This shows that the net present value of the cost of the three options are:

Cheaper copier 7 185

Expensive copier 6 772

No copier 7 988

This gives a preference for the expensive copier, with a low cost per page. A way of interpreting the figures is that if we outlaid $7 185 now, and spent some of this on buying the copier, some on copies as needed, and let the balance accumulate interest at 8 percent, then we would exactly finance the operation. And so on with the other two options.

Here it is worth mentioning the uses and abuses of forward estimates, which are cash flow projections used in many government agencies. They should not be used for evaluation. If we had been looking simply at cash outlays over four years we would have found that the cheaper copier would have involved far less cash outlay. This is an example of why we should not use forward estimates, which are no more than cash projections, for comparative purposes.

We should, of course, subject our model to sensitivity analysis. What if we do only 15 000 copies a year (buy the cheaper copier), if only 10 000 copies (no copier)? What if our discount rate rises to 10 percent (no change in decision). We should look at the effect of varying the discount rate. At lower discount rates the expensive copier continues to dominate; at higher rates the cheaper copier dominates, and at very high rates it's best to have no copier. These zones are shown in the graph below. As a general rule low discount rates favor investments with a high ratio of up-front to running cost, and as the discount rate rises we are willing to incur high ongoing costs in the future in favor of lower outlays now. (To generate a graph similar to this you will need to use the Data Table function in Excel.)

This sort of decision is common in the public sector. We have little choice about whether to undertake an activity or not; we need therefore to find the lowest cost means possible to achieve a given outcome. (It is a form of cost-effectiveness analysis, in answer to the question "what is the lowest way of financing a given number of copies?".) In more commercial situations, where we can put a price on our output, we follow the fundamental rule of benefit-cost analysis, and select the project with the highest net value.

DCF analysis is a standard technique, even for minor decisions (e.g. down to $1000) in the private sector. It is used in the public sector, but its use is confined mainly to large projects. There is no reason for not using it, however; the reasons it is not used seem to be a lack of familiarity with the technique and often an undue obsession with immediate cash outlays.

DCF analysis is a fundamental tool in benefit-cost analysis, except in the cases where costs and benefits occur all in the same year. Sometimes discounting is done on the costs and benefits separately; that is, a NPV of costs is determined, and a NPV of benefits is determined, and the two are compared. The NPV following such a manipulation is no different from the NPV of net cash flows, in which costs and benefits are offset before discounting. Finding the NPVs of the two streams, however, does allow for calculation of benefit-cost ratios.

Also, although the term "discounted cash flow" is used, this is more to describe a technique rather than to imply that the costs and benefits have to be actual cash outlays or receipts. In the private sector projects can usually be evaluated simply by considering cash flows. In the public sector, however, while costs and benefits may be tangible and measurable in dollar terms, there is often no cash flow. In a transport model we would include travel time savings, for example, as if these were actual cash benefits to the users concerned, although there would never be a cash transaction. Similarly, if our project involved some loss of environmental amenity, we would try to give it a dollar value as an outlay, even though we never make the transaction. Restricting our analysis to actual cash flows is to restrict our evaluation to a financial evaluation; public policy should rest on economic evaluation.

This exercise is typical of the exercises we need to do in comparing in-house bids with external bids. It illustrates two issues in the comparison:

(1) Low volumes tend to favor the "contracting" out option (no copier).

(2) A high discount rate tends to favor contracting out.

We have dealt with the first of these issues. On the second, there is often a tendency to use a higher than necessary discount rate in finding internal costs; such an error tends to introduce a bias in favor of contracting out.

Leases

Government agencies are often attracted to leasing as a means of conserving scarce cash resources. Basically, there are two forms of lease - an operating lease and a finance lease. In an operating lease the leasing firm provides not only the leased equipment but also all associated services. For example, an agency could enter an operating lease deal to have a firm provide all its cars, including maintenance and repairs, in much the same way as one may rent a car from Hertz or Budget. A finance lease is simply a financing deal. The asset passes into the care of the agency, and there are usually no associated services.

