Derivatives Pricing Approaches to Valuation Models: Sensitivity Analysis of Underlying Factors

Derivatives Pricing Approaches to Valuation Models:

Sensitivity Analysis of Underlying Factors

A. E. Baum+

C. J. Beardsley*

C. W. R. Ward+

+Department of Land Management and Development,The University of Reading,Whiteknights,Reading,RG6 6AW,

June, 1999

*ISMA Centre, University of ReadingWhiteknightsReadingRG6 6AW

Phone: +44 (0) 118 931 8178, Fax: +44 (0) 118 931 8172 E-mail:

[email protected]

Derivatives Pricing Approaches to Valuation Models:

Sensitivity Analysis of Underlying Factors

Abstract

Derivatives pricing techniques are now being applied to real estate

in the US, UK and Australia. However, the UK lease structure offers

a particular opportunity to explore this means of analysis because

of the complex option-like characteristics of the typical

institutional lease.

The standard upward-only rent review introduces the possibility of

splitting the cash flow from the property into separable components.

These are, (1) a simple annuity for any time period of the contract

and (2) the potential uplift(s) in rental tranches at future rent

reviews;

The simple annuity is easily analysed. The variability of future

rent reviews is more usefully understood with reference to

derivatives pricing. This approach introduces a number of new

variables and techniques into real estate investment analysis, the

two most interesting of which are (1) an ability to value stochastic

processes affecting the underlying property, particularly

2

volatility, and (2) the use of risk-neutral valuation techniques to

give correct values for cash flow derivatives which depend on the

underlying property

The range of assumptions that may be made by model builders is wide

and also includes assumptions about the variability and the term

structure of interest rates, plus the existence of tradable assets

which can be used to map the expected rental growth rates into a

risk neutral framework.

The paper uses Monte Carlo simulation techniques that examine how

these factors affect the values produced for standard UK leases. We

will examine the impact of relaxing the upward-only rent review

assumption both as a means of extending the application of the paper

and also to measure the importance of the upward-only rent review vis-

à-vis alternative forms of rental cash flows.

3

Introduction.

The modelling of option pricing stretches back to the beginning of

the 20th century when Bachelier (1900) developed the first valuation

model assuming that stock prices followed arithmetic Brownian

Motion. After an intervening period of neglect, advances were made

by Sprenkle (1964), Boness (1964) and Samuelson (1965) who

recognised that a more applicable model could be derived from

consideration of the geometric Brownian motion with drift approach.

Boness (1964) in particular recognised that the value of a call

option in a stock would be a function of the present values of the

expected share price and the exercise price. Given a market, in

which the share was traded, the present value of the expected share

price was represented by the current share price.

Boness’ formula could be represented by the expression

C = S0 Prob(1) – X Prob(2) e-rt ()

Where

Prob(1) = sensitivity of the Call option price to changes in

the Share price

Prob(2) = Probability of the share price exceeding the Exercise

price at expiration

S0 = the current share price

X = the exercise price

r = the expected return on the share price

t = the time to expiry of the option

4

While this was demonstrably wrong in that it used, as a discount

rate for the option, the return on the share, it could form the

basis of a normative valuation model provided ad hoc adjustments

were made to the discount rate to reflect the higher risk associated

with the option.

A huge step forward was made by Black and Scholes (1973) who

provided the first explicit general equilibrium solution to the

option pricing problem of valuing European call and put options.

They were able to simplify the choice of discount rate by

recognising that a riskless portfolio consisting of shares and

options could be constructed and valued using the risk free rate of

interest. Intuitively since the value of the portfolio could be

estimated, and the value of the share constituent could be observed

in the market, the value of the other constituent – the option could

be derived. This valuation approach based on the assumption of

arbitrage provided an analogous model to the Boness approach but

with the dramatic result of using the risk free rate as the discount

factor. In the Boness formula above, (1), the prob(2) was no longer

the probability of the share price being above the exercise price

but instead became the equivalent probability if the future share

price was modelled in a risk-neutral world. Whilst this is a neat

mathematical transmogrification, it does produce some difficulties

if the users of the model wish to use the model normatively rather

than to estimate the market value implicit in the market prices

observed. For example, the Black and Scholes model will reveal to

investors, the “fair” value of a call option, given today’s share

price. It is more difficult to use the model to value a call option

for an investor who believed that the share price was wrongly valued

by the market and who wished to increase the expected future price

of the share (by, for example, imposing a specific growth rate in

5

the share price). However one has to recognise that, just as

investors use the CAPM in valuing stocks, so they might wish to use

the positive model of Option Pricing to incorporate their own prior

expectations with the information contained in market prices. To

reflect the normative application, we use in this paper two versions

of the underlying model, the first in which the parameters reflect

market prices and a second in which the expected return from the

underlying asset may contain a mis-priced valuation.

