# Valuation Maths. TIME VALUE OF MONEY: COMPOUNDING & DISCOUNTING.

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Valuation Maths

TIME VALUE OF MONEY: COMPOUNDING & DISCOUNTING

Introduction Investment depends on time value of money - rather have money now than later –Why is £1 today not equivalent to £1 in a year’s time? Compounding and discounting measure the value of money over time in opposite directions Relationship between capital invested, future cash flows and time forms the basis of investment appraisal Two main types of payments to consider: single and multi-sum –E.g. buy a property today and sell it in five years’ time –Pay rent each quarter –Pay back a loan instalment each month

Time value... A sum of money to be received in the future is not worth its face value but a sum less than that - the actual amount depending upon the interest rate (or, more specifically, the discount rate)

Compounding... Calculating the interest that would be earned by the original capital and adding interest each year Thus, the size of the future sum is determined by a combination of the compound rate (i.e. the rate at which the original sum could earn interest) and the deferment period (the delay in payment)

Discounting... Discounting is the opposite of compounding and is a basic concept underpinning valuation

Single sum payments (the return is assumed to accrue in arrears in the following formulae) Future value, FV: Present value, PV: where: –n is the number of time periods over which the investment is held –r is the rate of return –PV is the present value of the investment –FV is the future value if the investment after n periods

Multi-sum payments Level annuities (in arrears) where PMT is the regular payment also known as a sinking fund also known as the PV £1 per annum or Years Purchase (YP), i.e. the number of years that will pass before purchase price is recouped As well as discounting individual sums, we can discount a series of future sums (an annuity) This forms the basis of property valuation since a property bought as an investment will produce a series of rental incomes and the sum paid will reflect the amount and timing of these rents

Present value of real estate annuities Your client proposes to purchase a property which will produce a rent of £5,000 pa for the next five years Assuming a discount rate of 10% pa, what is the capital value of these rents? We can use the PV of an annuity formula to value or ‘discount’ each rent payment in turn. Looking at the latter... RentReceivableDiscounted Value @ 10% £5,000immediately£5,000 after 1 year£4,545 £5,000after 2 years£4,132 £5,000after 3 years£3,757 £5,000after 4 years£3,415 Total discounted value£20,850

A note on PVs... The size of the present value will depend upon the time period involved and the interest rate (in this case known as the discount rate) The differences in discounted value of incomes are more pronounced in the short term than in the long term –E.g. notice in the table that the difference between the present value of year 2 and 3 rents was £375 whereas if the table had been extended to include rents receivable in say 20 years, the difference between the present values of the rents in year 19 and 20 would be only £74 (assuming the rent payable was still £5,000) Value Discounted Value Time

So, in valuation terms, it is the rents receivable in the early years that primarily dictate the overall value of the interest unless for some reason there is a substantial reversionary value expected e.g. when a landlord is able to regain possession after a long lease and perhaps redevelop the property The discount rate is crucial too: the five incomes above discounted at 9% pa would have had a capital value of £21,198 In selecting the appropriate discount rate a valuer must be mindful of all of the following factors: –type of property –status of the tenant –nature of the lease (particularly the rent review) –yield on alternative investments –anticipated rental growth –when income is first received & frequency

Annuity £1 will purchase

Constant growth annuity (in arrears) For a fixed term In perpetuity

Periodic growth annuity in arrears Where g is grown every pth year at g rate per annum. So becomes

EQUIVALENT RATES

Equivalent rates Relationship between rates over different time periods –Generally:(1+EAR) = (1+NAR/m) m –So annual to monthly:(1+i a ) = (1+i a /12) 12 –Annual and quarterly:(1+i a ) = (1+i a /4) 4 Rearranging we get: –EAR = and NAR = –So i a = (1+i m ) 12 – 1and i m = (1+i a ) 1/12 – 1 –Andi a = (1+i q ) 4 – 1and i q = (1+i a ) 1/4 – 1 Where: EAR = equivalent annual rate NAR = nominal annual rate m = number of compounding periods per annum i/m = r = actual simple interest rate being applied per compounding period

TIMING OF RECEIPTS: IN ARREARS OR IN ADVANCE

Cash flows in arrears and in advance Level annuity in advance –multiply the in arrears PV formula by (1+r) If level annuity is received in perpetuity and in advance

