Example0 The EU decided to remove the incandescent lamp gradually from the market, because there are other lamps that are more energy efficient now and therefore cheaper in the long run. In order to check this we make use of a basic principle within engineering economics (Park 2009): All that counts is the difference among alternatives. Fig. 1. Incandescent lamp. After 2012 these are banned, and shops may only sell their remaining stock.(Photo: Wikimedia Commons). Graphical plots on this page illustrate a household's risk and surplus when changing an old lamp (Fig. 1) to an energy efficient lamp. It is necessary to draw such plots in order to get a complete overview of the investment. The plots depend on the prices of the old and the new lamp, the annual savings, the hours of usage, and lifetime estimates of the old and the new lamp. The example on this page is introductory, and it illustrates the type of analysis which is fundamental for the appraisal (valuation) of all the engineering projects on this website. Fig. 2. Compact fluorescent lamp, CFL. (Photo: Wikimedia Commons). Contents Introduction Cash Flows Incremental Cash Flows Internal Rate of Return Balance of the Project Account Payback Period Surplus Reinvesting the Savings Conclusions Data External Links Introduction The Danish Energy Agency urges the consumers to change their incandescent lamps (Fig. 1) into something more energy efficient (Fig. 2) arguing that it will be cheaper in the long run. But what does 'cheaper in the long run' actually mean? An incandescent lamp costs only 12 DKK (1.60 EUR) while an equivalent energy saving compact fluorescent lamp (CFL) of medium durability costs 55 DKK (7.33 EUR). On the other hand, a CFL uses only 1/4 of the energy in comparison. Since there are some annual savings, it is intuitively clear that the CFL could 'catch up' with the old type of lamp. How long this takes is a measure of risk, because the sooner that period is over, the sooner the risk vanishes. Whether there is also a good surplus in the end depends on the lifetime of the lamps and the time horizon of the investment; should the family move to another home in the near future, then the investment could be more or less lost. Cash Flows Let us choose a time horizon of ten years. An incandescent lamp burns maybe 1 000 hours, before it breaks, and let us choose an alternative CFL with an estimated lifetime of 10 000 burning hours. The CFL lifetime is clearly much better, and it has the advantage that the lamp must be replaced less frequently. If we assume that the usage of the lamp is 1 000 hours per year -- almost 3 hours per day -- then the incandescent lamp must be replaced once every year while the CFL lasts 10 years. Our appraisal of the investment strongly depends on this good lifetime. Fig. 3. Two cash flow streams. The incandescent lamp is cheaper to buy, but the CFL is cheaper to operate. All amounts are negative, because they are expenditures. Fig. 4. Incremental cash flow stream. It is the difference between the two cash flow streams in the previous figure. Now some amounts are positive, because they are savings. The internal rate of return is 146%. Fig. 5. Cumulative cash flow diagram. This is the successive accumulation of the amounts in the previous figure starting from the left. The initial payment is negative, because the CFL is more expensive, but the cumulative cash flow becomes positive after one year. The surplus after ten years is almost 78 euro. The diagram in Figure 3 is a cash flow diagram. When expenditures and receipts are in cash, the net receipts at a single time instant (a column) is termed cash flow, and the series of flows over the whole time horizon is a cash flow stream (Luenberger 1998). Each cash flow is related to the end of the calendar year. For example, the initial investment is made at the end of year 0; all further investments are assumed to be made at the end of year 1, 2, ..., 10; and all expenditures of electricity are assumed to be calculated on an annual basis at the end of the year. The diagram gives an overview of the expenditures of the two alternatives, each in a different colour. The diagram depends on estimates of the energy consumed and the energy price, gathered in Table 1 below. This diagram is already more informative than the mere selling prices of the two lamps. Incremental Cash Flows Naturally, we view the CFL as an alternative which saves energy in the long run, and we shall apply the above quoted principle: All that counts is the difference among alternatives. We therefore take the incandescent lamp as our reference scenario and compare the CFL with that. Figure 4 is an incremental cash flow diagram. It shows the consequences of moving from the incandescent lamp to the CFL. It is the increment in the cash flow: The CFL cash flow stream minus the incandescent cash flow stream, mathematically speaking. The plot clearly shows that there are savings over most of the time horizon, and there seems to be more savings (positive cash flows) than expenditures (negative cash flows). Internal Rate of Return The incremental cash flow stream (Fig. 4) can be viewed as an investment with an initial payment at the end of year 0, the principal, which returns positive cash flows every year thereafter. It is equivalent to a deposit in a bank account with some unspecified interest rate. We would like to know this interest rate in order to compare the investment with other investments, for example in bonds. That interest rate is the internal rate of return (IRR). It is internal in contrast to the external rate of return, which is the market return rate for an investment under existing market conditions. The Excel function IRR calculates the internal rate of return, given the incremental cash flow stream as input. In our case the internal rate of return after ten years is 146%; that indicates a very profitable investment. The internal rate of return is equivalent to the interest rate of a bank account which would just finance the project; the withdrawals would be repaid exactly by the deposits generated by the project plus interest. If we are able to finance the project through an actual bank loan at a lower interest rate than the IRR, then there will be a surplus. Balance of the Project Account Inflows and outflows of the cash flow stream are related to an imaginary ideal bank account (Luenberger 1998), the project account. It is ideal, because it applies the same rate of interest to both deposits and withdrawals, and it has no service charges or transaction fees. It may apply different interest rates to different kinds of transactions, though. The project account is associated with the outside financial market. If we sum the cash flows of a whole stream we obtain the net future value