14.3.1 Sensitivity analysis

Where uncertain market conditions or other variable factors could have a material impact on the valuation, it may be prudent to provide a sensitivity analysis to illustrate the effect that changes to these variables could have on the reported valuation. Sensitivity analysis investigates the impact of uncertainty on key input variables by examining the degree of change in the valuation caused by a change in one or more of the key input variables. Usually, a change of 5 or 10% either side of the expected values of the key variables is tested to measure the effect on value. A more sophisticated analysis may apply more realistic variations to the key variables, for example, more upside variation in rent in a rising market. Or different positive and negative percentage changes may be applied depending on the variable, for example plus or minus 10% for rental value and plus or minus 2% for rental growth.

Sensitivity analysis does not consider the likelihood of outcomes and the input variables are usually altered one at a time. Although simplistic, the process does require the valuer to think about the realistic limits on shifts in the input variables and produces a range of valuations within which the actual price would be expected to fall.

Sensitivity analysis should be accompanied by a narrative describing the cause and nature of uncertainty. Analysis does not mean forecasting worse‐case scenario. It should address the impact of reasonable and likely alternative assumptions. There is a need to consider interdependence or correlation between significant inputs, otherwise the degree of uncertainty may be over‐estimated.

Univariate sensitivity analysis seeks to quantify the effect of changes in the values of certain input variables on the output variable, one variable at a time. Bivariate sensitivity analysis extends univariate analysis by examining the impact of changes to two variables at the same time. For example, Table 14.1 shows a sensitivity analysis of the effect on the land value when key input variables are altered.

Bivariate analysis does not take account of any possible correlation between the input variables; instead, they are assumed to move independently. But logic suggests that as rents rise, yields should fall and vice versa. Some of the output is repeated from the univariate sensitivity analysis but bivariate analysis provides more information about what happens when changes coincide, such as an increase in yield and a drop in rent.

A variation of sensitivity analysis is break‐even analysis. This is the recalculation of a valuation in which the output is set to zero by altering key inputs. Table 14.2 illustrates a simple example from a development valuation where the developer's profit is set to zero by altering each of the listed inputs one at a time.

Sensitivity analysis models uncertainty in a very simplistic way but it does encourage the valuer to think about how assumptions and point estimates of key input variables might vary.

14.3.2 Scenario modelling

Scenario modelling examines the value impact of changes in several inputs at the same time. The valuer constructs several scenarios that reflect different possible futures, perhaps corresponding to optimistic, realistic and pessimistic circumstances, and then examines the impact on value of each scenario. The difference between sensitivity analysis and scenario testing is that the latter examines the impact on value of simultaneous changes to several variables and therefore begins to give a more realistic representation of how the key variables might respond to economic changes. It creates specific pictures (scenarios) of the future as a means of reflecting uncertainty.

Extending the example in Table 14.2, scenarios for combinations of values of rent, yield and building costs can be created. Numerous scenarios using different combinations of values can be constructed, but it is perhaps better to think carefully about practical combinations of values rather than try and input every permutation. Table 14.3 reports land value under three scenarios: realistic, best and worst case.

Scenario modelling allows the valuer to ‘bookend’ the valuation, but it still does not give any idea of the likelihood that any of these discrete outcomes might occur. To do that, we need to assign some measure of probability or likelihood to each scenario. For example, assume 40% probability for realistic outcome, 30% for the worst and 30% for the best outcomes. The mean or expected outcome is calculated as follows:

Neither the distribution of valuations nor the probabilities themselves need be symmetrical about the middle or realistic valuation.

A drawback of this type of analysis is a lack of market evidence on which to base selection of probabilities, but the process does focus the mind on the likelihood of achieving predicted returns. For example, a prime shop property and an old factory may yield the same return but how likely is the latter to be achieved relative to the former? In other words, how volatile or uncertain is the return? Discrete probability modelling does not properly reflect the uncertainty or risk that might be associated with the expected cash flows – it calculates an expected value rather than a measure of variation or uncertainty. To illustrate what this means, consider two property investments, Property 1 and Property 2, in Table 14.4.

