Future and Present Worth

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In Energy: Past, Present, and Future, we gave a few examples of how a quantity such as savings in a bank, population, or energy consumption grows exponentially. With no penalty for early withdrawal, a deposit of $100 in a 10-year CD (certificate of deposit) receiving a 10% interest, grows to 100 (1.10) = 110 after a year, 100 (1.10)2 = 121 after the end of second year, and 100(1.1)10 = 259 dollars at the maturity date. Mathematically speaking, if the present value of a quantity is P and the rate of growth is r, then the future worth of a present value after time n years have passed is (F/P) given by:

F = P(1+r)n (1)

Rearranging this equation, we can find the present value of a quantity if we know its future worth, i.e. the present worth of a future value (P/F) is:

P = F(1+r)-n (2)

For convenience, values of F/P and P/F are tabulated for various n and r values and are given in Tables 1 and 2.

Table 1. Future Worth of a Present Value
F/P=(1+r)n
r/n 1 2 3 4 5 10 15 20 25 30
0.01 1.01 1.02 1.03 1.04 1.05 1.10 1.16 1.22 1.28 1.35
0.02 1.02 1.04 1.06 1.08 1.10 1.32 1.35 1.49 1.64 1.81
0.03 1.03 1.06 1.09 1.13 1.16 1.44 1.56 1.81 2.09 2.43
0.04 1.04 1.08 1.12 1.17 1.22 1.48 1.80 2.19 2.67 3.24
0.05 1.05 1.10 1.16 1.22 1.28 1.63 2.08 2.65 3.39 4.32
0.06 1.06 1.12 1.19 1.26 1.34 1.79 2.40 3.21 4.29 5.74
0.07 1.07 1.14 1.23 1.31 1.40 1.97 2.76 3.87 5.43 7.61
0.08 1.08 1.17 1.26 1.36 1.47 2.16 3.17 4.66 6.85 10.06
0.09 1.09 1.19 1.30 1.41 1.54 2.37 3.64 5.60 8.62 13.27
0.10 1.10 1.21 1.33 1.46 1.61 2.59 4.18 6.73 1.083 17.45
Table 2. Present Worth of a Future Value
P/F=(1+r)-n
r/n 1 2 3 4 5 10 15 20 25 30
0.01 0.99 0.98 0.97 0.96 0.95 0.91 0.86 0.82 0.78 0.74
0.02 0.98 0.96 0.94 0.92 0.91 0.82 0.74 0.67 0.61 0.55
0.03 0.97 0.94 0.92 0.89 0.86 0.74 0.64 0.55 0.48 0.41
0.04 0.96 0.92 0.89 0.85 0.82 0.68 0.56 0.46 0.38 0.31
0.05 0.95 0.91 0.86 0.82 0.78 0.61 0.48 0.38 0.30 0.23
0.06 0.94 0.89 0.84 0.79 0.75 0.56 0.42 0.31 0.23 0.17
0.07 0.93 0.87 0.82 0.76 0.71 0.51 0.36 0.26 0.18 0.13
0.08 0.93 0.86 0.79 0.74 0.68 0.46 0.32 0.21 0.15 0.10
0.09 0.92 0.84 0.77 0.71 0.65 0.42 0.27 0.18 0.12 0.08
0.10 0.91 0.83 0.75 0.68 0.62 0.39 0.24 0.15 0.09 0.06

Example: According the US Department of Energy data, the world production of crude oil increased from 60 million barrels per day in 1981 to 74 million barrels per day in 2001. What is the average rate of increase of the world’s crude over this period? Calculate the expected volume of crude in 2020 if the rate of production continues to increase at 1% per year.

Solution: We can think of the value in 2001 as the future worth of the value in 1981, i.e. F/P = 74/60 = 1.23. Consulting Table 1, we can see this corresponds to a rate of increase of about 1% per year over the past 20 years. The same table can be used to extrapolate future production from 2001 data. Interpolating for n = 19, we can find F/P = 1.21. The 2020 consumption of crude can be estimated as 74x1.21 = 90 million barrels. It is left to the student to verify these results by applying equations 1 or 2 instead of using tables.