of the cash flow stream at the end of the time horizon, say, K years from now. This would be the final balance of the project account. It is the net future value, because both negative and positive cash flows contribute to the sum. If we do it successively for each year k on the way (k = 1, 2, ..., K), the result is the cumulative cash flow diagram in Figure 5. This is a picture of the balance of the project account. Each column in the plot is the successive accumulation of the cash flows (Fig. 4) starting from the left. The plot shows at each time instant k the value of the investment k years from now. The project balance provides good support for the decision to invest or not. Initially there is a negative cash flow, but the accumulated savings compensate for that, and the balance becomes positive quickly. A positive value means there is a surplus, the project gains more than it looses. Payback Period There is a distinct point in time, the breakeven point, where the cumulative cash flow changes sign from negative to positive. The duration up until this event is the payback period, defined as the time it takes before the savings outbalance the expenditures. In this case breakeven happens after one year. One year is a very short payback period, given that the time horizon is ten years, and we can therefore conclude that the risk is low. Surplus The payback period does not tell what happens after the breakeven point, and it is necessary to know the expected surplus at the end of the time horizon, in order to valuate the investment. The annual savings, less the initial investment, add up to an amount, which is the expected surplus. This is in fact the same as the final value. The height of the last column in the project balance (Fig. 5) shows the expected surplus. The exact amount is 77.93 EUR (Table 2 in the Data section). This is very good considering that the initial investment is only 7.33 EUR. It is convenient that the diagram (Fig. 5) shows both the risk -- in terms of the payback period -- and the surplus in one diagram. Reinvesting the Savings The project account accumulates interest, whether it be negative or positive. For example, if energy saving light bulbs save some expenses on the electricity bill, the houseowner withdraws less money from the savings account and thus gains more interest. In case of expenses, the houseowner looses interest. In general, a bank deposit of a principal P euros at an interest rate i generates interest of the principal plus the accumulated interest that has been added to the account. There is thus an advantage of compound interest, which includes interest on interest. The future value F in k years from now of a principal P is F = P(1+i)^{k} We may apply this to both positive and negative cash flows, and even a whole cash flow stream. In the case of a positive cash flow we gain interest from reinvesting the cash in the project account at an interest rate i, and in the case of a negative cash flow we assume we loose interest from a foregone opportunity to invest at the interest rate i. Figure 6 shows what happens to the balance of the project account when considering interest. Due to interest on the first negative cash flow the payback period is prolonged somewhat, but this is hardly visible in the diagram. On the other hand all positive cash flows increase by the achieved interest, and the final value with interest at year ten is clearly larger than the simple final value. The diagram in Figure 6 is the balance of the project account -- with or without interest -- as time progresses. For example, assume that five years have passed, then the bar at year five shows the balance at the end of the year considering all cash flows from the previous years. The reinvestment interest is the interest we obtain in the project account. The simple version -- without interest -- corresponds to just spending the yearly savings. The favourable result of increased surplus is specific for this case. Generally it depends on the amount of negative cash flows and their timing relative to the positive cash flows. The result also depends on the magnitude of the interest rate i. In our case i = 0.04 (four percent). Fig. 6. Cumulative cash flow streams. For comparison the diagram shows the cumulative cash flow from the previous figure (simple), and the same with a 4% interest when reinvesting the savings (with interest). Conclusions It is safe to conclude that it pays in the long run to switch from an incandescent lamp (40 W) to a CFL (11 W). The extra cost is paid back already after a year, and from then on surplus builds up during the longer lifetime of the CFL. There are other decision criteria to consider, for example the colour of the emitted light, the delay when the CFL is switched on, and the content of mercury. Economy is not all. Our central conclusion is that the cumulative cash flow diagram is very informative, since it shows four pieces of information in a single diagram: The initial investment, the payback period, the lifetime, and the surplus (final value). For an investor such a diagram is a valuable decision support, because it gives an overview instantly of expenditures and earnings. One could wish it was mandatory to include in all sales material. For any engineering project a diagram could be made on the basis of the numbers available at the selling time (a pre-calculation), and as time passes another one could show how it actually went (a post-calculation). As a result of the Samso project, we have several cases of such before-and-after data and cumulative cash flow diagrams. The current analysis concerns actual euros only. It can be made more realistic by including inflation and the preference to receive money earlier than later. The next chapter considers their effect. Data Table 1. Assumptions regarding lamp types (Go Energi). Feature Incandescent CFL Lifetime (h) 1 000 10 000 Price (EUR) 1.60 7.33 Price (DKK) 12.00 55.00 Usage (h/year) 1 000 1 000 Wattage (W) 40 11 Marginal electricity price (EUR/kWh) 0.23 0.23 Marginal electricity price (DKK/kWh) 1.75 1.75 Table 2: Cash flow streams. Year 0 1 2 3 4 5 6 7 8 9 10 Incandescent (EUR) (A) -1.60 -10.93 -10.93 -10.93 -10.93 -10.93 -10.93 -10.93 -10.93 -10.93 -10.93 CFL (EUR) (B) -7.33 -2.57 -2.57 -2.57 -2.57 -2.57 -2.57 -2.57 -2.57 -2.57 -2.57 Incremental cash flows (EUR) (B-A) -5.73 8.37 8.37 8.37 8.37 8.37 8.37 8.37 8.37 8.37 8.37 Incremental cash flows with 4% interest (EUR) -5.73 2.40 10.87 19.67 28.82 38.34 48.24 58.54 69.25 80.38 91.96 Simple cumulative cash flows (EUR) -5.73 2.63 11.00 19.37 27.73 36.10 44.47 52.83 61.20 69.57 77.93 External Links Wikipedia Cash flow Wikipedia Compact fluorescent lamp Wikipedia Incandescent light bulb Wikipedia Internal rate of return Wikipedia Payback period