The weighted average valuations are identical and, at first glance, the most probable outcome for Property 2 is £3 500 000 compared to £3 125 000 for Property 1, but closer inspection reveals that the range (volatility) of valuations for Property 1 is £500 000 and for Property 2 it is £4 680 000 and with a 5% probability of making a loss. Property 1 is likely to be more attractive to a risk‐averse investor.

If a valuer can reasonably foresee different values arising under different circumstances, another approach would be to provide alternative valuations based on special assumptions reflecting those different circumstances, but only if they are realistic, relevant and valid in connection with the circumstances of the valuation (RICS 2011: VS 2.2).

This is a slight improvement on scenario modelling, but input variables are individually probabilistic so probabilities for specific combinations may be unrealistic. It is also important to consider the entire distribution of future possibilities and recognise that input values are uncertain, not discretely but continuously.

14.3.3 Simulation

It is unrealistic to assume a small number of discrete possible valuation outcomes. There is likely to be a range of outcomes and this would be best represented by a continuous probability distribution. If the probability distributions for predicted valuation outcomes for Properties 1 and 2 in Table 14.4 are assumed to be ‘normally distributed’ around their mean values, Property 1 would have a narrower, more peaked distribution indicating lower volatility whereas Property 2 would have a flatter, wider distribution indicating higher volatility. Standard deviation measures this volatility; the smaller the standard deviation of a distribution, the less volatile it is.

For example, assume 50 valuers have been asked to value Properties 1 and 2 and the mean valuation for Property 1 was £3 200 000 with a standard deviation of £500 000 and for Property 2 the mean valuation was £3 500 000 but with a much higher standard deviation of £1 000 000. The ‘coefficient of variation’ measures standard deviation relative to the mean and is a useful measure of volatility because allows comparison of valuations whose mean values are not equal. It is calculated by dividing the standard deviation by the mean. The coefficient of variation for Property 1 is 15.63% and for Property 2 it is 28.57%. Property 1 is less volatile by both standard deviation and coefficient of variation measures.

The example above shows how valuations from many valuers can be represented by a continuous probability distribution. However, usually, it is not the valuation output that is the focus of risk analysis, it is the inputs into the valuation itself. A technique known as simulation refers to the modelling of probability distributions for input variables. Simulation enables valuers to assign probabilities to input variables in the valuation and run simulations of likely combinations of values of these inputs to produce a probability distribution and associated confidence range for the output valuation. Statistics that summarise the uncertainty surrounding the valuation output can then be calculated. Usually, these would be the mean valuation and a measure of dispersion, such as the standard deviation.

Simulation involves a series of steps:

1. Build a valuation model and identify key variables

   The valuation would be undertaken using the best estimates of the input variables. These input variables can be classified as either deterministic, which can be predicted with a high degree of certainty, or stochastic, which cannot be predicted with a high degree of certainty. Generally, the stochastic variables that have a significant impact on the valuation are the ones on which simulation is likely to be run. Deterministic variables might include the rent review period, purchase costs and management costs. Key stochastic variables will include the yield, the market rent, lease‐break and lease‐renewal options, including void periods and associated costs.

2. Ascribe a probability distribution for each key stochastic input variable

   Each stochastic variable needs to be represented as a probability distribution rather than a point estimate. A probability distribution is a way of presenting the quantified risk for the variable. Ideally the estimation of probability distributions would be based on empirical evidence, but often data are not available in a sufficient quantity to allow this. A pragmatic alternative is to gather opinions of possible values of each variable, along with their probability of occurrence, from experts. These expert opinions could then be used to select an appropriate probability function, of which there are many. The probability functions that are typically chosen are the continuous ‘normal’ distribution (in which case a mean and standard deviation would need to be specified) and the closed ‘triangular’ distribution (in which case the mode, minimum and maximum values would need to be specified). A useful characteristic of the triangular distribution is that, unlike the normal distribution, symmetry does not have to be assumed; the maximum and minimum values do not have to be equally spaced each side of the mode. In this way, the triangular distribution might offer a more realistic representation than the normal distribution if more upside or downside risk is expected.

