of the inexpensive and the expensive options are typically as shown in Figure 4.1 of Open Design, a Collaborative Approach to Architecture (page 49). The probability that the return will be above a minimum threshold is as shown in Figure 4.2 of Open Design, a Collaborative Approach to Architecture (page 49).

If the investor prefers a moderate but sure return, he should choose option 'X'. Conversely, if he wishes to go for a more ambitious return and is willing to accept the associated higher risk, then he should choose option 'Y'. The difference between the two risk profiles is caused by the associated probability curves of the rent users will be prepared to pay. Since the variables investment I, cost of capital r, net exploitation revenues E, and rest value V as defined in the preceding section, are stochastically independent, a Monte Carlo simulation can be conducted based on ranges specified for these variables.

We have developed a software package for this purpose. The input consists of the lifetime m and the ranges (specified by three values) for the variables investment, cost of capital, yearly net exploitation E, and rest value V. The output gives the probability distribution for the profit p and a sensitivity analysis based on the best guess estimates. The internal rate of return (IRR) distribution is obtained by setting the input variable cost of capital r at zero (three times).

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