We've run over time yet again, and Mr Brennan will have to return another day after his brief stint this afternoon went unfinished.

"As is becoming rather clear, this takes a bit longer than everybody anticipated," Sir Patrick comments wearily.

Everyone heads for the car park, ready to return tomorrow for a session with CEPA, the consultancy that prepared an economic appraisal in 2011 - join us at 10:30 GMT.

Have a great evening.

With Ms McAfee's evidence at an end, Mike Brennan, who briefly gave evidence before lunchtime, returns to the Senate chamber, and Joseph Aiken replaces Mr Scoffield in posing the questions.

If you've joined us since then, you'll need to know that Mr Brennan is from the Department of Finance and heads its central expenditure division.

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He says that the department would have taken "a radically different approach" to assessing the business case for the RHI scheme if it had known what it does now.

Reiterating an earlier point, he says the approval of the case only lasted until March 2015 and there had been a request for a review of the RHI scheme in 2014, which made for a "safety net".

But he adds that the safety net became "completely and utterly irrelevant".

Dr MacLean asks about civil servants managing the work of consultants.

"You don't need to be trained as an economist to pick up a lot of the problems," he says.

He wonders whether there is a "reluctance to test any of the numbers because it's been done by the magic spreadsheet" rather than just working out whether or not they roughly make sense.

Image source, RHI Inquiry

Ms McAfee says that in some cases the work of consultants can be very detailed.

Dr MacLean says the consultants appear to have entered the wrong numbers into the table, which was then used in the business case.

When he looked at the tables, he says the first thing he did was to get his calculator out "and I couldn't get the number that they said was the answer".

Ms McAfee says she would have been checking the basic arithmetic: "If there's maybe a calculation that doesn't make sense in a table but ultimately it doesn't impact on the overall position, it might not be something that I would raise in a note."

Looking at Ms McAfee's witness statement regarding the CEPA report and the business case and their reasons for not including tiering of tariffs, Mr Scoffield makes an observation.

"You know now that the rationale for not including tiering appears to be contradictory - that's not something you identified at the time?"

"No," Ms McAfee replies.

It's well known by now that the tariff on offer in the RHI scheme was higher than the cost of the fuel used to power the biomass boilers that claimants installed through the initiative.

That flaw meant the scheme was wildly overgenerous, was one of the key reasons why the scheme went so badly wrong, and, as we heard yesterday, it was clearly laid out in the business case.

Sir Patrick questions Ms McAfee as to why she, in her role with DFP - "the public spending guards" - did not raise issue with it when she reviewed the case.

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"I can only say that it's an oversight," says Ms McAfee.

Dr Keith MacLean, the adviser to the inquiry, asks whether Ms McAfee reviewed the "basic arithmetic".

"I'm just interested whether it would be par for the course to actually check the basic numbers in there as part of the challenge function and the kicking of the tyres," he adds.

She says she would have looked at that but "wouldn't have appreciated the technical detail" of the model used to calculate the tariff.

Mr Scoffield takes Ms McAfee through some aspects of the CEPA report, including comparisons of the costs of two potential options for the RHI - a continuing subsidy scheme and a challenge fund option, with grants awarded up front in a competitive application process.

"The challenge fund was well over £200m cheaper over the life of the scheme than the option which DETI was recommending," he says.

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He asks Ms McAfee if she appreciated this when she was considering value for money.

She say she does not recall in detail, but she would have seen it in the report.

The value of the finance department's role in business case reviews is questioned by Mr Scoffield, given that some officials involved in the process have said they do not have sufficient expertise to challenge some of what is contained in the documents they assess.

Given that individual departments should have the specific knowledge in the areas in which they work, he asks what is added by the finance department's review.

Ms McAfee says she believes that the function of her branch is to look at the business cases "from a different perspective... from a fresh set of eyes".

A review of the RHI scheme that was planned for 2014 was "crucial", says Ms McAfee.

Mention was made of it in DETI's business case for the initiative, but ultimately it did not happen, and that has been identified as one of the critical mistakes in the scheme's lifespan.

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Mr Scoffield asks if the original plan for review contained within the business case would mean she would take a "lighter touch" in her scrutiny of it because problems could be spotted further down the line.

Ms McAfee says her level of examination would be the same whether or not a review was planned, but she would have "taken comfort" from the assurance that it would happen.

Had one not been planned, she says she would have asked for one.

Mr Scoffield asks Ms McAfee about RHI tariff calculations in the business case, quoting an email she sent making reference to certain parts of the business case.

"The method of calculating the tariff rates is the primary difference between the NI and GB schemes," she had written.

Image source, RHI Inquiry

Mr Scoffield asks what Ms McAfee what she meant by the comment.