   The input variables may also be independent or dependent. An independent variable is unaffected by any other variable in the model whereas a dependent variable is determined in full or in part by one or more other variables in the model. Different degrees of interdependence can significantly affect the simulation result. It is therefore necessary to specify the extent to which the input variables are correlated.

3. Run simulation

   Having selected the key variables and their probability distributions, a simulation run can begin. A run refers to the process whereby a distribution of valuation outcomes is generated by recalculating the valuation many times, each time using different randomly sampled combinations of values from within the parameters of the probability distributions of the key stochastic variables. In this way, the process selects input values in accordance with their probabilities (values are more likely to be selected from areas of the distribution that have higher probabilities of occurrence) and correlations (more likely combinations of values will be selected). Havard (2002) illustrates how this process works in the case of two variables, rental growth rate and exit yield, to which discrete probabilities have been assigned, as shown in Table 14.5.

   The simulation program randomly selects from the cumulative probability distribution for each variable. If we assume 22 was randomly selected for rental growth and 67 for the exit yield, this would equate to 3% rental growth rate and an exit yield of 9.25%. These sample values are then input into a run of the valuation model.

   In other words, because some values of key variables will have a greater probability of being achieved than others, the sample‐selection procedure ensures that they appear more frequently. This simulation process determines the range and probability of the valuation outcome.

   Annual rental growth rate			Exit yield
   %	Probability	Cumulative probability	%	Probability	Cumulative probability
   0	2	1–2	7.75	1	1
   1	5	3–7	8.00	4	2–5
   2	7	8–15	8.25	7	6–12
   3	10	16–25	8.50	10	13–22
   4	15	26–40	8.75	15	23–37
   5	21	41–61	9.00	21	38–58
   6	15	62–76	9.25	15	59–73
   7	10	77–86	9.50	10	74–83
   8	7	87–93	9.75	7	84–90
   9	5	94–98	10.00	5	91–95
   10	2	99–100	10.25	5	96–100

4. Output

   When setting up the simulation program, the output variable in the valuation model would have been specified and, invariably, this will be the valuation figure. The simulation results will provide information about the distribution of the valuation, including its central tendency (mean, median, mode), spread (range, standard deviation) and measures of symmetry (skewness) and peakedness (kurtosis). Regression analysis can also be undertaken to rank the input variables in terms of their impact on the output valuation.

For example, consider a freehold property investment that is let on a lease that has a break clause in two years' time. The rent passing is £200 000 per annum. Rather than a point estimate, the market rent is modelled as a stochastic variable that has a normal distribution with a mean value of £210 000 per annum and a standard deviation of £10 000. In addition, the maximum rent is considered to be £225 000 and a minimum of £190 000 per annum. It is not known whether the tenant will exercise the break option but if it is exercised, then there is likely to be a void period. Again, this is modelled as a stochastic variable but this time as a triangular distribution with a mode of one year, a maximum of two years and a minimum of six months.

It is also possible, and advisable, to consider correlations between these stochastic variables. This ensures that when combinations of input values are selected, they do so in accordance with their correlation. For example, if the two variables were market rent and yield, a negative correlation would be expected between these two variables, because a market upturn might be expected to stimulate demand in both the occupier market, thus raising rents, and the investor market, thus lowering yields. In the valuations below, correlations are not considered necessary between the two relatively unrelated variables of market rent and break option exercise. Therefore, the inputs for this valuation are as follows:

The market rent of £223 077 and the void period of 1.53 years are selected at random from within the parameters set by their probability distributions. Another run of the valuation would generate different values for these two inputs. So, using these inputs, the two valuations, one assuming the break option is not exercised and the other assuming that it is, would be as follows:

To handle the uncertainty as to whether the break option would be exercised or not, market evidence is examined to see how many tenants on average do exercise a break. Assume the result is 30%, so this percentage is used to weight the two valuations above and this produces a weighted average valuation of £4 094 666, i.e. (0.7 × £4 148 948) + (0.3 × £3 968 007). However, this valuation does not reflect the uncertainty surrounding the market rent and void period. To do this, the valuations must be run many times.