Example: A new substitute to Middle Eastern oil is expected to be found in the next 20 years. The cost of production of energy equivalent to one barrel of oil is estimated at $100. What should the price be of a barrel of oil sold today in a competitive market? In a market with substantial monopoly owner? Solution: Assuming the market is efficient that no cost is associated with the extraction of oil from the Middle East, and that the cost of production remains the same, the oil should be priced at a value equal to its future price discounted to present time:

100x{(P/F)n=20, r=0.05} = 100x0.38 = $38. (1)

It is very difficult to predict the cost of future technologies with certainty. Monopolies usually price their commodities at the cost of substitutes today, which could be substantially higher than $38 at today’s prices.

Example: The US consumed 7.3 billion barrels of oil (bbo) in 2003, 2% more than it consumed in 2002 (1). Assuming the consumption continued to increase at the same 2% per year, how much petroleum would be left in US reserves at the beginning of 2006? The total US petroleum reserves were estimated to be 100 billion barrels at the start of 2003.

Solution: In 2004, we would be consuming 7.3x1.02 = 7.5 billion barrels of oil (bbo). The remaining reserves would be 92.5 bbo. In 2005, we would be consuming 7.5x1.02 = 7.6 bbo. The amount of oil that would remain at the end of 2005 is 84.9 bbo.

Contents

Doubling Time

Assuming a quantity grows in an exponential manner at a rate of r per year, the quantity grows after n years as given by equation 1. Of interest is the amount of time it takes for a present value to double for a given rate of growth, i.e. F/P = 2. The solution can be found by searching in the table of F/P factors for a given r values. For example, for a rate of 8% per year, the doubling time is 9 years. In general, it can be shown that doubling time is approximately calculated by dividing 72 by the percent interest rate (annual percentage growth rate). This approach is called the Rule of 72 and is expressed algebraically as:

T2 = 72 / R (2)

where T2 is the doubling time in years, and R is the annual percentage growth (R = 100 r) (a).

Example: How long does it take for a deposit of $5,000 in a saving account to yield a total of $20,000 in principal and interest? The bank pays 6% interest.

Solution: At the rate of 6% (r = 0.06), it takes 72/6 = 12 years to double the deposit. So the investment doubles to $10,000 after 12 years and doubles again to $20,000 after 24 years.

Cumulative Value of a Series of Future Payments

Also of interest is the cumulative value of a series of installments paid in equal intervals. For example, the total principal and interest paid on a home mortgage is the sum of all monthly payments over the course of a 30-year loan. Similarly, the cumulative value of energy consumption is the sum of energy used annually from the original reserves.

The analysis presented in the previous section can be used to forecast energy consumption from the present rate of consumption assuming a certain growth rate. Of greater consequence is the amount of energy remaining at a future point in time. We calculated the remaining US oil for the years prior to 2006. That procedure can be used until the oil reserves are depleted. A better way is to find a formal relationship for the cumulative consumption is by adding all future consumption by summing up the series:

C = P[1+(1+r)+(1+r)²)+(1+r)³)+….+(1+r)n-1] (3)

The mathematical series converges to give:

C / P =((1+r)n - 1)/r (4)

Equation 4 can also be interpreted as the future value of the stream of annual payments, accumulating interest compounded over time. The C/P factors are tabulated in Table 3.

