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.

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1. Rationale and Approach

“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

“Do I need to know anything first?”

• No - not really!

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?

“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 How much cash will be paid out?

2 When will it be paid out?

3 How much cash will be received? 4 When will it be received?

5 What is the required return from the investment?

2. The basics of investment return

“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

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.

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 –

• But - we need to have a go.

3. Compounding – calculating the future value of money

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…

After one year £10,000times 1 plus growth rate (10%) 1.1 = £11,000

After two years £11,000times 1 plus growth rate (10%) 1.1 = £12,100

This is the compounding effect. You get interest on the previous interest.

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 £14,641times 1 plus growth rate (10%) 1.1 = £16,105

After five years, it’s grown to £16,105

There’s an easier way!

It’s to use a compounding formula: 1 plus the interest rate to the power of the number of periods: (1+i)n

You need to be able to find and find the xy button on your calculator

In this case (1+0.10)5 = 1.61051 so multiplying our original £10,000 by 1.61051 gives us £16,105.10.

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

After the first year’s growth, there’s another £1000 that can also grow.

0.1 is 10%

4. Discounting – calculating the present value of future money

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…

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

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.

Discounted for 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

This may be new. Discounting is simply the process of estimating the present value of future cash flows

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.

Since discounting is the opposite of growing, it is the inverse of the growth formula – it could also be (1.1)-1

Today We have £10,000Next year It has grown to £11,000The year after It has grown even more £12,100Three years time It continues to grow £13,310Four years time You get the picture £14,641Five years time We end up with £16,105

Five years in the future We hope to receive £16,105Brought back one year It is worth £14,641Brought back two years It is worth £13,310Brought back three years It is worth £12,100Brought back… It is worth £11,000Today It is worth £10,000

Compounding to obtain the future value of a current cash flow

Discounting to obtain the present value of future cash flow

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.

5. Calculating Net Present Value

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?

Period012345

First set out period of cash flow

Capital Outlay-£1,000,000

£0£0£0£0£0

Set out how much cash is going out and when it goes out

Income £0

£50,000£50,000£50,000£50,000£50,000

Set out how much income is coming in and when it comes in

Capital receipt£0£0£0£0£0

£1,200,000

Set out how much capital is returned and when it is returned

Net Cash Flow-£1,000,000

£50,000£50,000£50,000£50,000

£1,250,000

Now we have the all important Net Cash Flow generated by the investment

PV factor @ 10%1.00000.90910.82640.75130.68300.6209

This is where I’m taking my return by discounting each cash flow at 10% p.a.

Present Value-£1,000,000

£45,455£41,322£37,566£34,151

£776,152

Everything is converted to PV terms

NPV -£65,355

Is the sum of the present values

Is negative! The investment should be rejected.

How much should I offer assuming I want a 10% return and my assumptions are correct?

The current market value grown by 20%

Surprise- there’s an easier way

Excel has an NPV function that will find the NPV of the NCF automatically. You should find out how to use it…moving on

Basically, it’s the £1,000,000 minus £65,355.

That’s £934,645This 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.

For simplicity, I’m assuming that the rent is paid annually n arrears

5. Internal Rate of Return

Period Capital Outlay Income Capital receipt Net Cash Flow PV factor @ 10% Present Value0 -£934,645 £0 £0 -£934,645 1.0000 -£934,6451 £0 £50,000 £0 £50,000 0.9091 £45,4552 £0 £50,000 £0 £50,000 0.8264 £41,3223 £0 £50,000 £0 £50,000 0.7513 £37,5664 £0 £50,000 £0 £50,000 0.6830 £34,1515 £0 £50,000 £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)

Period Capital Outlay Income Capital receipt Net Cash Flow0 -£1,000,000 £0 £0 -£1,000,0001 £0 £50,000 £0 £50,0002 £0 £50,000 £0 £50,0003 £0 £50,000 £0 £50,0004 £0 £50,000 £0 £50,0005 £0 £50,000 £1,200,000 £1,250,000

NPV @ 10% -£65,355

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

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.

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.

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…

IRR 8.38%

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.

0 -1000000 -1000000

1 50000 50000

2 50000 50000

3 50000 50000

4 50000 50000

5 50000 1200000 1250000

NPV £0.00

IRR 8.38%

You tend to get return from two sources – income (rent from properties) and capital (changes in the value of the asset)

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

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