One thousand iterations of the valuations were run and the summary statistics for the output weighted average valuation are shown in Table 14.6. The optimistic skew of the exit‐yield distribution has increased the mean valuation of both properties approximately £15 000 above the original point estimates. In both cases, the standard deviation around the mean was just under £100 000. Figure 14.1 and the skewness value in Table 14.6 reveal that both output distributions are positively skewed, the property let under standard lease terms slightly more so. This is because the exit yield, which is itself positively skewed, explains more of the variation in value of the standard let investment.

It should be noted that specifying the forms of probability distributions for the inputs and their correlations is a challenge. The problem with this sort of analysis is being unable to confidently predict distributions and correlations of input variables. Statistical confidence requires sample sizes that are significantly larger than the typical pool of comparable evidence available when valuing a property. More research is needed to confidently base the choice of probability distributions and selection of co‐relationships between variables on empirical evidence.

1. A development site has planning consent for 2000 square metres (gross internal area) of office space with an efficiency ratio of 85%. Research indicates a build cost of £969 per square metre. An estimated £120 000 should cover external works (highways, landscaping, car parking). Professional fees are estimated at 13% of building and external works. Miscellaneous costs are assumed to be in the order of £80 000 and a figure of 3% of all these costs is assumed for contingencies. Comparable evidence suggests a rent of £200 per square metre and an investment yield of 7%. Disposal costs are 5.75% of NDV and site acquisition costs are 5.75% of the acquisition price. Site preparation costs are estimated to be £25 000. Fees for a full planning application are currently £2.50 per square metre of gross internal area. Building regulation fees are £20 000. The lender's legal fees, loan arrangement fee and developer's legal fees total £95 238. The letting agent's fee is 10% of the first year's rent, the letting legal fee is 5% of the first year's rent and the marketing fee is estimated to be £10 000. A lead‐in period of six months is considered appropriate, and finance has been secured at 10% per annum. Comparable evidence from developments schemes at Bristol Business Park indicates a building period of 15 months and a void period of 3 months can be assumed. Assume a developer's profit requirement of 20% of all costs.
   1. Calculate the value the site.
   2. Calculate the amount of developer's profit.
   3. Calculate the percentage return on NDV and development yield (rent as a percentage of total costs).
   4. Calculate the payback period, rent cover, interest cover and rent:debt ratio. Also, calculate the rent that would be required if the land was purchased for £1 m and all other revenue and cost inputs remained the same.
   5. Conduct the following risk analyses:
      1. Sensitivity analysis: Model the effect on developer's profit of 5 and 10% shifts in rent and 5 and 10% shifts in yield.
      2. Scenario modelling: Construct a pessimistic scenario (5% upward shift in yield, 5% downward shift in rent and six‐month void period) and an optimistic scenario (5% upward shift in rent, a yield of 6.75% and no void) and report effect on developer's profit.
2. The tables below provide an outline of revenues and costs for a mixed‐use development, together with the internal rate of return (IRR) and other performance metrics.

   Inputs
   Revenues	£/m2	NIA (m2 )
   Residential – market dwellings (price, area)	2000	5000		£9 756 098
   Residential – affordable dwellings (price, area)	1000	1000		£975 610
   Commercial space (rent, area, yield)	200	4000	5.00%	£15 609 756
   Other revenue				£3 000 000
   Costs
   Site acquisition price, including acquisition costs		£5 281 642	6.80%	£5 640 793
   Site preparation, infrastructure, utilities				£1 000 000
   Residential – market dwellings (£/m2, gross: net)	1000	80%		£6 250 000
   Residential – affordable dwellings (£/m2, gross: net)	1000	80%		£1 250 000
   Commercial space	800	85%		£3 764 706
   Abnormal costs				£200 000
   Professional fees (% total construction costs)			10.00%	£1 146 471
   Contingency (% total construction costs)			3.00%	£343 941
   Planning fees				£20 000
   Building control, NHBC, etc.				£50 000
   S106 planning obligations				£500 000
   CIL				£500 000
   Other fees (e.g. legal, loan, valuation)				£200 000
   Marketing				£300 000
   Other assumptions
   Site acquisition costs (% acquisition price)	6.80%
   Sale transaction costs (% sale price)	2.50%
   Letting transaction costs (% annual rent)	15.00%

