The time value of moneyMoney received today is worth more than the same sum received in the future, i.e. it has atime value.This occurs for three reasons: potential for earning interest/cost of finance impact of inflation effect of risk.Discounted cash flow (DCF) techniques take account of this time value of money when appraising investments.CompoundingA sum invested today will earn interest. Compounding calculates the future or terminal value of a given sum invested today for a number of years.To compound a sum, the figure is increased by the amount of interest it would earn over the period.Formula for compounding:To speed up the compounding calculation, we can use a formula to calculate the future value of a sum invested now. The formula is:F = P(1 + r)nwhere F = Future value after n periodsP = Present or Initial valuer = Rate of interest per periodn = Number of periods

4.6 TIME VALUE OF MONEY CONCEPTS4.6.1 IntroductionMost people have an intuitive sense of the time value of money. Given a choice between $100 today and $100 one year from today, almost everyone would prefer the $100 today. Why is this the case? Two primary factors lead to this time preference associated with money; interest and inflation. Interest is the ability to earn a return on money which is loaned rather than consumed. By taking the $100 today and placing it in an interest bearing bank account (i.e., loaning it to the bank), one year from today an amount greater than $100 would be available for withdrawal. Thus, taking the $100 today and loaning it to earn interest, generates a sum greater than $100 one year from today and thus is preferred. The amount in excess of $100 that would be available depends upon the interest rate being paid by the bank. The next section develops the mathematics of the relationship between interest rates and the timing of cash flows. The second factor which leads to the time preference associated with money is inflation. Inflation is a complex subject but in general can be described as a decrease in the purchasing power of money. The impact of inflation is that the basket of goods a consumer can buy today with $100 contains more than the basket the consumer could buy one year from today. This decrease in purchasing power is the result of inflation.

Relevance:It is the fact that cash flows occur over the investments life that requires the introduction of time value of money concepts to properly evaluate investments. If multiple investments are being evaluated and if the lives of the investments are not equal, special consideration must be given to the issue of selecting an appropriate planning horizon for the analysis. A convenient way to display the revenues (savings) and costs associated with an investment is a cash flow diagram. By using a cash flow diagram, the timing of the cash flows are more apparent and the chances of properly applying time value of money concepts are increased. With practice, different cash flow patterns can be recognized and they, in turn, may suggest the most direct approach for analysis. Time value of money problems involving compound interest are common. Because of this frequent need, tables of compound interest time value of money factors can be found in most books and reference manuals that deal with economic analysis. The factor (1+i)n is known as the single sum, future worth factor or the single payment, compound amount factor. This factor is denoted (F|P,i,n) where F denotes a future amount, P denotes a present amount, i is an interest rate (expressed as a percentage amount), and n denotes a number of years. The factor (F|P,i,n) is read to find F given P at i% for n years. Tables of values of (F|P,i,n) for selected values of i and n are provided in Appendix 4A. The tables of values in Appendix 4A are organized such that the annual interest rate (i) determines the appropriate page, the time value of money factor (F|P) determines the appropriate column, and the number of years (n) determines the appropriate row.

Time value of money factors are useful in economic analysis because they provide a mechanism to accomplish two primary functions:(1) they allow us to replace a cash flow at one point in time with an equivalent cash fl ow (in a time value of money sense) at a different point in time and (2) they allow us to convert one cash fl ow pattern to another (e.g., convert a single sum of money to an equivalent cash fl ow series or convert a cash fl ow series to an equivalent single sum).

Wind CharacteristicsSitingThe wind characteristics given in Figure 16.21 are simple average values. The wind is almost always quite variable in both speed and direction. Gusting is a rapid up-and-down change in wind speed and/or direction. An important characteristic of the wind is the number of hours that the wind exceeds a particular speed. This information can be expressed as speed-duration curves, such as those shown in Figure 16.25 for three sites in the United States. These curves are similar to the loadduration curves used by electric utilities. Because the power density of the wind depends on the cube of the wind speed, the distribution of annual average energy density of winds of various speeds will be quite different for two sites with different average wind speeds. A comparison between sites having average velocities of 13 and 24 miles/hr (5.8 and 10.7 meters/sec) is given in Figure 16.25. The area under the curve is indicative of the total energy available per unit area per year for each case. Sites should be selected where the wind speed is as high and steady as possible. Rough terrain and the presence of trees or building should be avoided. The crest of a well-rounded hill is ideal in most cases, whereasa peak with sharp, abrupt sides might be very unsatisfactory, because of fl ow reversals near the ground. Mountain gaps that might produce a funneling effect could be most suitable.