Option Pricing using Monte Carlo Simulation.

For some options, the value of the option depends on the past

behaviour of the underlying asset. For example, options can be

traded on the average price of a security during a specific period.

These options are described as being path-dependent and generally,

they are more difficult to value than the conventional put and call

options. In property lease terms, the exercise price may depend on

the value of the rent that is being paid prior to the exercise date.

Whilst solutions can be found for some path dependent options,

property leases can be characterised as consisting of multiple

options. For example, a common lease in UK commercial property

implies that rents should be reviewed upwards every five years. The

rent can only be increased so at the end of each five-year period,

the tenant and landlord have to agree on the appropriate adjustment

to make in the light of market rental movements for comparable

properties. Consider, in 1990, a property in an area that has not

experienced change in property rents since the property was first

let in 1985. The rent for the specific property might not be

increased in which case the next question to answer is whether rents

in 1995 will increase to more than the level originally fixed in

6

1985. Alternatively the rent might be adjusted in 1990 and the

option inherent at 1995 is whether the rent should be increased

above the 1990 level. It will be seen that each subsequent rent

review is a multiple option depending on what level of rent has

already been reached for the specific property in any of the

preceding number of years. One can thus identify in a 25-year lease

ten different options (5, 01; 10, 0; 10, 5; 15, 0; 15, 5; 15, 10; 20,

0; 20, 5; 20, 10; 20, 15). Of these ten options, six have (at the

beginning of the lease) unknown exercise prices in that the rent

that would be paid at the time at which the option could be

exercised would not be known at the beginning of the lease.

For this reason, although it might be possible to derive analytical

solutions to the valuation of the option-like characteristics of the

lease, the simplest approach is to use a Monte Carlo method. This

approach has been widely used in valuing options. Boyle (1976) used

Monte Carlo simulation for the first time to value options;. valuing

European call options on dividend paying stock using 5,000 trials

per estimate for various values of the underlying parameters. In

some circumstances, this technique can be used to derive a pragmatic

formulation of the specific valuation model. Chidambaran and

Figlewski (1995), for example, used Monte Carlo iterations of

100,000 varying the underlying parameters and then derived a quasi-

analytical solution by regressing the consequent valuations on the

parameters varied in the simulation.

1 In this list, the first number denotes the year in which the rent is

reviewed, the second indicates the year in which the existing rent was

fixed. Therefore, 20, 10 implies the option of raising the rent in the 20th

year when the rent has not been altered since it was set in the 10th year.

7

In this paper, we used the Monte Carlo method known as Latin

Hypercube Sampling – another name for this is stratified sampling

without replacement. Using this method, probability distribution is

split into n intervals of equal probability where n is the number of

iterations to be performed on the model

In the first iteration one of these intervals is selected using a

random number

A second random number is then generated to determine where

within that interval the probability function should lie

Then the value of that probability is generated

The process is repeated for the second iteration but the interval

used for the first iteration is marked as used and will not be

selected again

This process is repeated for all of the iterations. Since the

number of iterations is equal to the number of intervals each

interval will only have been sampled once and the distribution

will have been reproduced with predictable uniformity over the

range of probabilities.

Latin Hypercube sampling tends to force convergence of a sampled

distribution in fewer sample. As more iterations are run, the amount

of change in the statistics becomes less and less until the changes

meet a required minimum threshold (in our case the mean and

standard deviation of results were less than 0.5%). In addition, we

use a common random number generator when comparing results to

reduce the variance in the sampling procedure (Law and Kelton,

1991).

