# 10. Step 10: Discount benefits and costs, calculate summary results

## Steps

**10.1** Choose a discount rate.

**10.2** Assemble benefits and costs by time period.

**10.3** Calculate the NPV.

**10.4** Calculate the BCR.

**10.5** Calculate the NPVI (if required).

**10.6** Calculate the incremental BCR (if applicable).

**10.7** Calculate the internal rate of return (IRR) (if required).

**10.8** Calculate the first-year rate of return (FYRR).

## 10.1 Choose a discount rate

Discounting is necessary because a dollar of benefit in the future is worth less than a dollar of benefit today. There is a variety of views and approaches to selecting the social discount rate. The technical literature is large and complex. There is no definitive answer on which experts will agree. For a background discussion on discount rates see NGTSM 2006 Volume 5, Section 2.10.1.

In practice, use the discount rate nominated by the funding jurisdiction. For example, at the time of publication: Infrastructure Australia requires the use of a real rate of 7 per cent with 4 per cent and 10 per cent used for sensitivity testing; the Commonwealth Department of Transport and Regional Development requires the use of real rates of both 4 per cent and 7 per cent.

## 10.2 Assemble benefits and costs by time period

Adopt an end-of-year convention for discounting purposes, where all benefits and costs are assumed to occur at the end of the year in which they occur.

Set ‘year zero’ at the time of the commencement of construction. Any costs occurring in year zero are not discounted. Costs incurred during year one will be discounted by one year. Discount forward any costs (e.g. avoidable planning and design costs^{1}) incurred prior to the commencement of year zero (years minus one, minus two and so on) by multiplying by (1+r)^{t}.

With the life of the initiative assumed to commence at completion of construction, the number of years over which discounting occurs will be larger than the initiative’s life. For example, if an initiative takes two full years to construct and has a 30-year life, there will be 32 years of benefits and costs to discount - the final year’s net benefits being discounted by 32 years.

## 10.3 Calculate the NPV

The summation of all annual discounted present values of a stream of benefits or costs is called the ‘present value’ of that stream. The net present value (NPV) of an initiative is the difference between the discounted stream of benefits and the discounted stream of costs. The NPV is given by:

$NPV=\sum _{t=0}^{n}\frac{{B}_{t}-{OC}_{t}-{IC}_{t}}{{\left(1+r\right)}^{t}}$

where:

- t is time in years
- n is number of years during which benefits and costs occur
- r is the discount rate
- B
_{t}is benefits in year t - OC
_{t}is infrastructure operating costs in year t - IC
_{t}is investment costs in year t.

A positive NPV means that the initiative represents an improvement in economic efficiency compared with the Base Case.

Use the NPV to compare:

- Mutually exclusive options for the same initiative
- Alternative combinations of related initiatives (where implementation of one affects the benefits and/or costs of another)
- Alternative implementation timings for the same initiative.

The Incremental BCR in Section 11.6 below is an alternative tool for these situations.

## 10.4 Calculate the BCR

The BCR is the present value of benefits minus operating costs divided by the present value of costs. There are two alternative definitions depending on whether one puts infrastructure operating costs in the numerator or the denominator.

$BCR1=\frac{PV\left(B\right)}{PV\left(OC+IC\right)}$

$BCR2=\frac{PV\left(B-OC\right)}{PV\left(IC\right)}$

where $PV\left(x\right)=\sum _{t=0}^{n}\frac{{x}_{t}}{{\left(1+r\right)}^{t}}$

BCR1 puts costs and (savings in costs) that impact on government budgets in the denominator and everything else in the numerator. BCR2 puts costs that occur before completion of the initiative in the denominator and benefits and savings in costs that occur after completion of the initiative in the numerator. Benefits of deferred capital expenditure and costs of capital expenditure brought forward, discussed in Section 8.6, belong the denominator in BCR1 and the numerator in BCR2.

A BCR greater than one implies a positive NPV.

The BCR measure is used:

- As a convenient way to express the economic worth of an initiative
- To rank initiatives from an economic efficiency perspective where there is a budget constraint. BCR2 is the theoretically correct measure to use for this purpose because it is short-term funds being allocated. As long as operating and maintenance costs are small in relation to benefits and investment costs, BCR1 and BCR2 will be close and ranking by BCR1 should not lead to significant errors.