Operating leases often make good sense, especially if an agency doesn't have scale economies in maintaining equipment. Finance leases, on the other hand, are simply a way of borrowing money. When finance leases are offered one can use DCF analysis to calculate lease versus buy options, and can find at what rate of interest the NPV of the finance lease equates to the outright purchase option. One usually finds that the effective rate of interest on a finance lease is well above what a government agency pays on its borrowings.

4.3 Annuities and sinking funds

Discounting provides a means of taking flows of costs and benefits over time and converting them to a single lump sum at the beginning of the period so we can compare different projects. Suppose, however, we have a different sort of problem. We may know the NPV of the in-house option over n years, but have to compare that with a regular outlay for an external contractor. Or, if we are thinking of setting a price based on cost, what is the annual cost on which to base our pricing when our operations involve lumpy costs? Two techniques an help us solve such problems.

Annuities are about the process of converting a lump sum into a constant flow over a period. Sinking funds are not about bottom-of-the-harbor tax evasion; they are about converting a lumpy set of outlays into a smooth flow, or accumulating enough money for an expected future outlay.

Annuities

Many people, on retirement, choose (in fact are strongly persuaded by superannuation and tax legislation) to convert lump sums into annuities. If we retire with a lump sum we would like to spin that lump sum out over our remaining life. We can certainly draw the real interest component, but how fast can we draw down on the capital component? A benefactor may leave an endowment to a university to fund a chair for, say, twenty years. How fast can the university draw on the fund? Basically an annuity is an investment in which we invest a sum S now, or at time 0, and every year, for n years, draw off a certain amount A. Usually we know S, n, and r, the rate of return; our task is to find A.

To illustrate, imagine we have accumulated $200 000 in a superannuation policy, and expect to live for twenty years. What can we draw off that policy each year while making sure we leave nothing to the next generation? The discount rate is 5 percent.

We know that if we were to live forever, or if we wanted to leave $200 000 in our will, we could draw off $10 000 a year. We should be able to do better than that however. We know that if we simply drew down our capital we could draw $10 000 a year. But we cannot draw down capital and take interest on the full amount - we cannot have our cake and eat it. The amount of drawing we can sustain over twenty years has to be somewhere between $10 000 and $20 000.

Now the lump sum amount S is simply the present value of our sustainable cash flow of A for n years, starting at the end of this year.⁽⁹⁾ Then we can simply expand this out as a standard net present value formula:

In the case of a twenty year annuity, substituting n = 20, r = .05, and S = 200 000, we get A = $16 050.We can see from the formula that as n gets very large then the denominator approaches 1. The formula then becomes simply A = Sr, which is the familiar simple interest formula.

If we were not sure we were going to live twenty years, we could, for a management fee, buy a lifelong annuity. These are sold by funds managers, and annual payments are based on statistical life expectancy. If we live for ten years only, bad luck. If we last for thirty years we spend our dotage subsidized by those who went before.

We may find the annuity formula looks a bit formidable. We may find it easier, and more transparent, to use a spreadsheet, and to set up a difference equation model, manipulating the draw off amount A until we have exactly zero after twenty years - a 'suck it and see' approach, which is quite valid mathematically. (Mathematicians use the term iteration, rather than "suck it and see".) For each year we start with an opening capital sum, add the interest, subtract the drawing, and get the residual sum, which becomes the capital for the following year. Using the goal seek function in a spreadsheet we can ask the spreadsheet to do the iterations for us.

Sinking funds

A sinking fund is a fund to which we make regular payments to meet large expenses in the future. It is a common way of accumulating funds for major building maintenance and refurbishment.

We can develop an accumulation spreadsheet for a sinking fund for a once-off major outlay, but if our outlays are lumpy and irregular (e.g. paint every five years, re-roof every thirty years), then our easiest approach is to use a set of difference equations in a spreadsheet.

An example illustrates the point.

In the new Canberra suburb of Gorton, just south of Yass, the water supply authority has installed new pipes, header tanks, pumps etc.

