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AN INSURANCE ECONOMETRIC ASSETS VALUATION MODEL, FOR THE NATIONAL TECHNICAL UNIVERSITY OF ATHENS.

Dr. Costas Kyritsis Prof. Petros Kiochos

Software laboratory Department ofStatistics

National Technical and Insurance Science

University of Athens University of Pireas

Summary

In this paper we describe an econometric asset valuation model for the National Technical University of Athens. It was created in order to insure the buildings and scientific equipment .We make use of statistics and stochastic differential equations (geometric Brownian motion). It is described through tables the structure of the study , the buildings and the evolution of the assets of the Institution during the last twenty years, taking also into account the average rate of change of the value of money due to inflation and bank rates changes .

Key words: Insurance, Econometrics, Finance, Stochastic Differential Equations.

Introduction

In 1995 the administration of National Technical Univesity decided to carry out a study concerning the description of the Buildings and Equipment. The goal was to insure the University fortune which is in Patission and Zografou. The study would serve as a reference with which the Insurance Companies could give their offers.The study was undertaken by the first of the authors and was completed during September 1996.

The present paper is a research-oriented outline of the results and methods of the study based on the principles of Econometrics and Insurance.

1. Valuation of the assets through a sample.

If the University were keeping 3rd category accounting books (accumulative accounting) the task would be easy. We would have only to read the lasts year’s accumulative value of the Universities Assets. But the accounting, which is kept, is only incremental (Expenses only) and going back in time is only in paper and extremely voluminous.

Therefore we had to make use of Statistics and Econometrics for a fair estimation of the assets of the University. The sources of the data were:

a) The Technical Services of NTUA for the Buildings their area and diagrams of them

b) The accounting department of NTUA that keeps track of any new item that is purchased.

c) The secretaries of the Departments and Laboratories.

d) The directors of the Laboratories and Departments

e) Direct inspection of the places, buildings and Equipment .

The data are from a sample of Departments and Laboratories of a size of 6.69% only.

We should mention that the Assets of NTUA as a not for profit Organization is only Equities and does not Include Liabilities. Therefore there are not cashflows to any investors or discrimination between the Valuation of Assets or valuation of Equities .

As the valuation is for insurance the value of interest is not the depreciated or purchase value of the assets at the present but rather their replacement value at the present.

From the incremental accounting that is kept it was possible to extract the new investment in every year from 1973 to 1993 for this sample of 6.69% of Laboratories and Departments .It was used the assumption (after discussion with the accounting department) that, in the average, for a time interval of 20 years the equipment and furniture are completely depreciated and result to a zero value. In other words that the average asset life of an item is 20 years. Thus by adding up the new investment of every year for 20 years we could have a coarse estimation of the total accumulated value.

In addition as only the replacement value is of interest, instead of using deprecations rates, we would have to use the real time-changing value of money for 20 years (average rates of banks after tax minus inflation rate) in order to adjust the replacement values to final value of the last year.

The study among other data included the next items:

Each Building had its dossier. Each dossier contained

a) A report for the placement of the Building in the general area.

b) Indicative photos of the building outside and inside

c) An indicative diagram of the building

d) Data for the date hat was built ,its area etc. Number of rooms ,purpose ,use ,area, material of construction

e) Description of the means to protect the building from fire ,water, earthquakes ,thieves etc.

f) Table with the equipment proposed for insurance.

g) General comments

A control panel for all the buildings that contained.

a) a list of all buildings

b) A list with all the means of protection.

c) An organization plan of the University and its departments with useful information.

The estimation of the replacement value of the buildings is much easier because there is not new investment on them as fixed assets and the only change is the depreciation .In this paper we describe mainly the econometric model that was used to estimate the replacement value of the Equipment and furniture.

In the next table is estimated the area of each building.

BUILDINGS ZOGRAFOU	NOT NET AREA IN M^2
1. SOLID STATE	12,650.
2.PHYSICS	5.200,00
4. SOUND TECHNICS	921
5.HYDRAYLIC (OLD)	5.886,
9. PORTS	5.975,00
14.ADMINISTRATION	8.500,00
7.NAVAL-HYDRODYNAMICS	7.220,00
11.EARTHQAKES TECHNOLOGY	1.507,00
15.MINING	21.250,00
8.GENERAL DEPT.	13.920,00
12.CHEMICAL ENGINEERING	30.500,00
3.TOPOGRAPHY	9.173,00
6.STUDENTS
10.COMPUTER	3.850,00
13.RESTAURANT	1.510,00
16.TURBO MACHINES	180
17.TECHNOLOGY	180
TOTAL	128.422,00
PATISSION BUILDING
1.CIVIL ENGINEERING (GINI)	4.128,00
2.ARCHITECTURE (AVEROF)	4.128,00
3.MECHANICAL ENGINEERING (NEW BUILDINGS)	23.500,00
4.LABORATORY OF PAINTING	410
5.OLD ADMINISTRATION	2.600,00
TOTAL	34.766,00
GENERAL TOTAL M^2	163.188,00

2. The investment sequence from 1973 to 1993.

As we mentioned in the pervious paragraph we have a sequence of data which are the new investment each year and from the year 1973 to the year 1993. This sequence of numbers is only for a sample of 6.69% of all the University (relative to value). The number 6.69% was estimated as percentage of the new investment in these Departments and Laboratories in the year 1993 to the total new investment in 1993 of NTUA.