Tablel 3. Cumulative Value of a Series of Future Payments
C/P=[(1+r)n -1]/r
r/n 1 3 4 5 10 15 20 25 30
0.01 1.00 2.01 3.03 4.06 5.10 10 16 22 28 35
0.02 1.00 2.02 3.06 4.12 5.20 11 17 24 32 41
0.03 1.00 2.03 3.09 4.18 5.31 11 19 27 36 48
0.04 1.00 2.04 3.12 4.25 42 12 20 30 42 56
0.05 1.00 2.05 3.15 4.31 5.53 13 22 33 48 66
0.06 1.00 2.06 3.18 4.37 5.64 13 23 37 55 79
0.07 1.00 2.07 3.21 4.44 5.75 14 25 41 63 94
0.08 1.00 2.08 3.25 4.51 5.87 14 27 46 73 113
0.09 1.00 2.09 3.28 4.57 5.98 15 29 51 85 136
0.10 1.00 2.10 3.31 4.64 6.11 16 32 57 98 164
0.15 1.00 2.15 3.47 4.99 6.74 20 48 102 213 435
0.20 1.00 2.20 3.64 5.37 7.44 26 72 187 472 1,182
0.25 1.00 2.25 3.81 5.77 8.21 33 110 343 1,055 3,227
0.30 1.00 2.30 3.99 6.19 9.04 43 167 630 2,349 8,730

Example: Given the data, calculate how long US petroleum reserves will last. Repeat the calculation for the total world petroleum reserves. The remaining world oil reserves are estimated at approximately 2,000 billion barrels; the current rate of consumption is 30 billion barrels annually and is expected to rise at a rate of 4% per year.

Solution: In this example, C is cumulative consumption equal to total US oil reserves in 2003 (C = 100 bbo) and P is the total oil consumed in 2003 (P = 7.3 bbo), so C/P = 100/7.3 = 113.7. Assuming that the annual rate of consumption continues to increase at 2% a year, we can refer to Table 3 to find 10 < n < 15 years. Interpolating, a better estimate will give a value of n ≈ 12.3 years. Similarly, for the world, C/P = 2,000/30 = 66.6. For r = 0.04, we can estimate we will have petroleum for another 40 years.

Annuities (Net Present Value)

Investors are interested in the return that their capital investment brings to them. It is also understood that the greater the risk they take, the higher their expectations for returns will be. The minimum that any investor expects as their rate of return is the interest that they could collect if they had deposited their investment into a bank or other financial institution. Depending on the perceived risk that different projects have and the tolerance of the potential investor to the loss of capital, different investors may expect different rates of return.

Investors do not necessarily put all the capital up front. Rather, it is common that they make a series of payments as the project proceeds. To make a wise decision on the profitability of the investment, all future revenues and costs must be discounted to the present. The discount rate is commonly chosen to be the expected rate of return.

The present value of a stream of annual payments or annuities, A, is calculated by summing Equation 2 over all future payments:

P = A [1 + 1/(1+r) + 1/(1+r)2+ ... + 1/(1+r)n]

The series converges to give:

P / A =((1+r)n - 1)/(r(1+r)n) (5)

The P/A factors are calculated and summarized in Table 4.

Example: A $10,000,000 lottery win can be awarded either over a 20 year period, by installments of $500,000 each, or as $5,000,000 in cash right away. Assuming the winner plans to deposit his money in a 20-year CD that pays 8% in interest, which option makes more sense?

Solution: The present value of the annual installments is {(P/A) n=20, r=0.08} = 10.59. The present worth of all future installments is $500,000x10.59 = $5,295,000 and is somewhat better than $5,000,000 in cash paid out today.

Example: The population of a large metropolitan area is expected to increase by 200,000 people within the next 5 years. To meet the future demand, a 200 MW wind farm is proposed for construction. The cost of construction is expected to be $5M in the first two years and an additional $2M for the following three years. The plant is expected to generate a net income (revenue minus maintenance costs) of $2M for 20 years after it is completed. Does this investment make economic sense? Assume an interest rate of 5%.