YIELDS

(Present) values and yields As n gets bigger the PV of a level annuity (also known as the YP or PV of £1 per annum) simplifies to 1/r When looking at property investment transactions that have recently taken place in the market it is possible to substitute r to identify the market rate of return, known as the yield y given a price P. The equation remains the same but the notation changes: or for any market rent (MR) other that £1

Yields A yield is the relationship between how much money is put in to an investment (initial cost and subsequent expenditure) relative to how much comes out (income and eventual sale price) The yield is a reflection of the attractiveness or risk attached to an investment All investors like to receive the highest possible yield compatible with the perceived risk. Yields are the main yardstick of property performance Investing institutions make buying/selling decisions on the basis of yields rather than cost The effect of market forces results in popular properties having lower initial yields than others

How do clients use information on yields in their decision-making process? When we advise a client that the value of a property is X, it mean that if he/she buys at that price they will, in effect, be buying an investment which will give them an annual yield of Y% (the yield you, the valuer, has used in your valuation) It is then up to the client to decide whether the yield is sufficiently attractive to induce them to buy the property If it is not they can try to negotiate a lower price (and thus increase annual yield) or walk away from the deal Investors base their decisions on yield rather than price and will drop out when the yield drops below what they would regard as an acceptable level This produces one of the great mysteries of valuation i.e. that good properties offer low yields which appears to be a contradiction in terms – it occurs because of the markets 'bidding' process.

Simple investment analysis: yields By rearranging the previous equation, we can isolate y: Ratio of annual income to capital value Used to describe the quality of an investment Used by valuers as a unit of comparison Known as an initial yield

Yields and valuation Formula can be rearranged to help value property investments Example 9,000m 2 office block in central Bristol Market rent is £250/m 2 What is the rental value? If investors require a return (or yield) of 8% What is the capital value? What if investors require a yield of 6%, 7%, 9%...?

What do you observe? An inverse relationship As yields increase values fall As yields decrease values rise Small changes in yield create large changes in value How does this impact land value?

NET INITIAL YIELD

Net initial yield Investors actually work out the Net Initial Yield They calculate the relationship between money invested i.e. the price paid plus acquisition costs and the rental income

Net initial yield Rent£410,000 Price Paid£8,553,100 The total money invested assuming acquisition costs of 5.7625%: £8,553,100 + £492,872 = £9,045,962 The Net Initial Yield is then: £410,000/£9,045,962 = 4.53%. Essentially, yields are calculated by adding acquisition costs onto the price paid Then when a valuation is done it is standard practice to adjust the valuation by taking off acquisition costs. Unfortunately this is not quite as simple as deducting 5.5% or 5.7625% from the valuation estimated gross of costs... A property has just been sold...

Net initial yield For example: Valuation (gross of costs) = £1,000,000 Less acquisition costs at 5.7625% = 1,000,000(0.942375) = 942,375 What is the problem with this calculation? If we start from £942,375 and then add 5.7625% acquisition costs £942,375 * 1.057625), the total is £996,679 not £1,000,000 This is because acquisition costs are calculated as a proportion of the actual price and not the gross of costs valuation. For example, Stamp Duty is 4% of the price paid (which the valuation is trying to estimate) and not 4% of the price paid including acquisition costs. The calculation is: Valuation net of acquisition costs =

Yields: a summary The main task of the valuer is to assemble known facts and to make judgments about the unknown variables in order to assess the overall risk or attractiveness of the assessment This overall view will manifest itself in the form of the yield. This yield should produce a value at which the property could be traded. Clearly, the more facts that are known the less the risk will be. Thus, a commercial property which is already occupied will be more attractive than one which is unoccupied (there will be no risk of losing income because of failure to find a tenant). Similarly, a tenant who is financially sound is better than one which may go bankrupt. Development sites are particularly risky since they may take many months or years to fruition by which time the demand may have evaporated. It is important to be as accurate as possible in allocating the yield since small errors can result in considerable differences in capital value.