We adjust for the real time-changing value of money for 20 years (average rates of banks after tax minus inflation rate) with the recursive formula y_n+1= r_n y_n + x_n+1, where r_n = i_n- f_n . I_n is the average bank rate after taxes (or the rate of central bank) at the year n and f_n is the inflation rate at the year n. In this way we get a sequence of data y_n which is the accumulated investment till the year n .

The data , together with the rates of change of the value of money and the inflation rates are presented in the following tables.

Year	% rate of deposit	After taxes	Inflation rate	Real rate not including tax	Real rate including tax
1974.	9,00.	9,00.	26,90.	(-)17,90.	(-)17,90.
1976.	7,00.	7,00.	13,30.	(-)6,30.	(-)6,30.
1980.	13,50.	13,50.	24,90.	(-)11,40.	(-)11,40.
1981.	13,50.	13,50.	24,50.	(-)11,00.	(-)11,00.
1982.	13,50.	13,50.	21,00.	(-)7,50.	(-)7,50.
1983.	13,50.	13,50.	20,20.	(-)6,70.	(-)6,70.
1984.	15,00.	15,00.	18,50.	(-)3,50.	(-)3,50.
1985.	15,00.	15,00.	19,30.	(-)4,30.	(-)4,30.
1986.	15,00.	15,00.	23,00.	(-)8,00.	(-)8,00.
1987.	15,00.	15,00.	16,40.	(-)1,40.	(-)1,40.
1988.	15,00.	15,00.	13,50.	(+)1,50.	(+)1,50.
1989.	15,00.*	15,00.	13,70.	(+)1,30.	(+)1,30.
1990.	18,00.*	18,00.	20,40.	(-)2,40.	(-)2,40.
1991.***	18,00.*	15,00.	19,50.	(-)1,50.	(-)3,60.
1992.***	18,00.*	15,30.	15,80.	(+)2,20.	(-)0,50.
1993.***	17,70.*,**	15,00.	14,40.	(+)3,30.	(+)0,60.
1994.***	16,73.*.**	14,20.	10,80.	(+)5,93.	(+)3,42.
1995.	13,40.	11,39.	8,10.		(+)3.29.
1996.	11,80.

1) * From 6/6/1989 the rate is free and determined from the banks.

2) ** From 8/3/1993 , the minimum bank rates does not exist.

3) *** From 1/1/1991 there is tax 10%.

	Year		money time-value real rate	New investment in 6.69% of NTUA	New investment xn in all NTUA	Money time value adjusted accumulated total investment yn
1973		82.1%		3,114,879	46,571,529
1974		82.1%		3,583,427	53,576,937	91,812,162
1975		93.7%		4,911,947	73,440,054	148,817,839
1976		93.7%		5,322,863	79,583,788,	219,026,103
1977		93.7%		5,612,321	83,911,565	289,139,902
1978		93.7%		9,262,475	138,486,158	409,409,423
1979		93.7%		4,128,340	61,724,101	445,340,731
1980		88.6%		5,575,558	83,361,910	500,646,175
1981		89%		16,333,439	244,206,351	687,778,862
1982		92.5%		10,750,790	160,738,421	772,861,608
1983		93.3%		8,522,440	127,421,664	842,318,651
1984		96.4%		12,479,452	186,584,187	972,467,489
1985		95.7%		10,009,381	149,653,383	1,087,112,042
1986		92%		2,017,565,428+18,565,428	2,017,565,428+ 260,012,089	3,317,943,741
1987		98.6%		15,765,176	235,571,007	3,288,079,249
1988		101.6%		30,124,821	450,405,614	3,477,617,147
1989		101.3%		70,141,652	1,048,709,762	3,983,664,635
1990		97.6%		65,025,568	972,217,590	5,084,161,737
1991		96.4%		35,205,014	526,361,167	5,934,359,445
1992		99.5%		68,026,548	1,017,086,181	6,247,083,672
1993		100.6%		47,502,135	710,219,266	7,232,934,435
1994						7,938,551,308

We notice the significant increase of investment in 1986 with the system to measure the Earthquakes that costs almost 2 mega drachmas.

3. The Equipment and furniture modeled by the geometric Brownian motion with a (bilinear) stochastic differential equation.

The data year per year are discrete as the accounting is complete at a year base. Posterior data are of course deterministic. But either as future data or as uncertain past data it can be considered as sampling sequence from a time series or as discrete sampling of a continuous time stochastic process ,described by a stochastic differential equation .