   Cash Flow		0	1	2	3	4	5	6	7	8
   Revenue
   Market dwellings	9 756 098	—	—	—	—	—	—	—	4 878 049	4 878 049
   Affordable dwellings	975 610	—	—	—	—	—	—	—	487 805	487 805
   Commercial space	15 609 756	—	—	—	—	—	—	—	7 804 878	7 804 878
   Other revenue	3 000 000	—	—	—	—	—	—	—	—	3 000 000
   Total revenue (development value)	29 341 463	—	—	—	—	—	—	—	13 170 732	16 170 732
   Costs
   Site acquisition	5 640 793	(5 640 793)	—	—	—	—	—	—	—	—
   Site preparation	1 000 000	(1 000 000)	—	—	—	—	—	—	—	—
   Market dwellings	6 250 000	—	(312 500)	(312 500)	(625 000)	(1 250 000)	(1 875 000)	(1 250 000)	(625 000)	—
   Affordable dwellings	1 250 000	—	(62 500)	(62 500)	(125 000)	(250 000)	(375 000)	(250 000)	(125 000)	—
   Commercial space	3 764 706	—	(188 235)	(188 235)	(376 471)	(752 941)	(1 129 412)	(752 941)	(376 471)	—
   Abnormal costs	200 000	—	(10 000)	(10 000)	(20 000)	(40 000)	(60 000)	(40 000)	(20 000)	—
   Professional fees	1 146 471	—	(57 324)	(57 324)	(114 647)	(229 294)	(343 941)	(229 294)	(114 647)	—
   Contingency	343 941	—	(17 197)	(17 197)	(34 394)	(68 788)	(103 182)	(68 788)	(34 394)	—
   Planning fees	20 000	—	(1000)	(1000)	(2000)	(4000)	(6000)	(4000)	(2000)	—
   Building fees	50 000	—	(2500)	(2500)	(5000)	(10 000)	(15 000)	(10 000)	(5000)	—
   Planning obligations	500 000	—	(25 000)	(25 000)	(50 000)	(100 000)	(150 000)	(100 000)	(50 000)	—
   CIL	500 000	—	(25 000)	(25 000)	(50 000)	(100 000)	(150 000)	(100 000)	(50 000)	—
   Other fees	200 000	—	(10 000)	(10 000)	(20 000)	(40 000)	(60 000)	(40 000)	(20 000)	—
   Marketing	300 000	—	(15 000)	(15 000)	(30 000)	(60 000)	(90 000)	(60 000)	(30 000)	—
   Total costs	21 165 911	(6 640 793)	(726 256)	(726 256)	(1 452 512)	(2 905 024)	(4 357 535)	(2 905 024)	(1 452 512)	—
   Net cash flow		(6 640 793)	(726 256)	(726 256)	(1 452 512)	(2 905 024)	(4 357 535)	(2 905 024)	11 718 220	16 170 732

   IRR	33.05%
   Equity invested (£)	19 713 399
   Profit (£)	8 175 552
   Equity multiple	1.41
   Profit on cost	39%
   Profit on value	28%

