1. WELCOME TO AN INTRODUCTION TO
FINANCIAL MATHEMATICS FOR REAL
ESTATE APPRAISAL
Before going on to financial modelling of potential real estate projects or assets, you
need to have some understanding of financial maths. Ideally, you also need to be able
to use Excel.
Press F5 and Page Down

2. 1. Rationale and Approach

3. “What is the point of this stuff?”
• BECAUSE THESE BASIC FINANCIAL MATHEMATICAL
TECHNIQUES PROVIDE YOU WITH THE KEY KNOWLEDGE
NEEDED TO:
– ESTIMATE THE VALUE OF REAL ESTATE ASSETS
– ESTIMATE HOW MUCH REAL ESTATE DEVELOPMENT
OPPORTUNITIES ARE WORTH
– PROVIDE ADVICE ABOUT THE WORTH OF REAL ESTATE
ASSETS AND PROJECTS
– PROBABLY MOST IMPORTANTLY, THEY PROVIDE THE BASIS FOR
THE APPRAISAL OF ANY FINANCIAL ASSET (SHARES AND
BONDS) AND ARE USED BY BUSINESSES TO DECIDE HOW TO
ALLOCATE THEIR CAPITAL MOST EFFECTIVELY
– DON’T FIGHT IT. DON’T BE FRIGHTENED OF IT. TAKE IT ON
AND BE POSITIVE. IT’S NOT THAT DIFFICULT

4. “Do I need to know anything first?”
• No - not really!

5. Does this link to any body of theory
that I should find out about?
• If you insist…it’s about capital budgeting theory
and how firms should make their investment
decisions in an optimal manner. It’s defined broadly as
• Warning – fairly important definition coming
•
•
•
•
The process in which a business determines whether projects such as buying
new equipment or investing in a company are worth pursuing. Usually, a
potential project's expected cash inflows and outflows are estimated in
order to assess whether the returns generated meet a target benchmark.
Hey – they could even be buying a property or carrying out a property
development project
They are many, many, many, many, many, many, many, many, many, many
many, many, many, many, many books on project appraisal, investment
appraisal, capital budgeting etc broadly covering this area.
Why do think this is?

6. “I’m tired. Couldn’t you just tell me the very minimum?”
• Well, the basic approach is that you try to get the
present value of all the project’s future cash flows
(i.e. costs and revenues) and add them together.
• Basically you need to estimate and set out the answers to
five questions
1
2
3
4
5
How much cash will be paid out?
When will it be paid out?
How much cash will be received?
When will it be received?
the required return from the
What is
investment?

7. 2. The basics of investment
return

8. “Could you tell me a bit more about this return thing?”
• Well done - you’ve spotted that it’s important
• Well, because investors are giving up their
money, they generally (albeit the financial world
seems to have gone a bit weird in the downturn)
want some return to compensate for
– Losing the ability to spend their cash –loss of
liquidity
– Losing purchasing power of cash due to inflation
– Not getting it back - risk

9. I’m afraid that I still don’t really get
this return thing.
• OK – let’s think of a really simple example.
• You buy a share for £100, get a £5 dividend and then sell it for £110
(ignore transaction costs etc). What was the return?
• Ummm
• Well, you made £15 – that’s the cash return
• It is usually expressed in % terms
• Is it?
• Yep - £15 is 15% of the initial investment of £100 – you made a 15%
return on your investment
• Did I?
• Yep – it’s important to know what return investors require in order
to work out whether an investment is worth making.

10. Can’t we find any signals in markets about what returns investors require for
projects?
•
•
That sounds like another suspiciously well-informed question!
Yes – the government bond market is supposed to provide a signal of investors’ required
return for loss of liquidity and for expected inflation but since, outside the
Eurozone, governments can print their own currency, there is no risk of not getting your
money from them. Govt bonds provide us with a signal of the risk
free rate of
return (an important concept in investment). There’s a joke that some govt. bonds in the
•
•
•
•
•
Eurozone provide return free risk.
Now I get it! Ah – so if I can get a return of 10% from govt bonds, I’ll want higher returns
from all other investments?
Normally, yes. A project’s target rate of return could be based on the risk free rate
of return from government bonds plus a risk premium related to the project’s risk
Alternatively – most companies also have a (weighted average) cost of capital. This is often
used as a basis for estimating their target (internal) rate of return. It’s also call the “required”
or the “hurdle” rate of return
Stealing from wonderful Wikipedia, the WACC is the minimum return that a company must
earn on an asset to satisfy its creditors, owners, and other providers of capital, or they will
invest elsewhere
To be honest (at last) – getting the right WACC for a project is tricky and it is similarly
problematic to estimate for the risk premium above bonds. Basically, in practice, it’s often
hard to robustly estimate the target rate of return for investments or projects –