Continuous time models have often simpler symbolic computation .In addition sometimes the exact discrete time maximum likelihood or least squares estimators of the parameters are intractable while the discrete approximation of continuous time maximum likelihood estimators are feasible. For this reason we chose a continuos time non-linear model. It is also a good opportunity to make explicit how the somehow advanced research on stochastic differential equations can be combined with very real, elementary and practical applications.

The model for the accumulated total investment that we chose is the (bilinear) geometric Brownian motion, described by the stochastic differential equation

,

Where B_t is a Brownian motion and r, s are constants. For the definition of the stochastic differential equations and the geometric Brownian motion see Oksental B. (1995), p121 Chpt. V p 60 ,exerc.7.9 ,p 121,example 5.1 p 60.

Some of the properties of this stochastic process are the following:

a) If r<(1/2)s², then X_t converges to 0 as t goes to infinite, almost surely (with probability equal to one).

The probability p to ever reach the value X starting from x₀ <X is

P=(x₀/X)^a where a=1-(2r)/ s ²

b)If r>(1/2) s² then X_t converges to as t goes to infinite almost surely.

The average time T that it takes to reach, for the first time X starting from x_{0 is}

T = log(X/ x₀)/(r-(1/2) s²)

c) The logarithm of this process X/x₀ is an ordinary Brownian motion with drift:

log(X/x₀) = (r-(1/2) s²)t + s dB_t .

The regression curve is the next:

The solution of this stochastic differential equation is given by the formula

The choice of the geometric Brownian motion is not the only model that give adequate results. We could as well have chosen the logistic stochastic differential equation, which is also non-linear (the stochastic Verhulst equation) or the Shiga equation. For applications, details and a discrete approximation to the continuous maximum likelihood estimator of them see P.E.Kloeden et al (1997) Chapter 6, E.pp 234-236 .

4. growth rate Estimation of the accumulated investment.

For the growth model of geometric Brownian motion we can have a least squares estimation of the parameters r and s. We simply take the logarithm of the regression curve of the process. It becomes an ordinary least squares line estimation .

LogE[X]=rt + logE[x₀]

The slope r of the regression line is estimated to be 0.219731823 while the intersection to the time axes is estimated to be 0.108481138. The correlation coefficient is 98.5% so it is quite a good fit . The sample variance is 1.912212413 . So a 95% confidence interval for the s is

S²(18)/x²_a/2 s² S²(18)/x²_1-(a/2) or 1.444808874 s 2.827783036, if we take it approximately 2.13. If we want a percentage rate of growth in a the year then log(1+ρ)= r or ρ=exp(r)-1 =24.57%. We remark nevertheless that this year rate of growth includes both the depreciation rate of the durable and the new investment growth rate.

We should not confuse the regression curve with an actual path of this stochastic process. The regression curve can be interpreted as the potential of growth or a best fit smoothing.

A simpler estimate of the rate of growth can be carried by interpolating the curve

log(X/ x₀)=(r-(1/2) s²)t +σ dB_t .

with initial value 0.091812162 and final value 7.938551308. This gives r=22.28% ,thus ρ=24.95%.

Acknowledgements. We would like to thank the administration,and the professors of the National Technical University for their support and data that they provided .

περιληψη

Αυτή η εργασία είναι σύνοψη μιάς οικονομετρικής μελέτης για την αξιολόγηση της περιουσίας το Εθνικού Μετσόβιου Πολυτεχνείου. Στόχος ήταν η ασφάλιση των κτιρίων και του επιστημονικού εξοπλισμού. Καθώς τα λογιστικά αρχεία δεν ήταν αρκούντως χρήσιμα πέρα απο μιά πενταετία, έπρεπε να ανατρέξουμε στην στατιστική και την οικονομετρία. Ετσι είχαμε την ευκαιρία να εφαρμόσουμε στοχαστικές διαφορικές εξισώσεις και την γεωμετρική κίνηση Brown. Τα μοντέλα αυτά έχουν ήδη εφαρμοστεί στην ασφάλιση με επιτυχία

Bibliography

Oksendal B.(1995) Stochastic Differential Equations Springer

Clarke R.G., Wilson B.D., Daines R.H., Nadault S.D. (1998) Strategic Financial Management

IRWIN

Howell Tong (1990) Non-linear Time Series Clarenton Press Oxford

Kiochos P.(1991) Techniques of Actuarial Studies (In Greek) Contemporary Publications Athens

Kiochos P.(1996) Actuarial mathematics Interbooks,Athens

Kiochos P.(1996) Methodology of Actuarial Studies Interbooks.Athens

Kiochos P.(1997) Methodology of the research Stamoulis P.Athens

Kloeden P.E.,.Platen E, Scurz H. (1997) Numerical Solutions of SDE Through Computer Experiments Springer .

Malliaris A.G., Brock W.A.(1982) Stochastic Methods in Economics and Finance North-Holland.

Malinvaud E.(1970) Statistical Methods of Econometrics North-Holland

Meigs R.F.,.Meigs W.F (1993) Accounting McGraw-Hill