REAL AND NOMINAL RATES OF RETURN

Real (effective) and nominal cash-flows and rates of return Dealing with inflation in investment appraisal Real cash-flows project at current prices (i.e. no account taken of inflation) whereas nominal cash-flows project future (inflation-adjusted) prices Real rate of return does not reward investor for bearing inflation whereas nominal rate does Nominal cash-flow n = real cash-flow n x (1+  ) n –Where:  = rate of inflation n = no. years until that cash-flow arises

Real and nominal cash-flows and rates of return Real cash-flow n = nominal cash flow n (1+  ) n Nominal cash-flows should be discounted at a nominal rate of return Real cash-flows, should be discounted at a real rate of return (i.e. exclude impact of inflation) using the Fisher Formula:(1+r n ) = (1+r y )(1+  ) Where r n is the nominal rate and r y is the real rate Rearranging:(1+r y ) = (1+r n )/(1+  )

VALUING REAL ESTATE INCOME STREAMS

Valuations: what are you trying to do? When you are asked to "value" a property, you are essentially trying to convert one type of asset into another –The purchaser will receive a property and in return the purchaser will hand over a lump sum of cash to the vendor In commercial property, the "return" to the purchaser comes in the form of an income for a set number of years (or indefinitely if it’s a freehold interest) –Another way of looking at it is that you are trying to put a lump-sum value on a collection of prospective future incomes.

Valuing property investments Property ownership and occupation are often separate interests and the capital amount paid for the freehold is a function of the income-producing potential Even when occupiers buy property for their own occupation they usually consider the opportunity cost of the capital and the financial return the asset may produce With such properties valuation is the estimation of the future financial benefits derived from the ownership expressed in terms of their present value The valuer needs to be able to estimate future net benefits and discount them at a suitable rate to calculate present value

Terminology Market value –Rental value –Capital value All risks yield –Initial yield (current rent ÷ price) –Exit yield –Reversionary yield (market rent ÷ price) –Equivalent yield (IRR assuming increase to ERV at next opportunity but no further growth) Other –Rack-rented –Reversionary

Review A valuation of an income of £5,000 pa for 5 years receivable in advance (discounted @10% pa) RentReceivedPV of £1Total PV £5,000now1£5,000 after 1 yr0.9091£4,545 £5,000after 2 yrs0.8264£4,132 £5,000after 3 yrs0.7513£3,756 £5,000after 4 yrs0.6830£3,415 Total discounted value£20,849

What are the limitations of using YP (PV £1 per annum)? Using a YP can save a great deal of time and effort in converting a series of incomes into a single capital sum (capitalisation) but it does have some limitations: a)the income capitalised must be constant b)the YP formula must match the rental income timing c)we must allow for rent reviews

a) Constant income The YP approach only works for periods when the rent remains constant. If the rent changes, we must treat that as a separate calculation. Thus, to maintain the benefit of the YP we must either: a)Assume the current rent will always remain at its present level b)Accept the fact that the rent will probably increase at each future review and use multiple YP calculations Although (a) seems to be an unrealistic way of approaching it, it is possible to make this assumption if the valuer can "compensate" for the lost future rental growth in some other way. This method of valuation is known as the all-risks or initial yield approach (b) assumes that we will attempt to identify the future pattern of rents and build these into our calculations and forms the basis of a discounted cash-flow approach to valuation

b) Rental timing These could be annually in arrears, annually in advance, quarterly in advance, etc. It is important to match the YP formula with the timing of the rent. For example, the YP figure for five fixed rents received annually in arrears will differ from the YP based on the same rents received annually in advance or quarterly in advance.

c) Rent reviews One of the main assumptions built into the typical valuation is that ‘blocks’ of rents will be ‘capitalised’ Where the rent is expected to increase later in the lease as a result of rent reviews (a normal assumption), the capital value of later blocks of rent not yet 'on-stream' must be ‘deferred’

VALUING BLOCKS OF INCOME

Because these incomes will be received over a long period, we cannot simply add up the face value of the incomes to be received and equate that to the "value" in today's terms......an income to be received in the future has a lower value in today's terms and thus the rent has to be "discounted" at an appropriate rate PeriodRents Expected Year 1£5000 Year 2£5000 Year 3£5000 Year 4£5000 Year 5£5000 Total "value" today£25,000