Previous Research in Real Estate

8

One of the fundamental principles in Finance Theory is that of Value

Additivity. This principle asserts that if there are two assets A

and B then the value of the portfolio combining A and B must be the

sum of the values of A and B. Indeed this lies at the heart of

modern portfolio theory in indicating that the only type of risk

that investors can seek to be paid for bearing is the systematic

component of risk (that is the risk which cannot be diversified

away). In property leases there are examples of valuation which seem

to be inconsistent with this principle, not least the paper by

Capozza and Sick (1991) who examine the inherent option of

redeveloping a property on the value of existing properties. They

showed that although the cash flow implications of not having a

perpetual income from the lease is only around 1% the removal of the

option to redevelop the property during the lease might be as large

as 20%. In other words, the option to redevelop the land might not

be reflected in the values of the existing rental income flows of

the property unless the right to redevelop the property was “owned”

by the land-owner or land-lord,

In general, under-utilised land can be viewed as a call option

because the holder has the right to erect a new property (subject to

planning or zoning restrictions). Geltner, Riddiough and Stojanovic

(1994) investigated the sensitivity of the redevelopment option to

the supply of available land and found that reducing the available

land uses in an area effectively stimulated development at a much

earlier stage than if alternative land uses were available to the

land owner. Capozza and Sick (1993) investigated the disparity in

value between urban and agricultural land using an option-based

approach. Speculative development is often subject to uncertainty

because of the uncertain timing and cost of any development. It can

be argued that development can be viewed as an Option since the

9

holder of vacant land can effectively exercise an option to develop

the land. Titman (1985), Paddock, Siegal and Smith (1988), Williams

(1991), Capozza and Sick (1993), Quigg (1993,1995), Capozza and Li

(1994) and Yamaguchi et al (1995) have all incorporated option

pricing theory to value such development options. The relationship

between land value, risk in the market for Real Estate and the

optimal development were identified by Williams (1991) in examining

the scale of development and who modelled the rental flow as a

geometric Brownian Motion with drift. Capozza and Li (1994) showed

that intensity interacts in important ways with timing, taxes, and

project values.

Perhaps the seminal article in this literature was Grenadier (1995)

who modelled the pricing of occupancy value of property as a

combination of two processes, first a GBM process that controlled

the occupancy value of rent if no construction was involved,

secondly a process that would control the supply of new property

space if construction did take place. In the elegant paper,

Grenadier demonstrates how the combination of the two processes

gives rise to lease–term structure effects in rents in an analogous

way to that of interest rates. However it should be noted that in

order to derive the solution to the equilibrium rent, Grenadier had

to estimate or assume a number of parameters, including a ratio of

space demand to construction costs that will trigger new supply of

space and the instantaneous correlations of the cost and demand

functions with the market portfolio. Clearly, the estimation of

these parameters presents major problems for investors/researchers

wanting to operationalise the model.

Empirical research in the applicability of option pricing models to

real estate has been limited because of the lack of appropriate

10

data. Quigg (1993) carried out an important study on the

development option value using 2,700 residential land transactions

in Seattle. She valued vacant urban land by accounting for the

option to wait to develop land and found that market prices reflect

an average premium of six percent of land value reflecting the

premium for optimal development. She also found that the standard

deviation of individual real estate assets ranged from 18 to 28

percent per annum.

Since the Black and Scholes paper, there has been a huge literature

of expanding the models of option pricing and its applications to

more exotic products and markets. Specifically there has also been

much development using the Binomial Option pricing model (Cox, Ross

and Rubinstein, 1979) which has been applied to real estate

valuation by Ward, Hendershott and French (1998) to analyse the

different types of rent adjustments observed in real estate leases

in the US, UK and Australiasia. The assumption in that paper was

that the property rental flows were spanned by existing assets in

the capital market and could therefore be valued using geometric

Brownian motion.

Geometric Brownian Motion

In our model, we use a discrete time analog of the Black-Scholes

option pricing model.

Letting:

Po = the price of property now – this is known

T = any future time (measured in years)

11

PT = the price of the property at time T – it is a random variable

and its value is not known until time T

Z = a standard normal variable (with mean = 0 and standard

deviation = 1)

= Mean percentage growth rate of the property (per year)

expressed as a decimal

= Standard deviation of the growth rate of the property

(per year) expressed as a decimal.

