For many years property valuation tables have provided valuers with an important tool for carrying out a variety of conventional investment valuation calculations quickly and accurately. Although some of the main tables in use have a long history even these have developed, both in the range and scope of the tables provided, more or less in line with the development of valuation technique.
So, while earlier publications consisted predominantly of compound interest tables, the latest editions all include discounted cash flow tables to facilitate the use of at least some of the growth-explicit techniques which are becoming more widespread, especially in the analysis of property investments.
It could be argued that valuation tables are less useful now than they were at a time, not so very long ago, when hand-held calculators were unheard of. Then, most valuers relied solely on the tables, used in conjunction with logarithms and long multiplication. For the simpler investment calculations, it has to be conceded that tables are not the necessity they once were. However, for some of the more complicated calculations, they can still offer an important saving in time.
The main general tables available are Parry’s, Bowcock and Rose. In addition specific tables for calculating constant rent (Rose) and equated yields (Donaldsons’) are also available. Donaldsons’ Investment Tables enable the calculation of an equated yield for a property investment which takes into account rent review patterns and future anticipated rental growth, facilitating the comparison between property and other investments. For a detailed consideration of the use of Rose’s Constant Rent Tables, the reader is referred to “Mainly for Students”, October 5, 1991, p 145.
The three general books of tables obviously overlap to a great extent, although they do contain different tables and different forms of presentation. Determining which is the most appropriate for general use is probably a matter of personal taste.
Students of valuation are faced with a choice. Some, on the one hand, may prefer to undertake all their own calculations by formulae, using a calculator. Others will prefer to rely solely on the tables. A worthwhile compromise is to use a combination of tables and calculator and this has much to recommend it. Being fluent in the use of formulae may well be important in dealing with combinations of interest and tax rates which are not specifically covered in the tables, and it certainly leads to a better understanding of the principles involved. However, using the tables can save time, particularly in the use of the more complex calculations required for contemporary methods, and can be used periodically to check that computation is providing the correct answer.
Ideally then, the student is well advised to become conversant with both the use of formulae and tables, and should also be aware of all the main books of tables, the differences between them and the bases on which they are constructed.
Parry’s
The first edition was prepared by Richard Parry in 1913. The latest, 11th edition, by A W Davidson, was published in 1989 and states clearly, in the introduction: “The purpose of this book is to provide a comprehensive set of tables available in one volume to meet the requirements of current practice.” Thus, the first edition included dual rate tables which were, at the time, coming into use for the valuation of terminable incomes.
The tables were originally constructed on the assumption that incomes were received annually in arrear, but the 11th edition now includes years’ purchase tables on the assumption that income is received quarterly in advance. A further development in the latest edition is the inclusion of internal rate of return tables on both non-growth and growth assumptions.
The effects of taxation were first included in the eighth edition when the famous coloured pages were introduced. I have to confess that, on buying my copy of the ninth edition (at a cost of £4!), the initial sense of excitement, stimulated by the coloured pages, soon gave way to confusion and it was some time before their true significance and application were fully understood.
In general, the tables are provided on the basis that the valuer is left to decide which are appropriate for use in a given circumstance. The introductory sections provide much information on the construction of the tables but avoid the temptation of entering into the controversy over which methods should be applied.
Bowcock
First published in 1978, these were probably the first to recognise the need for tables which reflected the widespread commercial practice of interest being paid half yearly. Thus, while the tables are headed by the nominal annual rate of interest, the computations have been carried out using the effective annual rate. This is unique to Bowcock’s tables and care should be taken to distinguish this approach from the practice of compounding on a quarterly in advance basis, which requires further adjustment.
The effective rate of interest, r, can be found by the formula:
Where j is the nominal rate of interest and m is the number of times a year that interest is added.
For example, taking the amount of £1 over one year at a nominal rate of interest of 10% would give a value of £1.10, but assuming that interest is added half yearly the following results:
The effective rate is thus 10.25% pa.
