In economic assessments what is essentially being compared is cash sums at different points in time. For example, an investment in a project now will produce a return at a later date. Is this a good return or would it have been just as good or better by putting the money into some savings account? There is a need to understand how to manipulate monetary sums over time using interest rates in order to make such comparisons. Comments: This slide shows that £1000 invested for 5 years at 10% yields £1610.50. Thus if interest rates are 10% then £1000 now and £1610.50 in 5 years are the same value. The present worth of £1610.50 in 5 years is £1000.00. £1610.5 × 0.62092 = £1000 Try repeating the calculation with different interest rates.

Slide 1 calculated the compound amount of a single sum. Slide 2 illustrates the calculation for a uniform series. In this example, using 15%, £875.30 is the equivalent of £100.00 per year for 5 years.

Comments: Interest calculations • Compound amounts: calculate the compound amount of a sum for a given interest rate. • Present worth: calculate the present worth of a future sum for a given interest rate. • Compound amount of a regular or uniform series: calculate the compound amount of, say, an annual sum each year for a number of years at a given interest rate. • Sinking fund deposit: calculate the sum to be set aside each year for a number of years to generate a specified future sum at a given interest rate. • Present worth of a uniform or regular series:, calculate the present worth of a regular sum each year for a number of years. • Capital recovery: calculate the sum that has to be recovered each year for a number of years to justify an investment at a given interest rate. Use the formulae in the chapter appendix to experiment with these calculations.

Comments: In making an economic comparison between two schemes the approaches are: • Calculate the present worth of both schemes. If this is an investment scheme the one with the higher PW is economically better. If it’s a scheme with expenditure only the one with the smaller PW is cheaper. • Present worth converts future cash flows to a single capital sum at the beginning of the project. Equivalent annual costs converts the cash flows of a project to regular annual sums to allow comparison. Present worth is the more popular approach.

Comments: DCF yield. One definition of DCF yield is that it is the‘maximum’interest rate that could be paid on borrowed capital assuming that all the capital needed to fund the project is acquired as an overdraft. ____________________________________________________________________________________________________ Year Project’s cash First trial Second trial flow (£) ____________________________ ____________________________ PW factors PW PW factors PW at 9% (£) (£) at 10% (£) (£) ___________________________________________________________________________________________________ 0 −1000 1.000 −1000.00 1.000 −1000.00 1 +400 0.917 366.80 0.909 363.60 2 +400 0.842 336.80 0.826 330.40 3 +400 0.772 308.80 0.751 300.40 Net present worth +12.40 −5.60 Interpolating between 9 per cent and 10 per cent gives the interest rate which results in a net present worth of zero.

The graph shows that the DCF yield is the interest rate which gives a net present worth of zero. Net PW itself can be used as a means of determining whether a project is profitable enough to be considered a worthwhile investment. DCF yield is the more powerful of the profitability measures.

Including inflation in the calculations of PW or DCF yield can be achieved by: • Adjusting the cash flows so that the cash flows are no longer set at year zero prices but are inflated. In this case the PW calculated will be larger reflecting the increase, inflated, cash flows. The DCF yield calculated will also be larger and it will include the inflation element. The inflation rate would need to be deducted from the DCF yield to determine the real, not inflated, rate of return. • Adjusting the interest rate. In present worth calculations the interest rate would be reduced to reflect inflation. The relationship between the real rate (ir), the apparent rate (ia) (based on inflated cash flows) and the inflation rate (id) is similar to the relationship used in the PW calculations and is: (1+ia) = (1+ir)(1+id) An important feature of determining profitability measures is the accuracy of the estimates of future cash flows.

Comments: Sensitivity analysis is to do several calculations using different assumptions of the cash flows or the interest rates to study the effects.

Risk analysis requires a ‘model’ of the cash flows that are used in the DCF calculation. This model consists of breaking down the cash flows until a level of detail is reached whereby estimates for the element can be produced.

Comments: Financial modelling is an extensive and wide ranging approach to studying the economics of projects. The two slides list some of the more common uses of financial modelling. A discussion of some of these would prove valuable.