This is a question about the CRF formula used in TIMES/MARKAL. I had assumed that it was the same as the formula that I've always used but in reality, it is quite different and I'm hoping someone can shed some light on why that is.

The formula that I know is:

(1) CRF = (r*(1+r)^n) / ((1+r)^n -1)

where r is the interest rate and n is the period length (lifetime)

This formula (I'll call CRF1) is what is used in mortgage calculations and is used in the PMT function in MS excel for calculating loan payments.

The formula in the TIMES/MARKAL documentation is:

(2) CRF = (1/(1+r))*r/(1-(1+r)^-n)

I've never seen this formula (CRF2) before but I assume that the developers are much smarter than me and there's a good reason for this other formula.

I've made a graph that shows the value of CRF using equations (1) and (2) for different values of the discount (i.e. interest) rate (and assuming a lifetime of 15 years). CRF(1) asymptotes to the discount rate (dashed line) as the discount rate increases. Whereas CRF(2) increases with discount rate but falls below the discount rate around 20% discount rate. This seems very strange to me as I imagine that there would be a loan where the annual payment (as fraction of inital capital) would be below the interest rate.

In our model we introduce high technology specific discount rates for some technologies in order to account for risk and other non-economic reasons why adoption of this technology wouldn't just be on an economic cost basis (i.e. a hurdle rate). Sometimes they are as high as 50%. In this case we can see there is a significant difference in CRF calculated using equations 1 and 2. In fact, I calculated the difference between the two formulas as 1/(1+r) so when r is small, there are very similar but when r is large (e.g. 0.5), we get a big difference (CRF2 is 2/3 the value of CRF1).

Any help on this matter woudl be greatly appreciated. Thanks. . .

Chris