Therefore "valuing" a property involves two stages: –Stage 1: Estimating the amount of rent to be received in each year –Stage 2: Converting (or discounting) those rents to be in terms of their present equivalent value PeriodRents Expected* Present Value (discounted @10%) Year 1£5,000 Year 2£5,000£4,545 Year 3£5,000£4,132 Year 4£5,000£3,756 Year 5£5,000£3,415 Total “value” today£20,848 * This example assumed that rents are received in advance

The first stage usually involves assessing the current level of rent and then applying a "rental growth" factor to estimate the future rents. Once you have identified the rents expected, you can adopt one of two approaches to convert them to their present day equivalent value (i.e. the discounting process). You can: –Discount each rent individually using the discounting (PV) table –Discount them in "blocks" using the YP (years purchase) table Discounting Rents IndividuallyDiscounting Rents in Blocks Advantages  Very logical  Can cope with rents in arrears or advance very easily  Quicker Disadvantages  Very time consuming if you have a long period to deal with  Initially very confusing  Requires 2 tables (YP and PV)  YP table must “match” the rental pattern (arrears/advance)

In compiling your PV table (called a discounted cash flow table), in addition to the rents, you need to know the discount rate and the rental pattern (i.e. receivable in advance or in arrear) The example shown above has rents paid in advance. This can be deduced because the first rent is actually not discounted and is worth its full face value. Where necessary, the table can be easily modified to reflect a rental pattern where rents are received in arrears. This would involve discounting the initial rent for one year and the second rent for two years and so on. Although this is quicker, it does initially cause some confusion because it involves a two-stage discounting process. It is particularly beneficial for properties held for longer terms.

Example Value a property that has a current rental value of £12,000 p.a. This has just been let on a 25 year lease with five-year review periods. Rents are paid annually in arrears. Assume a rental growth rate (to estimate future rents) of 2% p.a. and a discount rate of 7%. To identify the future rental values expected, essentially a compounding exercise, use the current rent in conjunction with compound interest table (amount of £1). These can then be set out as follows: Period Rent Expected 1-5£12,000 6-10£13,249 11-15£14,628 16-20£16,150 21-25£17,831

The first stage of the valuation is to "capitalise" or value each block of rents. This is done with the Years Purchase. This provides a multiplier with which you can convert the 5 (in this case) rents at each review into a single capital sum equivalent. However, there are two things that you have to be aware of: –Each YP is calculated on the basis of particular rental timing (i.e. rents in arrear or rents in advance) –The capital value produced by using the YP produces a capital value expressed at the beginning of the period when the rents were paid. For example, when we value the 15-20 rents (£16,150 p.a.), the resulting capital value will be expressed at year 15 When calculating a YP you are assigning the number of years over which the rents at that level are paid, not when they are paid. Thus, in this example, we are always using a YP for 5 years since we are dealing with 5 yearly blocks of rent. The adjustment for when they are paid comes later. Applying the YP figures to the table gives us...

Period Rent Expected YP 5 years @ 7% Future Capital Value 1-5£12,0004.100£49,202 6-10£13,2494.100£54,321 11-15£14,6284.100£59,975 16-20£16,1504.100£66,215 21-25£17,8314.100£73,107 Present total value Our stream of 25 incomes have now been converted into 5 lots of capital sum equivalents, spread out over the term of the lease Remember that the idea of a valuation is to produce a present capital value equivalent of the rents. At the moment only the first capital sum is expressed in present-day terms, so this needs no further modification. However, all the others are expressed as a future capital sum. We therefore need to convert that to its present-day equivalent.

This is the role of the present value function. Thus, the next column in the DCF table is the PV multiplier. You must be very careful to use the correct deferment period –In this example, the first capital sum does not need any deferring and is entered as 1 (or left blank). The second capital sum is currently expressed in year 5 and thus needs deferring for that period (i.e. a PV for 5 years - 0.713). The third sum needs deferring for 10 years and so on. When we have the PV figures we can multiply them by the actual future sum involved to find the present value. PeriodRent ExpectedYP 5 years @ 7%Future Capital ValuePV @ 7%Present Value 1-5£12,0004.100£49,202n/a£49,202 6-10£13,2494.100£54,3210.7130£38,732 11-15£14,6284.100£59,9750.5084£30,489 16-20£16,1504.100£66,2150.3625£24,001 21-25£17,8314.100£73,1070.2584£18,894 Present total value £161,318