11. 3. Compounding – calculating
the future value of money

12. We? OK, I’m signed up
•
•
•
•
•
•
Hmmmm…Suspiciously positive…OK – the first thing that you need to be able to do is to estimate
costs or revenues payable or receivable in the future.
How do I do that?
Well, often a starting point is what they are today and you then need to know how much they’ll
change in the future.
Let’s say that you think that you will need a hair transplant (or substantive cosmetic surgery) in five
years’ time (think of it as refurbishment). This currently costs £10,000. Due to growing
demand, the Institute of Cosmetic Surgeons is forecasting that the costs of such procedures will
increase at 10% per annum.
The question is – how much a hair transplant will cost in five years’ time assuming that the forecast
is correct?
Sorry I’m interrupting this question to announce that…
• Financial forecasts are hardly ever correct
•
•
That’s not really relevant now but it’s important to bear it in mind when we start to use forecasts
for projecting future rents, future costs, future sale prices etc. for real estate developments and
assets.
Moving swiftly on…

13. It’s to use a compounding formula: 1 plus the interest rate to
the power of the number of periods: (1+i)n
There’s an
easier way!
You need to be able to find and find the xy button on your
calculator
After one year
1.1
=
£11,000
After two years
£11,000times 1 plus growth rate (10%)
1.1
=
£12,100
After three
years
£12,100times 1 plus growth rate (10%)
1.1
=
£13,310
After four years
£13,310times 1 plus growth rate (10%)
1.1
=
£14,641
After five years
After the first year’s
growth, there’s another
£1000 that can also
grow.
£10,000times 1 plus growth rate (10%)
£14,641times 1 plus growth rate (10%)
1.1
=
£16,105
This is the compounding effect. You
get interest on the previous interest.
After five years, it’s grown to £16,105
In this case (1+0.10)5 = 1.61051 so multiplying our original £10,000 by 1.61051 gives us
£16,105.10.
0.1 is 10%
Think of it as getting a total growth of 61.105% over the five year period. The extra
11.105% over the 10% per year is the result of growth on the previous growth

14. 4. Discounting – calculating the
present value of future money

15. What about if I want to estimate the today’s (i.e. present) value of the right to
receive some money in the future
•
•
•
•
•
•
•
•
•
•
That’s a worryingly informed question.
I’m think of selling my inheritance to pay my tuition fees
Er, let’s take a more hypothetical example. Let’s say that you’re a cosmetic surgeon and you are pretty sure that
you’re going to get £16,105 in five years time.
However, you need money today to cover a negligence claim. So you decide to see how much someone would
give you NOW for this asset.
Yes! It’s an asset. Most investments or projects involve the right to get money in the future in return for spending
money in advance. To be honest – it looks like a pretty risky asset if they’re trying to settle a negligence claim.
It’s really the opposite of compounding – we convert future incomes to their present value. With compounding,
we’re converting a current income to a future value.
Let’s assume that the cosmetic surgeon’s father, having retired with a lump sum from his pension, has got cash
that he wants to invest. He requires a 10% return per annum (the per annum bit is often left unstated) from any
investment.
What will he offer?
Given a required return of 10% per annum, getting to the present value of the right to receive £16,105 in five
years time is roughly/approximately/nearly/almost like taking 10% off per year to provide the return to the father
As you can see overleaf…