They estimate the assets will last 30 years before needing replacement. Also, for the first few years, the assets will require virtually no maintenance. They estimate that the following outlays will be required:

Years 1 to 9 - no maintenance

Years 10 to 19 - $2 million a year maintenance

Years 20 to 29 - $3 million a year maintenance

Year 30 - complete replacement of system for $30 million

They envisage they will have to contribute somewhere between $1 and $2 million a year, and, in order to keep good relations with ratepayers, they want to make sure the contributions start straight away and are the same each year. They can get a real return of five percent for any funds they set aside in a sinking fund.

The first twelve and last three rows of the spreadsheet are shown below. The example below shows a trial contribution of $1.70 million a year, which gives an amount leftover of exactly zero at the end of the period.

A completed spreadsheet does not show the steps in constructing it. You start with the second last column - the drawings, and incorporate these into the difference equations - open balance, plus interest, plus regular contribution, less drawing, equals closing balance. How do we know a regular contribution of $1.70 million will exactly balance the fund? We don't. You have to experiment with various rates of contribution to get it right with a zero balance after you have drawn off the $30 million at the end of year 30. The answer is a regular annual contribution of 1.70 million.⁽¹⁰⁾

The benefit of a sinking fund for GBEs is that it allows them to accumulate reserves for projected maintenance and replacement works; they do not have to raise rates dramatically or go cap-in-hand to the finance authority when it comes time to make major outlays. It also ensures present users contribute to expenses they accrue but do not realize. The cost is that, unless the fund is well protected, the government can pillage it for current consumption or to close a budget deficit, especially if the entity is not off-budget. Replacement of ageing infrastructure is a major problem in many of the world's older cities; in new cities, like Canberra, the problem is more likely to be that present taxpayers are not paying adequately for future maintenance, upgrading and replacement.

4.4 Choice of discount rate

In most investment decisions the main items of sensitivity are the revenue and the choice of discount rate.

It is customary to use a real, rather than a nominal discount rate in such analysis. In brief, the real discount rate is the nominal rate less inflation.⁽¹¹⁾ In that way we can do our analysis without having to worry about inflating our cost and revenue elements - all costs and prices can be expressed in constant price terms.

If you are having trouble thinking about real, as opposed to nominal, rates, think about a pensioner living off a lump sum. If she had, say, $300 000 to live off, and did not wish to deplete her capital, how much could she draw each year if nominal rates were 12 percent and inflation was 7 percent? The answer is $15 000, or 5 percent of $300 000. Actual interest paid would be $36 000, of course, but inflation has eroded her capital base by 7 percent, or $21 000. She has to put that $21 000 back into the capital base to maintain its real value, leaving her with $15 000 to draw.

Real interest rates (upon which the real discount rate is based) have been volatile in Australia over the eighties, reflecting the use, by successive governments, of monetary policy as a major instrument of economic regulation. Over the long term they average around five percent, but in the short term financial regulators may manipulate rates to influence investment activity. (The effect is mainly on investments with short payback periods, for long-term investors are more interested in long-term interest rates.) Over much of the eighties the western world was gripped by a dogma that government projects were somehow of less value to the community than private enterprise projects; central agencies therefore set very high hurdles for government projects - as high as a ten percent real discount rate - to ration government capital expenditure.

It is questionable whether government authorities should include a premium for risk when choosing a discount rate. In commercial markets lenders always apply a premium for risk. Government securities bear virtually no risk premium; bank bills bear very little. Building society deposits, credit union deposits, unsecured notes, shares, all bear higher risk premiums, depending on the riskiness of the projects in which the institutions lend or invest.

Conventionally, governments used not to apply a risk premium, but now a risk premium is implicit in most discount rates set by central agencies. International credit rating agencies, like Moodys, assess the risk rating of governments, and these flow through to discount rates. Government projects obviously vary in their riskiness - an investment in a road in a congested city is far less risky than an investment as a capital injection to a government owned airline. In general GBEs should use commercial rates, including a premium for risk.

When real interest rates are high, there can be difficulties in choosing an appropriate discount rate. Taking an opportunity cost approach, we should use the opportunity cost of capital - that is, the rate we could get on long term risk free investments - as our benchmark rate. This is sometimes called the social opportunity cost of capital.