She replies that it was evident that there was a difference in what was being presented in the two schemes, "particularly in terms of tiering".

There was no tiering planned for the Northern Ireland scheme, which was a major factor that ultimately led to the project's overspend.

The questions Ms McAfee will be asked will focus on her role in approving the business case for the RHI scheme, and you can read her written witness statement over at the inquiry's website, external.

She explains that the finance department's economic appraisal branch, of which she is a part, provides economic advice on business cases and training for other departments on how to prepare them.

Image source, RHI Inquiry

Mr Scoffield rewinds to April 2012, when the RHI case was reviewed and approved by DFP.

Ms McAfee outlines her role, saying that she would check whether there is sufficient information within a case to justify the selection of a particular project and prove its value for money.

Back in the Senate chamber, the inquiry proceedings resume, and there's been a change of plan - Mr Brennan, who was being questioned before lunch, isn't in the witness chair.

Instead, Rachel McAfee, a Department of Finance economist who was involved in the approval of the business case for the RHI scheme in 2012, is here to take question, and she takes the oath.

There's also a change of counsel - Mr Aiken is gone, and senior counsel David Scoffield QC will be asking the questions.

Speaking of Mr Scoffield, here's an interesting tweet from some of his school pals...

The inquiry pauses there for lunchtime, so we'll get a quick cuppa tea and a stretch of the legs, and we'll be back at 14:00.

Mr Aiken asks if, even though there were unusual features to the funding set up for the Norther Ireland RHI scheme, any steps were taken to establish the Treasury "why this funding was in the form it was being described?"

"I dont know that there were specific questions or conversations about the new type of AME," Mr Brennan replies.

The type of funding formula for the RHI scheme was not appropriate, says Mr Brennan.

That is because the initiative did not have a "demand cap" - an essential feature to control AME spending - when it was set up.

But Mr Aiken notes that Stormont's finance department did not raise that issue when it gave permission to DETI to run the scheme.

Image source, RHI Inquiry

When DFP looked at DETI's business case for the scheme, it imposed two conditions, says Mr Brennan - the approval was only to last until March 2015 and DETI was to review the scheme in 2014.

Inquiry chair Sir Patrick Coghlin says he's finding it "difficult" to understand why the finance department - "the department responsible for spending public money" - did not ask more questions.

Mr Brennan says DFP took "comfort" that DETI's top civil servant had already rubber stamped the business case.

Mr Brennan explains that it is "incredibly difficult" to control AME spending because it is typically demand-led.

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He gives the example of welfare payments saying that initial forecasts could end up being wide of the mark if the economy changed dramatically and unemployment increased.

"Therefore welfare payments would go up," he said, adding that AME spending is "highly volatile, highly variable".

Mr Aiken draws Mr Brennan's attention to the settlement letter from the Treasury regarding funding for the NI RHI scheme, and asks whether the content of the letter "set off any alarm bells " for Mr Brennan.

"No," he replies.

Mr Aiken turns to the similar settlement letter sent by the Treasury regarding funding for the GB RHI scheme.

Image source, RHI Inquiry

It included the warning that "a condition of this is that DECC will need to contribute towards any overspend".

Mr Aiken asks Mr Brennan if such a sentence had appeared in the Northern Ireland letter: "What do you anticipate your reaction would have been, and why?"

"An alarm bell would have gone off instantly because the primary concern would have been to protect the integrity of the DEL block," says Mr Brennan.

The AME arrangement that was put in place for funding the RHI scheme had not been seen in Northern Ireland before, says Mr Brennan.

"There would've been no awareness, no previous expertise of its treatment," he adds.

Image source, RHI Inquiry

The Department of Finance and Personnel (DFP) issued guidance to the other Stormont departments in April 2011 saying that there was a "need to exercise caution" when it came to understanding how it operated.

He confirms that the way in which the RHI scheme was funded and operated was a new concept to Northern Ireland.

"I'd never come across anything like that before."

Up steps Department of Finance official Mike Brennan, the second witness to appear before the inquiry, to be questioned by Mr Aiken.

Image source, RHI Inquiry

Mr Brennan, the head of the department's central expenditure division, takes the oath.

You can read his written witness statement on the inquiry's website, external.

Mr Aiken outlines the "parity principle" as it applies to AME.

Under the Barnett formula, government funding remains in proportion to that enjoyed in GB, but this does not apply with AME funding.

"The fact that there's an AME programme in GB applying this policy, it ought not to mean on the face of it that Northern Ireland was entitled to say: 'Well, you've got an AME for this, we're entitled to our Barnett consequential of it,'" he explains.

Mr Aiken says that the DfE (Department for the Economy, formerly DETI) has explained to the inquiry "that it did not intend to have a scheme that was more generous than its GB equivalent".