   1. Use a spreadsheet to replicate the cash flow and calculation of the performance metrics.
   2. Set up random uniform distributions for the following inputs, run 1000 simulations and report the resulting mean IRR:
      • Residential market dwelling prices between £1900 and £2200 per square metre
      • Commercial rent between £190 and £215 per square metre
      • Commercial yield between 4.5 and 5.5%
   3. As (b) but this time use random normal distributions as follows:
      • Residential market dwelling prices have a mean of £2000/m² and a standard deviation of £50/m².
      • Commercial rent has a mean of £200/m² and a standard deviation of £10/m².
      • Commercial yield has a mean of 5% and a standard deviation 0.5%.
   4. Comment on the resulting IRRs from (a), (b) and (c).
   5. If you have access to a simulation add‐on for Excel, set up the inputs as follows:

      Simulation inputs	Mean	SD	Min	Max
      Residential – market dwellings (£/m2)	2000	50	1900	2200
      Commercial rent (£/m2)	200	10	190	215
      Commercial yield	5.00%	0.50%	4.50%	5.50%

      Correlation matrix	Market dwelling	Commercial rent	Commercial yield
      Market dwelling	1
      Commercial rent	0.75	1
      Commercial yield	−0.6	−0.4	1

      Run 1000 simulations and report the mean IRR and variance, together with the maximum and minimum IRR values. Also, show the resulting frequency distribution of IRRs.

1. 
   
   1. Residual valuation to calculate site value

      Development value
      Net internal area (NIA) (m2)	1700
      Estimated rental value (ERV) (£/m2)	200
      		340 000
      Net initial yield	7.00%	14.2857
      Gross development value (GDV) before sale costs (£)			4 857 143
      Net development value (NDV) after sale costs (£)			4 761 905
      Development costs
      Site preparation (£)		(25 000)
      Building costs (£/m2 GIA)	969	(1 938 000)
      External costs (£)		(120 000)
      Professional fees (% building costs and external works)	13.00%	(267 540)
      Miscellaneous costs (£)		(80 000)
      Contingency allowance (% construction costs)	3.00%	(72 166)
      Planning fees (£)		(5000)
      Building regulation fees (£)		(20 000)
      Planning obligations (£)		0
      Other fees, e.g. legal, loan, valuation (£)		(95 238)
      Finance on costs and fees for HALF building period @	10.00%	(160 993)
      Finance on costs and finance for void period @	10.00%	(67 131)
      Letting agent's fee (% ERV)	10.00%	(34 000)
      Letting legal fee (% ERV)	5.00%	(17 000)
      Marketing (£)		(10 000)
      Developer's profit on total development costs (%):	20.00%	(582 414)
      Total development costs (TDC) (£)			(3 494 482)
      NDV – TDC (£)			1 267 423
      Land costs (£)
      Developer's profit on land costs (%)	20.00%	(211 237)	1 056 185
      Finance on land costs over total development period	10.00%	2.00	0.8264
      Residual land value before purchase costs (£)				872 881
      Residual land value after purchase costs (£)				819 606

   2. Residual valuation to calculate developer's profit

      Development value
      Net internal area (NIA) (m2)	1700
      Estimated rental value (ERV) (£/m2)	200
      		340 000
      Net initial yield	7.00%	14.2857
      Gross development value (GDV) before sale costs (£)			4 857 143
      Net development value (NDV) after sale costs (£)			4 761 905
      Development costs
      Land price (£)		(819 606)
      Land purchase costs (% land price)	6.50%	(53 274)
      Finance on land costs for total development period @	10.00%	(183 305)
      Site preparation (£)		(25 000)
      Building costs (£/m2 GIA)	969	(1 938 000)
      External costs (£)		(120 000)
      Professional fees (% building costs and external works)	13.00%	(267 540)
      Miscellaneous costs (£)		(80 000)
      Contingency allowance (% construction costs)	3.00%	(72 166)
      Planning fees (£)		(5000)
      Building regulation fees (£)		(20 000)
      Planning obligations (£)		0
      Other fees, e.g. legal, loan, valuation (£)		(95 238)
      Finance on building costs and fees for HALF building period @	10.00%	(160 993)
      Finance on building costs, fees and interest to date for void period:	10.00%	(67 131)
      Letting agent's fee (% ERV)	10.00%	(34 000)
      Letting legal fee (% ERV)	5.00%	(17 000)
      Marketing (£)		(10 000)
      Total development costs (TDC) (£)			(3 968 254)
      Developer's profit on completion (£)			793 651