On the other hand, many desirable projects would simply not get off the ground at real discount rates as high as 5 or 8 percent. People want to make investments in schooling, environmental protection, or culture, returns from which may not be realized even for generations. In the past communities have embarked on projects which would take hundred of years to complete, such as medieval cathedrals. These decisions, if taken democratically, are governed by a social rate of time preference, which is never zero (our planet may be hit by an asteroid tomorrow), but is often as low as one or two percent, as is revealed by society's choices on education, training etc.

Any notion, therefore, that choice between competing projects, or the decision whether to undertake a project at all, can be resolved by recourse to technocratic reductionism, is incorrect. We cannot just take a directive from headquarters and plug it into our models.

Exercises

1. Returning to our hot water system. Hot water systems last around fifteen years. Solar systems are expensive to install, but reasonably low cost to run (they need boosting in winter and on dull days). Electric heaters cost much less to install, but they cost more to run.

Imagine a household has a choice between a solar system which would cost $2000 to install but only $100 a year to operate, and an electric system which would cost $500 to install but $300 a year to operate. Which system has a lower NPV of cost?

Modelling this choice over 15 years (years 0 to 14) you will find the solar system has a lower NPV at low discount rates - up to 10.2 percent. (This illustrates the general principle that as discount rates rise we tend to favor decisions which sacrifice future gains for immediate benefits.)

2. The Shire of Wanton Valley (a tax exempt organization) wants to put up a 35 bed private nursing home. It must be an economically justifiable project.

The revenue per resident it will get is $20 000 a year, and there will be no trouble filling it. The shire can plan on all beds being full and earning money once construction is completed, which should take about a year.

The architect has presented two options:

The first has a low construction cost ($1.4 million). Its running costs will be $430 000 a year once completed and occupied. It is a conventional building.

The second is a higher cost option ($2.3 million). Its running costs, however, will be $130 000 a year lower, as it will incorporate solar energy systems, low maintenance materials, a layout to make for easier cleaning, and a kitchen with significant labor savings.

Demographic projections suggest that in eight years time the north coast may have become yuppified. So in the interests of conservatism the Council's projections are going out no more than eight years from construction, which will take a year. At the end of the eight year period the cheaper building would have no value, but the other building would have a disposal value (for conversion to a motel) of $200 000. The projected cash flows in $'000 are:

The real cost of capital (the opportunity cost on other projects) is 5 percent. What is your advice to the shire?

You will find, at low discount rates, the preference is for the more expensive institution; above 6.8 percent and up to 10.8 percent the less expensive option has a higher NPV, and at higher rates neither has a positive NPV.

3. The VFT project between Melbourne and Sydney was found by the private sector consortium not to be a financially viable project. But many economists believed it was (and still is) an economically viable project. What explains this apparent difference?

4. In 1991, the ACT Electricity and Water Authority offered ratepayers the choice of paying water and sewerage charges in one hit, with a five percent discount, in late September, or spreading the full amount over four equal payments in late September, late November, late January and late March (three payments of $102, and a fourth of $104). If you were a ratepayer in the ACT, which would you choose? Set up a model to find the value or penalty in early payment. (As a hint, it's quite valid to use partial years in calculating discount factors. N does not have to be an integer.)

You will find that the discount rate which reduces the difference between immediate payment and the NPV of time payment to zero is 23 percent. What explains such a high discount rate? Was the opportunity cost of funds for the ACT Government really 23 percent, or are there other factors which may explain such a high figure?

5. In the sinking fund model, we assumed the fund was there to replace existing infrastructure, and found a contribution rate of $1.70 million was necessary to provide funds to cover maintenance and eventual replacement. Show that if the residents have to pay back a loan of $30 million for infrastructure already provided - that is, shifting the $30 million to the beginning rather than the end of the project, then the sinking fund contribution rate rises to around $3.1 to $3.2 million (depending on the precise timing of the $30 million).

6. A firm offers a photocopier on the following terms. Buy outright for $2950, or lease at $24 per week for three years with a ten percent residual payment ($295) at the end of the period. Show that this represents an effective rate of around 23 percent.

Notes

8. Some writers use the symbol r, others the symbol i, which strictly equates to the specific entity interest. The term r, short for return, is more general, and can refer to return in the form of interest, dividends or capital gain on loan or equity investment. It can also refer to non-financial returns.