Using this notation, the future price of the property may be

modelled as:

PT = Po * exp[(µ - )*T + * Z* ]

Chriss (1997) provides an intuitive description of the process of

geometric Brownian motion, which had its origin in a physical

description of the motion of a heavy particle suspended in a medium

of light particles. The light particles move around rapidly, and as

a matter of course, occasionally randomly crash into the heavy

particle. Each collision slightly displaces the heavy particle; the

direction and magnitude of this displacement is random and

independent from all the other collisions. The nature of this

randomness does not change from collision to collision. In the

language of probability theory, each collision is an independent,

identically distributed random event. The mean and standard

deviation of the large particle’s motion depend on the amount of

time that has passed. If the particle’s behaviour is very volatile

in the short run it will be proportionately volatile in the long

run. Imagine property prices as heavy particles that are jarred

12

around by lighter particles, trades. Each transaction moves the

market price slightly. The conclusions of the model are:

If P is a property with a value of Pt at the current time t, and if

P follows a geometric Brownian motion with mean µ and standard

deviation , then the return on P between now (time t) and a future

time T is normally distributed with:

1. Mean = *(T – t)

2. Standard Deviation = *

The pricing of the property is also referred to as an Itô process,

which has an expected drift rate of (1) above and a variance rate of

(2) above. There are two aspects to note from this model. First, the

standard deviation increases in proportion to the square root of

time. Second, the rate of return is adjusted downwards by a factor

of . What this says is that short run returns alone are not

good predictors of long run returns.

To see this, consider a process with no uncertainty such as the

growth of a bank deposit. At a rate of 10% compounded annually a

deposit with grow from a principal sum of, say, £100, to £110 and

then to £121 and then to £138.1 and so on. However, suppose that

there is a 50/50 chance that at the end of the year the return will

either increase by 5% or fall by 5%. There is now uncertainty with

two possible random ‘jolts’ that could occur at each time step. The

result of this is there is an equal probability of a return of (1 +

0.05) being followed by a return of (1 – 0.05) i.e. 0.9975.

Mathematically as the process unfolds this equates to an adjustment

of (1 + x)*(1 – x) = 1- x2.

13

Heuristically, the stochastic component in returns under the

geometric Brownian motion model is assumed to be normally

distributed with a mean of zero and standard deviation of . If x

is a random variable representing the stochastic component of

Brownian motion then:

Var [x] = E [x2] - E [x]2

But, Var [x] = and since E [x] = 0 the above equation implies

that E [x2] = . And since there is a probability of a positive

return being followed by a negative return the average amount the

drift rate will be depressed is because x2 itself represents the

result of two price movements.

In modelling property prices, the effect of geometric Brownian

motion is to produce a lognormal probability distribution for the

value of the property. Such a distribution for year 10 of our

normative model (described below) is shown:

14

Parameters in Option Pricing Model

The Black and Scholes (1973) model of option pricing requires the

following parameters; current asset price, exercise price, risk-free

rate of return, the time to expiry of the option and the volatility

of returns of the underlying asset. In the case of income paying

assets, the model can be modified to reflect either the present

value of the income received from owning the asset or the equivalent

continuous income yield of the asset. In either case, the value of

the option is reduced, as the capital growth of the underlying asset

is lower, the higher the income received by owners of the underlying

asset.

The crucial assumption in using the Black and Scholes model is that

the asset is spanned by existing market assets which can be used to

form an instantaneous hedge portfolio which is riskless in the short

term. This then permits the assumption of the asset price being

modelled by geometric Brownian motion (GBM) and the substitution of

the riskless rate of interest for the risk-adjusted return on the

asset with a great advantage of computational simplicity of the

model valuations. In the Boness model, referred to in the

introductory paragraphs of this paper, the use of the ‘wrong’

discount rate tended to produce valuations of the option that were

too high. However if the assumption of spanning markets is not

justified then the use of GBM is formally flawed and the consequent

valuation of the option-like characteristics of the property lease

is also only an approximation of the equivalent market prices.

15

As Dixit and Pindyck (1994) explain, the relaxation of the

assumption of spanning implies that the expected return on the

underlying asset is no longer the risk free rate but an adjusted

return. Grenadier echoes this discussion in his model,

“…In the case of leasing markets, where theunderlying assets such as office buildings aresubject to substantial transactions costs,indivisibility, and the inability to be soldshort, such arbitrage arguments areparticularly questionable.”

