Finance

Learning Objectives

• You will understand the basic concepts of the time value of money

• You will be able to calculate how financial instruments like loans and

  credit cards work

Feedback on the reading

Review

• Ideal circuit elements

• Kirchoff's Laws

• Equivalent resistors

Concepts

• Ideal bank

• Comparsion principle

Connection to energy efficiency

• Many energy projects are made attractive to consumers by providing

  financing so that people don't have to pay the entire cost at once.

Informal poll

Would you rather have

• $1K now or $500 in a year?

• $1K now or $1K in a year?

• $1K now or $2K in a year?

• $1K now or $5K in a year?

• $1K now or $10K in a year?

Equivalence principle

• Given a choice between money now and money later, most demand a larger

  value if money is provided at a later date

Loans

• With a loan we essentially rent money and pay a percentage of the

  amount we still owe the lender

Discount rate based on equivalence principle

• Banks have preferences that set their loan rates

• People have preferences that dictate their discount rate

Discount Rate and Net Present Value

$\textrm{Present Value (USD)} = \frac {\textrm{Future Amount (USD)}} {(1 + \textrm{Discount Rate})^{\textrm{number of years}} }$

$P = \frac {F} {(1 + i)^{n} }$

Learning Objectives

• You will be able to use common financial methods (NPV, IRR, CRF) to compare energy

• Capital Recovery Function (CRF)

• Net Present Value (NPV)

• Internal Rate of Return (IRR)

Guiding Questions

• How do we compare two options with different costs over time?

Two cash flows

Discount Rate and Net Present Value

$FV = PV (1 + i)^{n}$

$PV = \frac {FV} {(1 + i)^{n} }$

$\textrm{Present Value (USD)} = \frac {\textrm{Future Amount (USD)}} {(1 + \textrm{Rate})^{\textrm{number of years}} }$

Activity

• What is the present value of a $1000 payment 5 years from now with an

  interest rate of 10%?

Strategy

Activity

• What is the future value of $1000 payment today in 10 years at 7%

  interest?

Strategy

Net Present Value (NPV)

Net Present Value

• Provides a shorthand to calculate the present value of a stream of

  payments.

• Provides a method of comparison for two different cash flows

• Cash flows can be compared for equivalence

Cash flow

• We think of discrete events in time where cash is paid or recieved

Two options

Present Value

Single payment $PV = \frac{C}{(1+i)^n}$

Stream of payments $PV = C_0 + \frac{C_1}{1+i} + \frac{C_2}{(1+i)^2} + \dots + \frac{C_N}{(1+i)^N}$

Compact notation $PV = \sum_{n=0}^{N} \frac{C_n}{(1+i)^n}$

Two options

Internal Rate of Return (IRR)

Internal Rate of Return

• Tells us at what interest rate a cash flow has a net present value of

  zero

• We will look at this on a spreadsheet

Capital Recovery Function

Capital Recovery Function

Tells you how much the loan payment is for a given balance, number of payments, and interest rate.

Capital Recovery Factor

$CRF = \frac {i(1+i)^n}{(1+i)^n-1}$

Additional Metrics for Energy Projects

• Cost of conserved energy

• Cost of avoided carbon

Spreadsheet functions

• NPV(rate, payments) net present value

• IRR(payments, guess) internal rate of return

• PMT(rate, number of payments, principal)

Learning Objectives

• Use conserved cost of energy as an investment metric

Activity Objective

• Be able to use NPV, IRR, and CCE on calculator and computer

Review

• How did we determine the cost of a loan?

• What were our methods?

Capital Recovery Factor (CRF)

• Allows for the calculation of the annual or monthly payment to repay a

  loan

$CRF = \frac {i(1+i)^n}{(1+i)^n-1}$

Monthly payments

If payments are monthly, we divide the annual percentage rate (APR) by 12 and multiply the number of years ($n$) by 12.

$CRF = \frac {i(1+i/12)^{12n}}{(1+i/12)^{12n}-1}$

Exercise

• What is the annual payment for a $10K loan with an interest rate of 5%

  paid over a period of 7 years?

Solution

Exercise

• What is the CRF for a 10 year loan at 0% interest?

Solution

• 0.10 or 10%

Cost of conserved energy

• Allows you to calculate the value of an investment as a per kWh cost

• This allows easy comparison with the current or plausible future cost

  of electricity

CCE (Meier 1984)

$$ CCE = \frac{\textrm{Investment}}{\textrm{Annual Energy Savings}} \cdot \frac{d}{1-(1+d)^{-n}}

## Interpreting the CCE $$ CCE = \frac{\textrm{Investment}}{\textrm{Annual Energy Savings}} \cdot \frac{i}{1-(1+i)^{-n}} \cdot \frac{(1+i)^n}{(1+i)^n}

$$ CCE = \frac{\textrm{Investment}}{\textrm{Annual Energy Savings}} \cdot \frac{i(1+i)^n}{(1+i)^n - 1}

$$ CCE = \frac{\textrm{Investment}\cdot\textrm{CRF}}{\textrm{Annual Energy Savings}}

Cost of avoided carbon

• Allows you to calculate the value of an investment as the cost per

  amount of carbon not emitted

• Allows comparison with the current market price of carbon