Never use BCRs to choose between mutually exclusive options for the same initiative, because they remove the effects of different scales of the initiatives.

## 10.5 Calculate net present value per dollar invested if required

The net present value per dollar invested is defined as

$NPVI=\frac{NPV}{PV\left(IC\right)}$

The NPVI is exactly equal to BCR2 minus one. If BCR2 has been provided, there is no value added by providing the NPVI. However, if only BCR1 has been provided, the NPVI can be used for ranking initiatives subject to a budget constraint.

## 10.6 Calculate the incremental BCR (if applicable)

The incremental BCR (IBCR) can be used instead of the NPV in the three comparative situations listed in Section 11.3 above. The IBCR is defined as

$IBCR=\frac{PV\left({B}_{2}-{OC}_{2}\right)-PV\left({B}_{1}-{OC}_{1}\right)}{PV\left({IC}_{2}\right)-PV\left({IC}_{1}\right)}$

where the subscripts represent options 1 and 2, and option 2 has the greater investment cost. The IBCR is well-suited for comparing options involving different scales of initiative. Increases in the scale of initiative are worthwhile as long as the IBCR for each scale exceeds one.

IBCR comparisons can take account of budget constraints, unlike simple NPV comparisons.

- Posit a cut-off BCR at the level of the lowest acceptable BCR for initiatives competing for funds out the same budget. A cut-off BCR of 2 implies that each dollar of funds used to pay of the increment has an opportunity cost of funds of $2 of forgone benefit from other investment opportunities not taken.
- List the options in ascending order of investment cost and calculate the IBCR for each adjacent pair.
- An IBCR above the cut-off implies the increment is worth accepting. If the IBCR for a pair of options is below the cut-off, reject the higher cost option. Remove it from the list and use the lower cost option as the basis for the next increment.
- The economically best option is that with the highest investment cost which has an IBCR greater than or equal to the cut-off BCR
^{2}. (UK DFT 2006).

## 10.7 Calculate the internal rate of return (IRR) (if required)

Central agencies sometimes require reporting of the internal rate of return (IRR).

The IRR is defined as the value of the discount rate at which the NPV equals zero. It represents the minimum discount rate at which the initiative is viable in economic terms. There is no formula for the IRR. It needs to be found by iteration. Excel has a function to do this.

The IRR can be used in the same way as the NPV to indicate whether or not an initiative will be of benefit to society as a whole. It provides an indication of the economic worth of an initiative without requiring specification of a discount rate.

The IRR has no other uses. Never use the IRR to rank initiatives or to choose between mutually exclusive options as this amounts to comparing initiatives using different discount rates.

## 10.8 Calculate the first-year rate of return (FYRR)

The first-year rate of return (FYRR) is the level of benefits minus operating costs in the first year of operation of the initiative discounted to year zero, divided by the present value of investment costs. That is:

$FYRR=\frac{{B}_{{t}_{f}}}{{\left(1+r\right)}^{{t}_{f}}}/\sum _{t=0}^{{t}_{f}}\frac{{IC}_{t}}{{\left(1+r\right)}^{t}}$

where t_{f} is the first year of operation of the initiative.

The FYRR can indicate whether an initiative’s optimal implementation time is in the past or future, and hence whether deferral is warranted. Provided the assumptions underlying the criterion are met (see NGTSM 2006 Volume 5 for details), the optimal implementation time is the first year in which the FYRR is greater than the discount rate.

All initiatives should be subjected to the FYRR test and the result reported in the Business Case.

[1] Leave out any planning and design costs already occurred at the time of undertaking the CBA because they will not be affected by any decision to proceed with the initiative.

[2] Testing whether an option has an IBCR greater than a cut-off BCR, Î¼, is the same as comparing the NPVs for the two options with investment costs grossed up by the cut-off BCR to account for the opportunity cost of scarce investment funds. $IBCR=\frac{PV\left({B}_{2}-{OC}_{2}\right)-PV\left({B}_{1}-{OC}_{1}\right)}{PV\left({IC}_{2}\right)-PV\left({IC}_{1}\right)}>\mu $ is the same as $PV\left({B}_{2}-{OC}_{2}\right)-\mu PV\left({IC}_{2}\right)>PV\left({B}_{1}-{OC}_{1}\right)-\mu PV\left({IC}_{1}\right)$