Wind Speed PatternsWind speed patterns can be depicted as awind speed spectrum. A high value indicates a significant change in wind speed over the corresponding time period. Although this graph is obviously site-specific, there are distinctive similarities. A typical graph is shown on the right.The peaks in the wind speed spectrum account for annual, seasonal and daily patterns as well as short-term turbulences. A striking phenomenon is the spectral gap between time periods of 10 minutes to 2 hours.These patterns are important not only for yield estimations, but also for forecasting of wind power output.

Wind Speed Distribution: The Macrometeorological RangeLarge-scale movements of air masses account for 3 peaks on the macrometeorological side of the spectrum.1. Diurnal Patterncaused by different temperatures at day and night. This effect is more distinct at coastal sites than off-shore.2. Depressions and Anti-cyclonesusually occur with periods of about 4 days. Tthis phenomenon is more distinct in oceanic than continental regions.3. Annual Patternvaries with the degree of latitude and vanishes in close proximity to the equator.The distribution of hourly average wind speeds (i.e. excluding turbulence) can be described by a so-called Weibull distribution:

with ashape factor kand ascaling factor A. The dimensionless shape factor reflects the influence of the topography on wind speeds and ranges between 1.2 (mountains) to 4.0 (monsoon regions). The scaling factor A is roughly 125% of the average annual wind speed. Alternatively, the following relationship between average wind speed, shape factor and scaling factor can be used:

where (x) is the gamma-function. In practice, the wind distribution is measured first, and then the parameters are adapted and used for further calculations.

Micro-meteorological Range: TurbulenceOne of the main characteristics of wind its high temporal variations. Wind speeds can double or triple within seconds, meaning power increased 8 or 27 times! Turbulence intensity increases with obstacles such as buildings, tress or steep mountain tops. Sites with high average wind speeds tend to suffer less from turbulence.

Distribution of Wind Direction

Although not of interest for the site selection, the distribution of the wind direction is important for the layout of a wind farm. This is done in three steps:1. Measure the time wind blows in each direction - sector. One sector may cover 10 - 30. In the diagram, wind blows south more than 20% of the time, whereas south-east only 5%.2. Measure the mean wind speed in each direction.3. Combine both measurements by multiplying the time with the cubic speed for each sector individually to get the distribution of energy across all directions, as the energy content per sector is Time x v.

Wind Shear Profile

Typically, winds increase further off the ground, a microscale phenomenon calledwind shear. How much the wind speeds increase with height depends not only on prevailing wind speeds at other heights, but also on the type of surface. Given a wind speed (v1) at one height (h1), the wind speed at another height (h2) can be calculated as follows:

where z0is an index that describes the roughness of the surface. Values for the roughness index range from 0.01 for flat landscapes to 2.0 in town centres. Two important insights follow from this: In rough areas, especially built-up areas, the height of the turbine hub is much more important than off-shore, as wind speed change slower along the distance from the surface. For large turbines, the difference between the wind speed experienced by blade tips at top and bottom vary much more in rough areas - forces that cause additional wear and tear.

Solar CollectorsA wide variety of devices may be used to collect solar energy. A general classification of types is given in Figure 16.3. Tracking-type collectors are usually used where relatively high temperatures (above 250F) are required. These types of collectors are discussed at the end of this section. The more common fixed, flat-plate collector will be discussed first, followed by a discussion of tube-type or mildly concentrating collectors. The flat-plate collector is a device, usually faced to the south (in the northern parts of the globe) and usually at some fixed angle of tilt from the horizontal. Its purpose is to use the solar radiation that falls upon it to raise the temperature of some fluid to a level above the ambient conditions. That heated fluid, in turn, may be used to provide hot water or space heat, to drive an engine or a refrigerating device, or perhaps to remove moisture from a substance. A typical glazed flat-plate solar collector of the liquid type is shown in Figure 16.4. The suns radiation has a short wavelength and easily passes through the glazing (or glazings), with only about 10 to 15% of the energy typically reflected and absorbed in each glazing. The sunlight that passes through is almost completely absorbed by the absorber surface and raises the absorber temperature. Heat loss out the back from the absorber plate is minimized by the use of insulation. Heat loss out the front is decreased somewhat by the glazing, since air motion is restricted. The heated