(Grenadier, 1995, p304)

In these conditions, the expected return on the asset is reduced to

a level that is equivalent to the risk neutral return by subtracting

from the expected return a premium which reflects the systematic

risk of the non-tradable asset.

Or

( )

In this paper, we first use the heroic assumption that a tradable

property asset does exist and then we relax this assumption and use

a process as defined by (2) above to investigate how sensitive the

valuation is to changes in the assumptions of the model. This

methodology corresponds to factoring in the “market price of risk”

or Lambda discussed in Hull (1997) and Shimko (1992)

The UK Lease

16

In this paper, we are analysing the conventional UK institutional

lease that is specified for a maximum of 25 years with rent being

reviewed every five years. The rent is changed only upwards so if

market rents have fallen over a period of time, the tenant is left

paying a higher rent than available on other properties more

recently let. In practice there may be some concessions to tenants

such as break clauses but we ignore this issue in this paper. We

also ignore any reversionary value to the landlord at the end of the

lease since this contribution to the value of the property would be

the same regardless of the terms of the 25-year lease.

Following Grenadier, we assume that the asset value (which in our

initial case is assumed to be traded) is a perpetuity that offers a

constant income yield. The asset price is assumed to follow GBM. The

income from such an asset is then used to benchmark the rental value

of the individual property. Since we are not including the

reversionary value the lease can be thought of as the present value

of the annual rental flows. Thus in a risk neutral world the rental

flows are growing at a rate which is negatively related to the

initial yield of the asset. In terms of an equity investment this is

equivalent to saying that the dividend growth is inversely related

to the dividend yield.

We then create a lease which is based on the annual rent but fixed

for period of five years and then adjusted either to the higher

figure of the annual rent or remaining at the level attained at the

current point in time. Of course, we recognise that this will be

less valuable in present value terms than the annual rental flows.

We could have adjusted the five year rent to provide an equal

present value to the annual revised rent, but it is simpler to

17

represent the differences in terms of the total value of the lease

rather than an adjusted rent to provide the same values.

We arbitrarily fix the risk free rate at 5% and an initial yield of

4%. In terms of the five-year review, this provides an implicit

growth rate of 1.02% to provide the equivalent total return in a

risk neutral world.

Volatility

Property is usually characterised as a low volatility asset;

quarterly rents and yields often appear to move “stickily” in

response to changes in the economic environment. However over longer

horizons it is apparent that investors and developers appear

surprised at changes in rents and returns. This would suggest that

the underlying volatility is greater than that indicated by

quarterly returns. This suggestion is confirmed by estimates from

Lee and Ward (1999) who find, using a decomposition of the returns

from different portfolios, estimates of annual volatilities of

individual properties ranging from 16% to 32% and an average

correlation between properties of just over 0.15. This is a return

volatility which corresponds to the modelling of the asset price

returns used by Grenadier (op.cit,)

The volatility of returns was therefore specified to be a minimum of

zero (to correspond with the conventional DCF approach to valuing UK

property) and a maximum of 40%. The asset prices are then simulated

10,000 times using a common random number to provide different

rental paths, which are then summarised in the following table.

Table 1: The Effect of volatility on Lease ValueAll values expressed in £ million

18

Volatility 0% 10% 20% 30% 40%Value of Upwards-

only lease

60.

3

64.

9

71.

8

78.

9

86.

0Value of Annual

reviewed lease

62.

3

63.

3

62.

2

62.

1

61.

9Ratio of Upwards-

only to freely

floating

0.9

7

1.0

4

1.1

5

1.2

7

1.3

9

As expected, the effect of volatility is to increase the value of

option-like lease. In the case of zero volatility, the upwards-only

lease is less valuable than the annual review because the initial

rents are fixed at the same level. Clearly if there is any implied

rental growth, the step-pattern of the five year review pattern will

be less valuable than the annual review.

The normative case is shown in Table 2. In this case, the investor

believes that the parameters are misvalued and expects that the

property itself will outperform the rates implied by the market. In

the example used, we assumed that the rental growth rate was 2% more

than the rate implied by the present market prices. Whilst this

steps beyond the intrinsic arguments of the arbitrage based option

pricing model, it is a pragmatic attempt to deal with the present

market structure in the UK where the present yields in the property

market would appear to be so far out of line with the yields

available from bonds and equities that property would seem to imply

a negative rental growth or a huge risk premium.

