Pricing Energy and Energy Efficiency
Energy prices (volumetric) are usually listed in dimensions of money per energy. Gas costs about 5 USD per gallon and electricity about 0.30 USD per kWh.
To compute the value of an energy efficiency measure, we need to estimate the amount of energy saved and divide by the amount of money it costs to save that energy.
Often, we must take out a loan to pay for an energy efficiency measure like an insulation remodel or all new lightbulbs.
The equivalent energy price is the yearly cost of the loan divided by the yearly amount of energy saved.
We compute this yearly (or monthly) cost using the Capital Recovery Factor.
Capital Recovery Factor
Suppose we make a loan.
The way a loan is calculated is that the present value of all the discounted future payments is equal to the loan amount.
That is
\textrm{Loan Amount} = \sum_{n=1}^{N} \frac{\textrm{Payment}}{(1+i)^n}
Where i is the interest rate, and N is the total number of payments.
Note that the present value of each payment get smaller as the payments are farther out, even though the same amount is paid off each period.
This formula allows us to calculate this payment.
\textrm{CRF} = \frac{\textrm{Payment}}{\textrm{Loan Amount}} = \frac {i(1+i)^n}{(1+i)^n-1}
Monthly vs. Yearly Payments
- Many types of loans advertise a yearly interest rate, but require monthly payments and charge interest monthly.
- The yearly interest rate is the APR or annual percentage rate
- To find the monthly rate divide this by twelve
- i_y is the annual percentage rate
- n_y is the number of periods in years
The monthly numbers (i_m and n_m) are:
i_{m} = i_{y}/12 n_{m} = n_{y}\cdot 12
Cash Flow
This graph shows the value of the loan and payments at the time they were made. The sum of these loan payments will exceed the loan amount.
This graph shows the value of the payments at the time of the loan (the present). The sum of these loan payment present values will equal the loan amount and value.
Calculator syntax
Which of these is will be evaluated correctly by a calculator or spreadsheet?
\textrm{CRF} = \frac {i(1+i)^n}{(1+i)^n-1}
a) i*(1+i)^n/(1+i)^n-1
b) (i*(1+i)^n)/(1+i)^n-1
c) i*(1+i)^n/((1+i)^n-1)
d) i*(1+i)^n/((1+i)^n)-1