Solution: In the first five years that the plant is under construction, $16M are spent on construction. Starting year 6, the plant will generate revenues of $2M in annual income for the next 20 years after maintenance costs are deducted. A summary of the cost and income schedule is given below:

Year 1 2 3 4 5 6 7 25
Cost (M) 5 5 2 2 2 0 0 0 0
Income (M) 0 0 0 0 0 2 2 2 2

Please note that P/A factors assume that all payments are equal and start from year 1. As mentioned, there is no income for the first 5 years. To correct for this we subtract the net present value of income that would be generated for 25 years and subtract from it the lack of income opportunity for the first five years. Similarly, we can break down costs as annual cost of $2M for five years, in addition to annual costs of $3M a year for the first two years. This is best understood if the timeline table is rearranged as:

Year 1 2 3 4 5 6 7 25
Cost (M) 2 2 2 2
Cost (M) 3 3
Income (M) 2 2 2 2 2 2 2 2 2
Income (M) -2 -2 -2 -2 -2

The approach is to calculate P/A factor for a series of annual payments starting in year 1. Consulting Table 4, the net present values of income and expenses are:

NPV(income) = [2(P/A, 0.05, 25) - 2(P/A,0.05,5)] = 2(14.09)–2(4.33)=9.52 M

NPV (cost) = [3(P/A,0.05,2) + 2 (P/A,0.05,5)] = 3(1.86) + 2(4.33) = 14.24 M

NPV (income) / NPV (costs) = 19.52/14.24 = 1.37 ; thus the investment makes sense!

Capital Recovery Factor

The capital recovery factor (annualized cost) represents the series of annual payments to pay off a loan. This is the reverse of the previous problem -- we know the present worth of all future payments (i.e. the loan principal). The result is:

A/P = (r(1+r)n)/((1+r)n-1) (6)

A/P factors are tabulated in Table 5.

Example: What is the annual payment required to pay a $250,000 loan at 7% over 15 years?

Solution: Annual cost = loan amount x [(A/P)r=0.07, n=15] = ($250,000) x 0.11 = $27,500

Internal Rate of Return

A convenient way to evaluate the economic benefit of a project is to calculate the effective rate of return of all future transactions. Since payments and revenues take place at different times, a proper method of evaluation is to discount them to present value. The internal rate of return is computed simply by equating the NPVs for all future costs and incomes. NPV (cost) = NPV (income)

Table 4. Net Present Value of a Series of Future
Installments (Annuities)
P/A=[(1+r)n-1]/[r(1+r)n]
r/n 1 2 3 4 5 10 15 20 25 30
0.01 0.99 1.97 2.94 3.90 4.85 9.47 13.87 18.05 22.02 25.81
0.02 0.98 1.94 2.88 3.81 4.71 8.98 12.85 16.35 19.52 22.40
0.03 0.97 1.91 2.83 3.72 4.58 8.53 11.94 14.88 17.41 19.60
0.04 0.96 1.89 2.78 3.63 4.45 8.11 11.12 13.59 15.62 17.29
0.05 0.95 1.86 2.72 3.55 4.33 7.72 10.38 12.46 14.09 15.37
0.06 0.94 1.83 2.67 3.47 4.21 7.36 9.71 11.47 12.78 13.76
0.07 0.93 1.81 2.62 3.39 4.10 7.02 9.11 10.59 11.65 12.41
0.08 0.93 1.78 2.58 3.31 3.99 6.71 8.56 9.82 10.67 11.26
0.09 0.92 1.76 2.53 3.24 3.89 6.42 8.06 9.13 9.82 10.27
0.10 0.91 1.74 2.49 3.17 3.79 6.14 7.61 8.51 9.08 9.43
0.15 0.87 1.63 2.28 2.85 3.35 5.02 5.85 6.26 6.46 6.57
0.20 0.83 1.53 2.11 2.59 2.99 4.19 4.68 4.87 4.95 4.98
0.25 0.80 1.44 1.95 2.36 2.69 3.57 3.86 3.95 3.98 4.00
Table 5. Capital recovery factor
A/P=[r(1+r)n]/[(1+r)n-1]
r/n 1 2 3 4 5 10 15 20 25 30
0.01 1.01 0.508 0.340 0.256 0.206 0.106 0.072 0.055 0.045 0.039
0.02 1.02 0.515 0.347 0.262 0.212 0.111 0.078 0.061 0.051 0.045
0.03 1.03 0.524 0.353 0.269 0.218 0.117 0.084 0.067 0.990 0.051
0.04 1.04 0.529 0.360 0.275 0.225 0.123 0.090 0.074 0.064 0.058
0.05 1.05 0.538 0.368 0.282 0.231 0.130 0.096 0.080 0.071 0.065
0.06 1.06 0.546 0.375 0.288 0.238 0.136 0.103 0.087 0.078 0.073
0.07 1.08 0.552 0.382 0.295 0.244 0.142 0.110 0.094 0.086 0.081
0.08 1.08 0.562 0.388 0.302 0.251 0.149 0.117 0.102 0.094 0.089
0.09 1.09 0.568 0.395 0.309 0.257 0.156 0.124 0.110 0.102 0.097
0.10 1.10 0.575 0.402 0.315 0.264 0.163 0.131 0.118 0.110 0.106
0.15 1.15 0.613 0.439 0.351 0.299 0.199 0.171 0.160 0.155 0.152
0.20 1.20 0.654 0.474 0.386 0.334 0.239 0.214 0.205 0.202 0.201
0.25 1.25 0.694 0.513 0.424 0.372 0.280 0.259 0.253 0.251 0.250
0.30 1.30 0.735 0.549 0.461 0.410 0.324 0.306 0.301 0.300 0.300