   3. Return measures
      • Return on NDV: 16.67%
      • Income yield: 8.57%
   4. Risk measures
      • Payback period (costs/ERV), i.e. inverse of income yield: 11.67 years
      • Rent cover (profit/rent): 2.33 years
      • Interest cover (profit/annual mortgage payment on costs): 2.33 years*
      • Rent‐to‐debt ratio (rent/annual mortgage payment): 1.04
      • Rent needs to be £212/m² if the land price is £1m (found using goal seek).

      *Assuming mortgage term is 25 years at an interest rate of 7%, then the multiplier will be 0.0858 and the annual mortgage payment on the costs will be £340 518.

   5. Risk analyses
      1. Sensitivity analysis

         Rent
         		£180	£190	£200	£210	£220
         Yield	7.70%	(67 050)	146 850	360 751	574 651	788 551
         	7.35%	118 479	342 686	566 894	791 101	1 015 308
         	7.00%	322 561	558 106	793 651	1 029 196	1 264 741
         	6.65%	548 124	796 201	1 044 278	1 292 354	1 540 431
         	6.30%	798 751	1 060 751	1 322 752	1 584 752	1 846 752

      2. Scenario model

         		Current values	Pessimistic profit	Optimistic profit
         Changing cells
         	ERV	£200	£190	£210
         	Yield	7.00%	7.35%	6.75%
         	Void	0.25	0.50	0.00
         Result cells:
         	Developer's profit	793 651	248 468	1 306 381

2. 
   
   1. As above
   2. By way of example, here is one set of values:

      	£/m2	NIA (m2)	Yield	Value
      Residential – market dwellings	1919	5000		£9 360 976
      Commercial space	208	4000	5.10%	£15 915 830

      And a sample from the 1000 runs:

      	IRR
      Trial	32.71%	Ranked
      1	33.91%	26.94%
      2	36.18%	27.09%
      3	33.88%	27.42%
      4	35.94%	27.44%
      5	36.99%	27.54%
      6	38.49%	27.56%
      7	30.86%	27.69%
      8	36.32%	27.77%
      9	37.70%	27.86%
      10	29.63%	27.96%
      …	…	…
      990	32.08%	42.06%
      991	34.48%	42.07%
      992	33.91%	42.09%
      993	33.84%	42.09%
      994	31.39%	42.11%
      995	27.56%	42.14%
      996	30.05%	42.14%
      997	34.28%	42.19%
      998	35.94%	42.38%
      999	38.21%	42.40%
      1000	29.24%	43.04%
      Mean IRR	34.88%

   3. 	IRR
      Trial	38.75%	Ranked
      1	32.21%	12.08%
      2	38.94%	13.23%
      3	41.89%	15.17%
      4	34.64%	16.55%
      5	43.53%	18.30%
      6	33.19%	18.38%
      7	25.53%	18.42%
      8	27.86%	18.49%
      9	32.08%	18.60%
      10	20.70%	18.97%
      …	…	…
      990	38.33%	51.55%
      991	28.02%	51.75%
      992	40.52%	52.05%
      993	27.93%	52.89%
      994	26.38%	52.90%
      995	37.09%	53.33%
      996	40.44%	55.03%
      997	31.63%	55.49%
      998	42.44%	56.34%
      999	27.12%	56.61%
      1000	34.82%	57.61%
      Mean IRR	33.85%

   4. In (c), because the tails of the distributions have not been truncated, the model produces much lower and higher IRRs at the extremes compared to the uniform distribution model in (b), but the bell shape of the normal distribution results in a mean that is closer to the point estimate IRR in (a).
   5. Mean IRR = 33.68%

      Variance = 0.256%

      Maximum IRR = 46.39%

      Minimum IRR = 23.05%

      Frequency distribution of IRRs: