Help with maths

lisyloo

I want to make sure my logic here is correct - because it's important.

I am comparing an investment where you invest £8.8K for 20 years and leave it there with a Solar PV install which costs £8.8K and is written off at the end of the 20 years.

The expected money back from the government scheme plus electricity savings over the 20 years assuming 3% RPI is £27K

The installer is calculating the return as
(27-8.8)/8.8/20 = 11.07%
I'm sure this is wrong because it doesn't compound and any alternative (cash or S&S) would compound.

I just want to make sure I'm right with:
27/8.8 ^ (1/20) -1 = 6.01%

Can anyone confirm?

Stochasticity

It's more complex to calculate as the benefit of the £27,000 is spread out over the 20 years, rather than being received as a lump sum at the end.

Assuming the £27,000 comprises twenty annual payments which increase steadily in line with inflation at 3%, the annual return is something like 12.4% (before inflation).

lisyloo

Thanks.
Wow - I've missed somethng major there.

bowlhead99

*edit* Agreed with Stochasticity!

A couple of things:

1) With your maths, are you getting your % returns by doing your calcs on their real projection figures (i.e. unrounded) but simplifying using rounded figures when showing us?

e.g. 27/8.8 ^ (1/20) -1 = 6.01% according to you but my calculator gives more like 5.76%.

Similarly the other method (27-8.8)/8.8/20 = 11.07% per you, but I get 10.34%

2) As I don't have the actual unrounded figures I'm just going to go with the simple round ones in my maths. Anyway regardless of that, the question is, which is more of an appropriate method. I think actually their method is closer than yours.

Your way of doing it is making an assumption that you have put in 8.8k today and get back 27k on the last day of the project 20 years later. It's true that if those were the actual cashflows, you could say: "Well, the 27 I get back at the end of 20 years is about three times the 8.8 I invested at the beginning. Waiting 20 years for a 200% return is something like 5.7% compounded."

I agree that if you paid 8.8k into an account that paid a 5.7% return for 20 years you would end up with 27k to walk away with at the end. However, that's presumably not how it works. You are getting a payout (whether a saving on your energy bills, or a cash return from selling spare electricity or whatever), each and every year, and that return rises over time with inflation as the price of electricity rises. Potentially you would get something like (my simple assumptions to make maths easier):

Yr 0: (8,800) your spend
Yr 1: 1,000 your saving
Yr 2: 1,030 (your 1000 saving plus inflation)
Yr 3: 1,061 (1000 plus inflation plus inflation)
Yr 4: 1,091 (etc., etc.)

And so on until you get to Year 20 when you receive the last cashflow in (or electricity bill saving) of 1,750 and you've taken total payouts or savings of 26.9k during years 1-20, against your original 8.8k spend at year 0.

That is a completely different scenario from what led you to calculate that the return was only 6%ish. Your 6%ish came from the maths that assumes a model like:

Yr 0: (8,800) your spend
Yr 1: 0 your saving
Yr 2: 0 your saving
Yr 3: 0 your saving
Yr 4: 0
...etc...
Year 20: 27,000 your saving

That second scenario doesn't seem to mirror reality very closely. If you are getting cash each year, you can do something with that. For example the 1000 back in year 1 can be immediately invested into something else (even if you just put it into a bank account at 3% it would turn into 1750 over the remaining 19 year time horizon). Same for the year 2 cashflow and so on. So you need to give them credit for getting the cash back to you quicker than waiting to year 20.

If you have access to an Excel spreadsheet you can enter the cashflows (-8800,+1000,+1030,+1061 etc) and use the IRR or XIRR function to tell you the internal rate of return. Doing this gives me 12.4% - much closer to their presentation of it, than to yours.

Of course, it might be that their model of getting you your 27k total returns over the life of the project is already assuming that you take the annual savings or inflows and put them in a bank or other investment that generates a return that equals or betters inflation? Perhaps they're assuming the annual savings or cash generated is quite a lot less than the 1k(and growing) each year that I assumed, but you then get a return off that cashflow you have banked to gross up the total cash received over the life of the project to a 27k lump sum in year 27. If that were the case, your 'lump sum' method estimate of 6% would be the way to do it. But it would be sneaky on their part.

It should be pretty obvious how they have done it; if you can get an annual breakdown of their assumed cashflows you can use IRR formulae in excel like I suggested.

So, it might well be that you are doing them a disservice and actually the investment opportunity is a genuine double digit return over the course of 20 years. Of course, they have a vested interest in showing it in a good light. So you need to consider how the returns might vary:

What might it cost to maintain the panels? Or to replace them when they take damage or fall off your house in a 'once or twice in 20 years' storm? What do you make on selling off your unused electricity - is it guaranteed? Are the assumptions about hours of sunlight in line with your area's actuals over the last 10-20 years, or are they a national company who've taken an average over the length of the British Isles from Jersey to Inverness? Could anything reduce your income or savings, or increase your cost, due to changes in government regulation of the energy market? Have you leased your roof to the installation company who requires anyone buying your house to keep fulfilling a contract and reduces the market value of your property? Etc Etc.

alanq

Has the cost of replacing the inverter been factored in?