absorber plate also radiates energy back toward the sky, but this radiation is longer-wavelength radiation and most of this radiation not reflected back to the absorber by the glazing is absorbed by the glazing. The heated glazing, in turn, converts some of the absorbed energy back to the air space between it and the absorber plate. The trapping of sunlight by the glazing and the consequent heating is known as the greenhouse effect. Energy is removed from the collector by the coolant fluid. A steady condition would be reached when the absorber temperature is such that losses to the coolant and to the surroundings equal the energy gain from the solar input. When no energy is being removed from the collectors by the coolant, the collectors are said to be at stagnation. For a well-designed solar collector, that stagnation temperature may be well above 300F. This must be considered in the design of solar collectors and solar systems, since loss of coolant pumping power might be expected to occur sometime during the system lifetime. A typical coolant flow rate for flat-plate collectors is about 0.02 gpm/ft2 of collector surface (for a 20F rise). The fraction of the incident sunlight that is collected by the solar collector for useful purposes is called the collector efficiency. This efficiency depends upon several variables, which might change for a fixed absorber plate design and fixed amount of back and side insulation.These are:1. Rate of insolation2. Number and type of glazing3. Ambient air temperature4. Average (or entering) coolant fluid temperaturePaybackThe payback period is the time a project will take to pay back the money spent on it. It is based on expected cash flows and provides a measure of liquidity.FormulaConstant annual cash flows:

Uneven annual cash flows:Where cash flows are uneven, payback is calculated by working out the cumulative cash flow over the life of the project.Decision ruleWhen using Payback, the company must first set a target payback period. Select projects which pay back within the specified time period Choose between options on the basis of the fastest paybackExample using PaybackConstant annual cash flowsAn expenditure of $2 million is expected to generate net cash inflows of $500,000 each year for the next seven years.

What is the payback period for the project?

Uneven annual cash flowsA project is expected to have the following cash flows:

What is the expected payback period?

Payback is between the end of Year 3 and the end of Year 4. This is the point at which the cumulative cash flow changes from being negative to positive. If we assume a constant rate of cash flow throughout the year, we could estimate that payback will be three years plus ($500/800) of Year 4. This is because the cumulative cash flow is minus $500 at the start of the year and the Year 4 cash flow would be $800. Therefore payback is after 3.625 years.Advantages and disadvantage of PaybackAdvantages include: it is simple it is useful in certain situations: rapidly changing technology improving investment conditions it favours quick return: helps company growth minimises risk maximises liquidity it uses cash flows, not accounting profit.Disadvantages include: it ignores returns after the payback period it ignores the timings of the cash flows. This can be resolved using thediscounted payback period. it is subjectiveas it givesno definitive investment signal it ignores project profitability.Discount cash flow techniquesFeedbackWhen appraising capital projects, basic techniques such a ROCEandPaybackcould be used. Alternatively, companies could use discounted cash flow techniques discussed on this page, such as Net Present Value (NPV) and Internal Rate of Return (IRR).Cash flows and relevant costsFor all methods of investment appraisal, with the exception of ROCE, onlyrelevant cash flowsshould be considered. These are: cash flows that will happen in the future, and cash flows that will arise only if the capital project goes ahead. futureThe following should always be ignored in investment appraisal: sunk costs (costs that have already been incurred) committed costs (costs that will be incurred anyway) non-cash items allocated costs.The time value of moneyMoney received today is worth more than the same sum received in the future, i.e. it has atime value.This occurs for three reasons: potential for earning interest/cost of finance impact of inflation effect of risk.Discounted cash flow (DCF) techniques take account of this time value of money when appraising investments.

CompoundingA sum invested today will earn interest. Compounding calculates the future or terminal value of a given sum invested today for a number of years.To compound a sum, the figure is increased by the amount of interest it would earn over the period.Formula for compounding:To speed up the compounding calculation, we can use a formula to calculate the future value of a sum invested now. The formula is:F = P(1 + r)nwhere F = Future value after n periodsP = Present or Initial valuer = Rate of interest per periodn = Number of periodsDiscountingIn a potential investment project, cash flows will arise at many different points in time. To make a useful comparison of the different flows, they must all be converted to a common point in time, usually the present day, i.e. the cash flows are discounted.

The present value (PV) is the cash equivalent now of money receivable/payable at some future date.Formula for discounting:The PV of a future sum can be calculated using the formula:

