Economic Analysis of Energy Investments

Abstract

This unit introduces the concept of economic analysis of an investment project and highlights the differences between the economic and financial analyses of investments. It also introduces the commonly used indicators for project investment analysis and the basics of risk analysis.

Annex 7.1: Some Commonly Used Interest Formulae

1.1 Single Compound Amount Formula

If P dollars are deposited now in an account earning i% per period for N periods, then

$$ F = P(1 + i)^{N} $$

(7.6)

This expression is used to move any amount forward in time.

Example

A firm borrows $1,000 for 5 years. How much must it repay in a lump sum at the end of the fifth year? Assume interest rate is 5%.

With P = 1,000, i = 0.05, N = 5; F = $1276.28

1.2 Single Present-Worth Formula

The present worth P of a sum F which would be available N periods in the future is given by

$$ P = F\left[ {{\frac{1}{{(1 + i)^{N} }}}} \right] $$

(7.7)

Example

A company desires to have $1,000 8 years from now. What amount is needed now to provide for it, if interest rate is 5%?

Answer

F = 1,000, i = 0.05, N = 8; Hence the single payment present worth factor is 0.67684. The amount of investment required now is $676.84.

If the cash flow in the series has the same value, the series is called uniform series.

If the value of a cash flow varies by a constant amount G from the previous period, the series is called gradient series.

If the value of a given cash flow differs from the value of the previous period by a constant %, the series is called a geometric series.

Closed form expressions are available for these categories.

1.3 Uniform Series Compound Amount

If a uniform amount A, called annuity is deposited at the end of each period for N periods in an account earning i% per period, the future sum at the end of N periods is

$$ F = A[1 + (1 + i) + (1 + i)^{2} + \cdots + (1 + i)^{N - 1}] $$

(7.8)

$$ F = A\left[ {{\frac{{(1 + i)^{N} - 1}}{i}}} \right] $$

(7.9)

The expression $\left[ {{\frac{{(1 + i)^{N} - 1}}{i}}} \right]$is called uniform series compound amount factor.

Example

If 4 annual deposits of $2,000 each are placed in an account, how much money has accumulated immediately after the last deposit, if the rate of interest is 5%?

Answer

With N = 4, i = 0.05, the uniform series compound amount factor is 4.310. The future amount would be 2,000 × 4.310 = $8,620.

1.4 Uniform Sinking Fund Formula

A fund established to accumulate a desired future amount of money at the end of a given length of time through the collection of a uniform series of payments is called a sinking fund.

If F is the total amount at the end of N periods, the annuity A that has to be paid is given by

$$ A = F\left[ {{\frac{i}{{(1 + i)^{N} - 1}}}} \right] $$

(7.10)

The expression $\left[ {{\frac{i}{{(1 + i)^{N} - 1}}}} \right]$ is called sinking fund factor.

Example

How much should be deposited each year in an account in order to accumulate $10,000 at the time of the fifth annual deposit? Assume an interest rate of 5%.

Answer

The sinking fund factor is 0.180975. The annuity requirement is $1809.75 (or $1,810).

1.5 Uniform Capital Recovery Formula

This calculates the amount of annuity required to accumulate to a given present investment P, with given interest rates and number of periods.

Substituting F = P(1 + i)^N in Eq. 7.9 gives

$$ A = P\left[ {{\frac{{i(1 + i)^{N} }}{{(1 + i)^{N} - 1}}}} \right] $$

(7.11)

The expression $\left[ {{\frac{{i(1 + i)^{N} }}{{(1 + i)^{N} - 1}}}} \right]$ is called the capital recovery factor.

Example

What is the size of 10 equal annual payments to repay a loan of $1,000? First payment is 1 year after receiving loan. Interest on loan is 5%/year.

Answer

Here N = 10, i = 5%, Hence the capital recovery factor is 0.1295. Hence, the annuity required is $129.5 at the end of each year.

1.6 Uniform Series Present Worth Formula

The present worth of a series of uniform end-of-period payments is given by

$$ P = A\left[ {{\frac{{(1 + i)^{N} - 1}}{{i(1 + i)^{N} }}}} \right]. $$

(7.12)

The expression $\left[ {{\frac{{(1 + i)^{N} - 1}}{{i(1 + i)^{N} }}}} \right]$ is known as uniform series present worth factor.

Example

How much should be deposited in a fund to provide for 5 annual withdrawals of $100 each? First withdrawal is 1 year after deposit. Assume an interest rate of 5%.

Answer

With N = 5 and i = 0.05, the present worth factor is 0.43295. Hence, the amount required is $432.95 (Table 7.13).

Table 7.13 Summary of interest formulas for uniform series