"Inverters are an essential part of your PV system. They convert what the panels gather from the sun and convert it into usable energy. An inverter will need to be changed or repaired at least once in the life of the solar panels and they're around £1000 each. Factor this in to any costing. However, your installer may offer an extended warranty which covers this cost."

http://local.which.co.uk/advice/solar-pv-house-type-shade-roof-angle-direction

I wouldn't bank on the installer/manufacturer being around to honour the warranty in 20 years time.

Also you need to take account whether your house will gain or lose in value having had solar panels fitted.

Rollinghome

alanq wrote: »

Also you need to take account whether your house will gain or lose in value having had solar panels fitted.

That's arguably the most important factor and the most difficult to calculate. We've got a south-facing pitched roof but, whatever the sums, wouldn't want panels on our roof, or on neighbouring roofs, and would be a major downer on any house I wanted to buy. The higher the value of the house then the more likely the possible loss of value will outweigh any other gain.

Because I wouldn't consider them I haven't bothered looking into them but the other point that occurred to me is how far will the technology move on over the next 20 years? Will they become smaller and more efficient and by 2034 will those of 2014 look as trendy as a 1980's mobile?

bigadaj

As well as calculating the internal rate of return if you wanted to be thorough then running a sensitivity analysis on the input numbers would be useful. For example varying the inflation rate, the cost/ benefit of the electricity or some degradation in output efficiency of the panels themselves over time would allow a range of returns to be generated. This would then give you a good handle on the potential range and likely best and worst scenarios might be, though it would still be by no means guaranteed.

lisyloo

Hi bowlhead

by doing your calcs on their real projection figures (i.e. unrounded) but simplifying using rounded figures when showing us?

Yes, correct that would be why it's not exact.

You are getting a payout ... each and every year

I believe the payouts (FIT and export) are quarterly and the energy savings are effectively monthly if you think of it as a deduction from the monthly Direct debit.
And you are quite right - we have it wrong, because with the solar we have money coming back that could be invested elsewhere.

What might it cost to maintain the panels?

Impossible to say so I agree that's a risk.
They should last 25 years.
Manufacturing and workmanship defaults should be covered by a warranty, but there could still be costs of labour/scaffolding which aren't covered by warranties.

Or to replace them when they take damage or fall off your house in a 'once or twice in 20 years' storm?

Storm damage should be covered by home insurance. There would be the usual costs of an insurance claim such as excess, uplift in premium, loss of no claims etc.

What do you make on selling off your unused electricity - is it guaranteed?

There is an export tarriff. Its "guaranteed" for 20 years and rises in line with RPI if you believe that politicians will keep their promises.

Are the assumptions about hours of sunlight in line with your area's actuals over the last 10-20 years, or are they a national company who've taken an average over the length of the British Isles from Jersey to Inverness?

They are specific to the local area and conservative compared with real life situations I've checked.

Could anything reduce your income or savings, or increase your cost, due to changes in government regulation of the energy market?

The politicians might not keep their promises.
The sun might not shine enough.
Things could break.
Inflation could be worse or better than expected.
Energy price increases could be higher or lower than expected.

Have you leased your roof to the installation company who requires anyone buying your house to keep fulfilling a contract and reduces the market value of your property?

No we're buying the panels oursleves.
Any buyer would benefit from cheques from the export/FIT tarriff and reductions in energy costs.
I appreciate that still doesn't mean everyone will like them.

lisyloo

Hi Alan

Has the cost of replacing the inverter been factored in?

Not in those particular calculations/formuale.
That isn't because I don't recognise that risk, but because it's a little harder to quantify.

I wouldn't bank on the installer/manufacturer being around to honour the warranty in 20 years time.

The installers warranty is insured in case they go bust.
I'm looking to pay some by credit card for section 75 protection on the manufacturers warranties, but not 100% sure it applies.
I'm aware of the risk of companies going bust, but as you appreicate these things are dificult to put into forumlas.

Also you need to take account whether your house will gain or lose in value having had solar panels fitted.

In our opinion it will at least make it more attractive, but as always beauty is in the eye of the beholder so there is no way to know whether any particular seller will like it or not.

lisyloo

Hi Rollinghome

but, whatever the sums, wouldn't want panels on our roof

Could you tell me why?
Is it aesthetic? worries about maintenance? or something else?

and by 2034 will those of 2014 look as trendy as a 1980's mobile?

Maybe, but my calculations mean they are financially written off over 20 years, so they could be replaced.

lisyloo

if you wanted to be thorough then running a sensitivity analysis on the input numbers would be useful

Yes, thanks.
I have a spreadsheet from the installer so I can vary the inputs.
I've already tried some worse case scenarios
e.g. using 0% of the solar generated and it generates a positive return from the FIT/export tarrif.