This is just a re-arrangement of the formula used for compounding.(1 + r)-nis called the discount factor (DF).The cost of capitalIn discounted cash flow techniques, the rate of interest is required. There are a number of alternative terms used to refer to the rate of interest: cost of capital discount rate required return.Net Present Value (NPV)To appraise the overall impact of a project using DCF techniques involves discounting all the relevant cash flows associated with the project back to their PV (present value).If we treat outflows of the project as negative and inflows as positive, the NPV of the project is the sum of the PVs of all flows that arise as a result of doing the project.Decision rule:The NPV represents the surplus funds (after funding the investment) earned on the project, therefore: if the NPV is positive - the project is financially viable if the NPV is zero - the project breaks even if the NPV is negative - the project is not financially viable if the company has two or more mutually exclusive projects under consideration it should choose the one with the highest NPVAssumptions in calculating the net present valueThe following assumptions are made about cash flows when calculating the net present value: all cash flows occur at the start or end of a year initial investments occur T0 other cash flows start one year after that (T1).Advantages and disadvantages of using NPVAdvantagesTheoretically the NPV method of investment appraisal is superior to all others. This is because it: considers the time value of money is an absolute measure of return is based on cash flows not profits considers the whole life of the project should lead to maximisation of shareholder wealth.Disadvantages It is difficult to explain to managers It requires knowledge of the cost of capital It is relatively complex.

Calculating discount factorsDiscount factors can always be calculated using the formula(1 + r)-n.However in some special cases, time-saving techniques can be used.Discounting annuitiesAn annuity is a constant annual cash flow for a number of years.Where an investment appraisal involves a constant annual cash flow, a special discount factor known as anannuity factorcan be used.Theannuity factor(AF) is the name given to the sum of the individual DF. The formula for the annuity factor is:

Discounting perpetuitiesA perpetuityis an annual cash flow that occurs forever..The PV of a perpetuity is found using the formulacash flow=PV/ror

PV=cash flow x 1/r1/r is known as the perpetuity factorPresent Value Tables(1 + r)-nis called the discount factor (DF). This can be found from the formula, or from specialPresent Value tablesin which many discount factors have already been calculated. There are alsoAnnuity Tablesin which many annuity factors have already been calculated.Advanced and delayed annuities and perpetuitiesThe use of annuity factors and perpetuity factors both assume that the first cash flow will be occurring in one year's time. Annuity or perpetuity factors will therefore discount the cash flows back to give the value one year before the first cash flow arose. For standard annuities and perpetuities this gives the present (T0) value since the first cash flow started at T1.Be careful: if this is not the case, you will need to adjust your calculation.In some investment appraisals, regular cash flows may start now (at T0) rather than in one year's time (T1).Calculate the PV by ignoring the payment at T0when considering the number of cash flows and then adding one to the annuity or perpetuity factor.

Delayed annuities and perpetuitiesSome regular cash flows may start later than T1.These are dealt with by:(1)applying the appropriate factor to the cash flow as normal(2)discounting your answer back to T0.The Internal Rate of Return (IRR)The IRR is another project appraisal method using DCF techniques.The IRR represents the discount rate at which the NPV of an investment is zero. As such it represents a breakeven cost of capital.Calculating the IRR using linear interpolationThe steps in linear interpolation are:(1)Calculate two NPVs for the project at two different costs of capital(2)Use the following formula to find the IRR:

where:L = Lower rate of interestH = Higher rate of interestNL= NPV at lower rate of interestNH= NPV at higher rate of interest.

The Internal Rate of Return (IRR)The IRR is another project appraisal method using DCF techniques.The IRR represents the discount rate at which the NPV of an investment is zero. As such it represents a breakeven cost of capital.Calculating the IRR using linear interpolationThe steps in linear interpolation are:(1)Calculate two NPVs for the project at two different costs of capital(2)Use the following formula to find the IRR:

where:L = Lower rate of interestH = Higher rate of interestNL= NPV at lower rate of interestNH= NPV at higher rate of interest.The diagram below shows the IRR as estimated by the formula.

Decision rule: projects should be accepted if their IRR is greater than the cost of capital.Advantages and disadvantages of IRRAdvantagesThe IRR has a number of benefits, e.g. it: considers the time value of money is a percentage and therefore easily understood uses cash flows not profits considers the whole life of the project means a firm selecting projects where the IRR exceeds the cost of capital should increase shareholders' wealth.

Disadvantages It is not a measure of absolute profitability. Interpolation only provides an estimate and an accurate estimate requires the use of a spreadsheet programme. It is fairly complicated to calculate. Non-conventional cash flows may give rise to multiple IRRs which means the interpolation method can't be used.Difficulties with the IRR approachInterpolation only provides an estimate (and an accurate estimate requires the use of a spreadsheet programme). The cost of capital calculation itself is also only an estimate and if the margin between required return and the IRR is small, this lack of accuracy could actually mean the wrong decision is taken.Another drawback of IRR is that non-conventional cash flows may give rise tono IRR or multiple IRRs. For example a project with an outflow at T0 and T2 but income at T1 could, depending on the size of the cash flows, have a number of different profiles on a graph (see below). Even where the project does have one IRR, it can be seen from the graph that the decision rule would lead to the wrong result as the project does not earn a positive NPV at any cost of capital.