Table 2: The Effect of volatility on Lease ValueThe Normative case – Abnormal rental growth of 2%

All values expressed in £ million

Volatility 0% 10% 20% 30% 40%

19

Value of Upwards-

only lease

71.

5

74.

0

81.

0

89.

2

97.

6Value of Annual

reviewed lease

76.

9

77.

0

77.

1

77.

3

77.

4Ratio of Upwards-

only to freely

floating

0.9

3

0.9

6

1.0

5

1.1

5

1.2

6

In this case, as expected the option effects are less important

because the rental growth effect dominates the implied rental growth

rates in the positive model. The implication is that if property is

undervalued the premium values that might be backed out of market

prices are too high. Of course if it were really possible to

construct an arbitrage portfolio, the difference between the two

tables would disappear; but in the absence of a practical arbitrage

portfolio, the differences remind us that the assumption of tradable

asset is an important element of option pricing when applied in the

real world.

Interest Rates

In Tables 1 and 2, we assume that the riskless rate of interest in

known and constant but in practice we observe a term structure of

interest rates that imply changing discount rates over time. In this

paper we look at the sensitivity of the values to changes in the

modelling process of interest rates. We relax the assumption of

constant interest rates and instead assume a model based on Cox,

Ingersoll and Ross (1985) in which the yield curve converges from

the short-term rate to a longer term equilibrium rate.

20

The Cox, Ingersoll and Ross model specifies the mean reverting

process as the following:

where a, b, and are constants. The short rate is pulled to a level

b at rate a. Superimposed on this “pull” is a normally distributed

stochastic term that has a standard deviation proportional to .

This means that as the short term interest rate increases, its

standard deviation increases.

Since the present yield curve is downward sloping, we assume that

the yield converges to 4% with different variability. The volatility

of the underlying asset is kept constant at 20%. The parameter, a,

is kept constant at 6%.

Table 3: The Effect of Mean Reverting Interest RatesAll values expressed in £ million

Volatility 20%

no

yield

reversi

on

20%

Std.dev.

of yld=

0.5%

20%

Std.dev. of

yld= 1.0%

Value of

Upwards-only

71.8 81.2 81.3

21

lease Value of

Annual

reviewed lease

62.2 64.8 64.9

Ratio of

Upwards-only

to freely

floating

1.15 1.31 1.31

As the imposition of the downward sloping yield curve reduces the

average interest rates, the introduction of mean reverting interest

rates increases the value of the lease as the discount rate is

reduced. It will be seen that the effect of increasing the

variability of the interest rate shifts has little or no effect on

the valuations. This result suggests that the assumption of mean –

reversion in the yield structure can probably be adequately captured

by reducing (or increasing) the average rate of interest rather than

using a stochastic interest rate model.

Underlying Asset: Tradable vs. Non-Tradable

It is in this area that there is most scope for error. As explained

above, the assumption of tradable assets allows the researcher to

apply a risk-neutral rate of interest as the total return on the

underlying asset. Without this assumption, the total return on the

asset can diverge from the risk neutral return. Given the

characteristics of property, the adjustment is, to a large extent,

arbitrary, influenced both by conceptual and empirical decisions.

In the equation (2) above, the total return has to be adjusted by

subtracting from the expected market return, the product of the risk

22

market premium and the individual systematic risk. In this paper, we

take the FTA All Share Index as representing the traded asset on

which the property depends. To capture the systematic risk of the

property, we regress the IPD Annual Property Index (returns) against

the All Share Index (returns). However because the Property Index is

an appraisal-based index, it is treated as an illiquid asset and

the regression is run following the Dimson method, i.e.

(3)

Table 4: Estimated Beta (Dimson) of PropertyWith respect to Equities

Coeffici

ent

S.Error

.

t-

statist

icIntercept 4.03 3.72 1.08Beta0 0.116 0.068 1.70Beta1 0.133 0.072 1.836Beta2 0.155 0.067 2.306Aggregate

Beta

0.40

Adjusted R-squared = 0.15.In the regression, we used contemporaneous returns and two lagged returns in the regression on the assumption that the stockmarket ‘leads’ the property market.