16. This may be new. Discounting is simply the process of
estimating the present value of future cash flows
Discounted for
Since discounting is the opposite of growing, it is the
inverse of the growth formula – it could also be (1.1)-1
1year
16105times
1 divided by 1 plus discount rate
1/1.1
=
14641
Discounted for another year
14641times
1 divided by 1 plus discount rate
1/1.1
=
13310
Discounted for another year
13310times
1 divided by 1 plus discount rate
1/1.1
=
12100
Discounted for another year
12100times
1 divided by 1 plus discount rate
1/1.1
=
11000
Discounted for another years
11000times 1 divided by 1 plus discount rate
1/1.1
=
10000
Again - there’s
an easier way
It’s to use the discounting formula - 1 plus the
interest rate to the power of MINUS the
number of periods: (1+i)-n
Not really a surprise that we get to
£10,000. If we grow £10,000 at 10%
per annum for 5 years we get
£16105. If we discount £16105 for
five years at 10%, we get £10,000.
Compounding and discounting are
opposites.
In this case (1+0.10)-5 = 0.62092 so multiplying our original £16,105 by 0.62092 gives us
£10,000.
Note that, when discounting, approximately 10% per annum is being deducted from a
declining sum in the same way that, when compounding, 10% per annum is being added
to an increasing sum. Note also the similarities with the table (overleaf) showing how it
grows.

17. Compounding to obtain the future value of a current cash flow
Today
Next year
The year after
Three years time
Four years time
Five years time
We have
It has grown to
It has grown even more
It continues to grow
You get the picture
We end up with
£10,000
£11,000
£12,100
£13,310
£14,641
£16,105
Discounting to obtain the present value of future cash flow
Five years in the future
Brought back one year
Brought back two years
Brought back three years
Brought back…
Today
We hope to receive
It is worth
It is worth
It is worth
It is worth
It is worth
£16,105
£14,641
£13,310
£12,100
£11,000
£10,000

18. I really need a break now
•
•
•
•
•
•
…and you deserve it
But it’s really important that you get the concept of present value
The value of many assets is often calculated by estimating their future cash
flows, present valuing/discounting them (i.e. adjusting them for the required return)
and adding them up. This is the same for properties, bonds, shares, business
opportunities…the Net Present Value of the project or investment is estimated.
Actually, in an efficient market, the value of any investment should be the sum of the
present values of all the future revenues generated by that investment and..
Ok – I’ll stop and you can stop too
But if you’re a still bit hazy…have another look at the present value stuff when if you
come back for NPV and IRR
Sorry if I have laboured the point a bit – but once you ‘get’ these basic concepts, a lot of
real estate appraisal using cash flows analysis is extremely straightforward.

19. 5. Calculating Net Present
Value

20. OK – tell me about this NPV thingy, er, please
•
•
•
•
•
•
Won’t take long
Net Present Value is the standard technique in investment or project appraisal for deciding whether a project is
worth doing – or an investment is worth making
The NPV rule is that: if discounted at the target rate of return, the NPV is positive, then the project is acceptable.
Let’s take a hypothetical example
I am rich (it’s very hypothetical) and I’m thinking of buying a five bed-room house in London at a cost of
£1,000,000 (it’s only money) and letting it to students (yuk!).
I think that I will get £50,000 a year in rent (because I’m worth it!) and a forecaster states (let’s ignore what I said
earlier about forecasters) that house prices in London should rise by 20% (overall) over the next five years.
•
•
Five years is also my investment holding period.
I have a target rate of return of 10% per annum (A bit greedy?)
•
Let’s ignore all the “clutter” of agents fees, management costs, repairs and replacement, timing of rental payments, stamp duty, rental growth etc – albeit, in
practice, they will be important determinants of my return. We’ll do this later.
•
•
Should I make this investment?
If the NPV is positive, I should. If it is negative, I shouldn’t.
• What is it then?

21. First set out
period of cash
flow
Period
0
1
2
3
4
5
Set out how
much cash is
going out and
when it goes out
Capital Outlay
-£1,000,000
£0
£0
£0
£0
£0
For simplicity,
I’m assuming
that the rent is
paid annually n
arrears
Surprise- there’s
an easier way
Set out how
much income is
coming in and
when it comes
in
Income
£0
£50,000
£50,000
£50,000
£50,000
£50,000
Set out how
much capital is
returned and
when it is
returned
Now we have
the all
important Net
Cash Flow
generated by
the investment
This is where
I’m taking my
return by
discounting
each cash flow
at 10% p.a.
Capital receipt
£0
£0
£0
£0
£0
£1,200,000
Net Cash Flow
-£1,000,000
£50,000
£50,000
£50,000
£50,000
£1,250,000
PV factor @ 10%
1.0000
0.9091
0.8264
0.7513
0.6830
0.6209
NPV
Everything is
converted to PV
terms
Present Value
-£1,000,000
£45,455
£41,322
£37,566
£34,151
£776,152
-£65,355
The current market
value grown by 20%
Is the sum of the
present values
Excel has an NPV function that will find the
NPV of the NCF automatically. You should find
out how to use it…moving on
Is negative! The
investment should be
rejected.
How much should I
offer assuming I want
a 10% return and my
assumptions are
correct?
Basically, it’s the
£1,000,000 minus
£65,355.
This is actually called the Gross Present
Value. It’s the same as the NPV – but the
initial investment is ignored. In this case,
it represents what this investment is
worth to me given my expectations and
required return.
That’s £934,645