Review of the blocked income discounting process In this DCF table we have included a column to specifically show the future capital values. However, this is not strictly required and you could normally find the present discounted value by multiplying the rent expected by the YP and then by the PV figure. Note: The example shows the present day value of the 25 years worth of rents. A freeholder would, of course, be able to re-let the property for a further period and receive rents for that period too. Stage 1 Identify/calculate the rental values (using the current rent in conjunction with the Amt of £1). Stage 2 Look up and enter the YP figure (normally the same for each line). This converts the block rents into a single (future) capital sum. Stage 3 Look up and enter the PV figures to bring future capital values back to present day equivalent Stage 4Calculate present discounted value of each block. Stage 5Add up to find total discounted value.

As you may notice, everything in traditional valuation methods (by far the most commonly used in practice) is expressed in current terms. The rent paid is, er, the rent actually being paid. The Market Rent is the estimated amount that the property could currently let for. How is the fact that the rents will (almost certainly) change in the future not accounted for in the valuation? How is future growth taken into account?

Another example Your client wants to buy a property which is expected to produce a rent of £1,000 pa for 3 years followed by £1,500 for the following 3 years followed by £2,000 pa for the final 3 years of a 9 year lease. The rents are paid annually in arrears. Advise your client about the likely capital value of the property using a yield of 8% to discount the rents.

The 3 rd problem will become clearer as we proceed through the answer.. PeriodRentYP 3yrs @ 8% Cap.value at start of period Capital Value today 1-3£1,0002.5771£2,577 4-6£1,5002.5771£3,865 £2,577 £3,865 £3,068 0.7938 (3yrs)£3,068 Present Value @ 8% 7-9£2,0002.5771£5,140 0.6302 (6yrs) £3,248 Total present value£8,893 Problem – 3 different levels of rent Solution – Use 3 separate YP calculations

PeriodRentYP 3yrs @ 8% Cap.value at start of period Capital Value today 1-3£1,0002.5771£2,577 4-6£1,5002.5771£3,865 0.7938 (3yrs)£3,068 Present Value @ 8% 7-9£2,0002.5771£5,140 0.6302 (6yrs)£3,248 Total present value£8,893 In practice, the middle column is sometimes omitted Without using the YP, the alternative would look as follows...

YearRentPV @ 8%Present Dis. Value 1£1,0000.9259£925.90 2£1,0000.8573£857.30 3£1,0000.7938£793.80 4£1,5000.7350£1,102.50 5£1,5000.6806£1,020.90 6£1,5000.6302£ 945.30 7£2,0000.5835£1,167.00 8£2,0000.5402£1,080.40 9£2,0000.5002£1,000.40 Total discounted present value£8,893.60

Review The valuation process involves:- Identify the current rent and rent review pattern Identify/calculate the future rentals Capitalise each block of rents using a YP (to find their future CV) Discount the future CV of each block to find its PV Add up all the discounted PV’s to find the total present capital value of the property

Estimating future rents One of the big problems with valuation Can only be based on judgement and knowledge of your sector Usually based upon an anticipated annual growth of the current rent (i.e. a compounding exercise), but… be aware that the rent can only increase at the approved rent review intervals

Estimating future rents Example: A property has a current rental value of £25,000 pa. It has just been let on a 15 year lease with 5 yearly upward-only reviews. Calculate the rents likely to be received at each review based on an anticipated rental growth of 2.5% pa Rent at first review £25,000 x FV of £1 in 5 yrs @ 2.5% pa = £25,000 x 1.1314 = £28,285 pa Rent at second review £25,000 x FV of £1 in 10 yrs @ 2.5% pa = £25,000 x 1.2801 = £32,002 pa

Incorporating into the valuation These rents can then be incorporated into a normal valuation:- PeriodRentYP 5 years (@ 5%) Cap.value at start of period Present Value @ 5% Capital Value today 1-5£25,0004.3295£108,237 6-10£28,2854.3295£122,4600.7835 (5yrs)£ 95,947 11-15£32,0024.3295£138,5520.6139 (10 yrs)£ 85,057 Total present discounted value£289,241 £25,000 pa 5 1015 £28,285 pa £32,002 pa