Using this methodology, the estimated beta was 0.4, the mean return

on the Stock market was

21% with a standard deviation of 25% and the risk free rate was, as

before 5%. The results are shown in Table 5. These results could be

23

suspect because the historical risk premium of the equity market is

very high if measured over the 1971-1998 period. An alternative

result could be found if one used the risk premium of the equity

market over the longer period 1919-1978. This widely quoted premium

of 8% (see Brealey and Myers, 1998) is often used as an estimate of

the forward-looking risk premium on equities. This is incorporated

in the analysis and the results are reported in Table 6. As can be

seen, the results are sensitive to the equity premium chosen. Since

there is no theoretical or a priori justification for a particular risk

premium, this must be seen as a major issue when the option-pricing

model is applied to property.

Table 5: The Effect of Non-tradable AssetsNormative case assumptions above

Historical Risk Premium on Equity Market: 16.5%, 1971-1998

All values expressed in £ million

Volatility 20%

= 0

20%

= 0.5

20%

= 1Value of

Annual

reviewed lease

77.1 44.4 29.4

Value of

Upwards-only

lease

81 63.3 60

Ratio of 1.05 1.43 1.97

24

Upwards-only

to freely

floating

Table 6: The Effect of Non-tradable AssetsNormative case assumptions above

Historical Risk Premium on Equity Market: 8%, 1919-1978

All values expressed in £ million

Volatility 20%

= 0

20%

= 0.5

20%

= 1Value of

Annual

reviewed lease

77.1 65.8 56.7

Value of

Upwards-only

lease

81 75 69.6

Ratio of

Upwards-only

to freely

floating

1.05 1.14 1.23

However, one need not stop at the choice of one risk premium. In

principle, the traded portfolio could consist of any relevant

portfolio and an obvious case could be made for the choice of bonds.

We therefore estimated the Dimson beta for property against the

Long-term government bond index reported in the IPD Annual Report

1999. This is presented in Table 7.

25

Table 7: Estimated Beta (Dimson) of PropertyWith respect to Bonds

Coeffici

ent

S.Error

.

t-

statist

icIntercept 4.88 4.057 1.204Beta0 0.099 0.140 0.710Beta1 0.199 0.141 1.412Beta2 0.265 0.138 1.921Aggregate

Beta

0.56

Source: IPD Annual Index Report (1999)Adjusted R-squared = 0.08.In the regression, we used contemporaneous returns and two lagged returns in the regression on the assumption that the bondmarket ‘leads’ the property market

As can be seen, the property index has a higher Beta with respect to

the bond index than it does with equities. (The statistical

insignificance of the estimates is not an issue in the Dimson

methodology since one is only looking for unbiased parameter

estimates). However, it was found that the risk premium for bonds

was 8.5% over the 1971-1998 period, a figure that is quite close to

the equity risk premium for the 1919-1978 period. In other words,

the results from varying the market traded asset from equities to

bonds resulted in a higher systematic risk factor but lower risk

premium. Therefore the results are consistent with the case already

reported in Tables 6 and 7; the important variable appears to be the

market risk premium.

26

Conclusions

The purpose of this paper was to explore the sensitivity of the

Option pricing model lease valuation to changes in the parameters

commonly used. The importance of volatility is readily understood

but the more crucial aspect is shown the estimated risk premium on

the traded asset if the assumption of property leases being traded

is relaxed.

In summarising the results of this investigation, it is convenient

to refer to Table 8 which presents the differing estimates of the

premium on a upwards-only rent review under different assumptions.

Table 7: Summary of Tables

Source Table 1 Table 2 Table 3 Table 5 Table 6Ratio of

Upwards-only

lease to

freely

floating

1.15 1.05 1.31 1.43

for

Beta =

0.5

1.14

for

Beta =

0.5

As can be seen the range of plausible premia is very wide and is

dominated by the distance between Table 2 (in which the investor

modifies the rental growth rate) and Table 5 (in which the investor

takes an optimistic interpretation of the historical return on the

“market portfolio”. Both of these adjustments are subjective and

27

both violate the possibility of creating a riskless hedge portfolio.

We are thus left with the uncomfortable conclusion that the

arbitrage-based model requires subjective inputs before it can be

applied.

28

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32