23. 5. Internal Rate of Return

24. Period
0
1
2
3
4
5
Capital Outlay
-£934,645
£0
£0
£0
£0
£0
Income
£0
£50,000
£50,000
£50,000
£50,000
£50,000
Capital receipt Net Cash Flow PV factor @ 10% Present Value
£0
-£934,645
1.0000
-£934,645
£0
£50,000
0.9091
£45,455
£0
£50,000
0.8264
£41,322
£0
£50,000
0.7513
£37,566
£0
£50,000
0.6830
£34,151
£1,200,000
£1,250,000
0.6209
£776,152
NPV
£0
If I pay £934,645, I get my 10% return per annum and nothing is leftover. The NPV is zero. If the NPV was
positive, I’d get more than 10% return per annum, if it was negative, I’d get less than 10% return per
annum.
Let’s go back and look at these returns in more detail assuming that I’ve paid the original purchase price of
£1,000,000 (only because the numbers are more tractable)

25. Period
0
1
2
3
4
5
You tend to get return from
two sources – income (rent
from properties) and capital
(changes in the value of the
asset)
Capital Outlay
-£1,000,000
£0
£0
£0
£0
£0
Income
£0
£50,000
£50,000
£50,000
£50,000
£50,000
Capital receipt
£0
£0
£0
£0
£0
£1,200,000
Net Cash Flow
-£1,000,000
£50,000
£50,000
£50,000
£50,000
£1,250,000
NPV @ 10%
-£65,355
IRR
8.38%
Let’s start to think in terms of my percentage returns on my £1,000,000
What’s my income return per period as percentage of capital invested?
Fairly straightforward - £50,000 is 5% of £1,000,000
0 -1000000
-1000000
1
50000
50000
The capital return is more tricky. I invest £1m and get £1.2m back. That’s 20% over five years. That’s just about
4% per year – but a bit less because of the compounding. (Annoyingly) receiving 4% per year would give you more
than 20% over five years. It’s actually 3.71% per annum.
2
50000
50000
3
50000
50000
4
50000
50000
So, I’m getting 5% per annum for income return and 3.71% capital return, I don’t see any other return. So my
total return must be close to 8.7% per annum. This is a good approximation – but just an approximation – of my
internal rate of return or IRR. There one thing that you really need to remember about the IRR. It’s the discount
rate at which the NPV is zero.
5
50000 1200000 1250000
Excel has an IRR function that you can use to get the IRR of any cash flow. Applying it to the cash flow above, gives
us…
We can get a fairly good approximation here because the numbers are so simple. The IRR is quite tricky to work out mathematically and manually. It
requires an iterative process (i.e. some guessing) – and it will only be an approximation. Excel does it pretty effortlessly and more accurately. The
important thing is that you understand the principles – it is broadly the total return per period received on capital invested. However, there are many
technical issues with its underlying assumptions (reinvestment rate), mathematical quirks (you can get two IRRs when you have a number of sign
changes in the net cash flow) and what to do when the NPV rule (choose investment with highest NPV) - and IRR rule (choose investment with highest
IRR) are in conflict. But hopefully, you’ve read this far to be reassured that you’ve got the essentials of it here.
NPV
£0.00
IRR
8.38%

26. G
THAT’S IT! HONESTLY, I THINK YOU’VE DONE
ENOUGH FINANCIAL MATHS NOW TO MOVE ON TO
LOOK AT CARRYING OUT CASH FLOW APPRAISALS
OF REAL ESTATE ASSETS AND DEVELOPMENT
OPPORTUNITIES