Example: An investor is considering investing in the wind farm project described above. The investor also has an opportunity to purchase a government bond which is expected to earn a 10% return. Which option would be advantageous?

Solution: To have a fair basis for comparison, the investor should compare the internal rate of return for the project and the rate of return from the bond. The internal rate of return can be calculated by equating the NPVs for income and cost.

NPV (income) = NPV (cost)

2(P/A, r, 25) - 2(P/A, r, 5) = 3(P/A, r, 2) + 2(P/A, r, 5),

or

Δ (NPV) = 2(P/A, r, 25)-4(P/A, r, 5) -3(P/A, r, 2) = 0

The problem must be solved by trial and error. Assume a rate of return and evaluate the equation above until it converges.

r Δ (NPV)
0.07 2(11.65) -4(4.10) -3(1.81) = 1.47
0.08 2(10.67) -4(3.99) -3(1.78) = 0.04
0.09 2(9.82) -4(3.89) -3(1.76) = -1.20

The effective rate of return for investing in the wind farm project is about 8%, which is less than the 10% that the bond would earn. Although the investment in the bond is a better choice financially, an environmentally contentious investor may still consider the investment in the wind farm.

References

(1) US DoT, http://www.bts.gov/publications/national_transportation_statistics/2003/html/table_04_01.html

(2) Toossi Reza, "Energy and the Environment:Sources, technologies, and impacts", Verve Publishers, 2005

Additional Comments

(a) In this example, we have assumed that the interest rate is compounded yearly. If the interest rates were compounded monthly, or even better, daily, then our money would have doubled in a shorter time. Taken to the extreme, exponential growth would occur on a continuous basis as it does naturally in population, energy consumption, etc., further shortening the doubling time, i.e. N = N0e-rt (See equation 1, Appendix B). The doubling time is found by substituting N/N0 = 2

Further Reading

Colander, D. C., Economics, 3rd E., Irwin-McGraw-Hill, 1998.

Bosselman, F., Energy, Economics and the Environment, Second Edition, Foundation Press, 2005.

Energy Economics, Science Direct Elsevier Publishing Company. Publishes research papers concerned with the economic and econometric modeling and analysis of energy systems and issues.

External Links

United States Association for Energy Economics (http://www.usaee.org).

International Monetary Fund (http://www.imf.org).

The World Bank (http://www.worldbank.com).