Further tables are provided for conversion to alternative interest periods of four and 12 times per year. However, the tables which are concerned with rental income are based on the additional assumption that incomes are normally received quarterly in advance and not annually in arrear. Even here it should be noted that, though the rate of interest at the top of the page is the nominal rate of interest, the calculations are based on the effective rate.
As a consequence, a comparison of what appear to be the same Years’ Purchase tables in both Parry’s and Bowcock will produce different results:
Rose
Rose’s Property Valuation Tables (not to be confused with the constant rent table by the same author) reproduce, in part, Inwood’s Tables, which were first published in 1811 and which included, apparently for the first time, dual rate years’ purchase tables. The multipliers in these and other tables became known as “Inwood’s factors”, and are still referred to as such by realtors in the United States, as distinct from the British preference for “years’ purchase”.
The author suggests, with some justification, that this latter appellation is to be regretted, as it is responsible in part for a certain amount of confusion. In the explanations which accompany the tables an attempt is made to clarify this confusion:
A man believed he could pay £100 for a farm let at £5 pa, and if he saved up all his rents for 20 years he would at the end of that period have bought the farm outright.
Thus the term “years’ purchase” has become associated with the idea of a pay-back period, when of course it is necessary to take account of the interest accruing on the rent received so that the farm is actually paid for in something less than 20 years.
Recognition of the effect of interest on rental income led to the inclusion of the “Present value of £1 pa” tables (Inwood’s factors), which could be applied to any income-producing, annuity-type investment, whether perpetual or terminable. The other type of investment recognised by Inwood is the reversion, valued using combinations of the present value and present value of £1 pa formulae but which in Rose are referred to simply as “reversion tables”.
One thing that remains unique about these tables is the side-by-side arrangement of the compound interest multipliers: Amount of £1, present value of £1, Amount of £1 pa and present value of £1 pa, known as “consolidated compound interest tables”. These are constructed on an annual-in-arrear assumption, but they are followed by present value of £1 pa tables constructed assuming interest to be payable quarterly in advance.
Basic compound interest tables
These basic formulae appear in all the tables and should be familiar. Using Parry’s notation:
The annuity £1 will purchase on a dual-and single-rate basis is found using the formula;
Annuity = i + s.
where s is the annual sinking fund instalment required to repay £1 of capital.
Years’ purchase tables on an annual-in-arrear basis will also be familiar, again using Parry’s notation:
However, the quarterly in advance formulae are less so:
The above formulae use Parry’s notation where a is the sinking fund rate of accumulation, r is the effective yield which assumes that income received during the year is available for re-investment quarterly at the same rate and i is the nominal rate of interest. Although Bowcock uses an almost identical formula, it is important to recognise that the effective rate of interest, r, is arrived at on different assumptions.
Tax-adjustment factors
In many instances it is necessary to make some allowance for the effects of taxation. Where it is necessary to calculate the true net income after tax has been deducted, this can be found by multiplying the income by the net adjustment factor (Tn) allowing for tax at the appropriate rate. The tax grossing factor (Tg), when multiplied by the true net income, will show the equivalent grossed-up income which must be achieved before tax.
The most important application of the tax factors is, of course, the dual rate tables, where allowance for taxation on that part of the income which must be invested in a sinking fund, to replace the capital outlay, is often required. The tax-grossing formula is added to the sinking fund formula which becomes:
Present value tables with allowance for taxation also appear in all tables. This allows the valuer to take account of the taxed income forgone during the waiting period. Thus if a purchaser buys a reversion with a five-year deferment, assuming a discount rate of 10%, the price paid would be:
This reflects the loss of income on the capital expenditure over the deferment period. However, if that income were to be taxed, the true loss of income is actually less than 10% pa. Adding to the net tax adjustment figure will take account of this difference. Assuming a tax rate of 25%:
Rose’s Tables also include reversionary tables which make allowance for subsequent payment of capital gains tax on disposal.
Discounted cash flow
With growth-explicit valuation techniques gaining increasing acceptance, especially in investment appraisal, these tables become more important, especially as many of the DCF equations are such that they cannot be solved by ordinary algebraic methods. Parry’s tables include internal rate of return tables which assume either no growth in the rental income on reversion or growth at various rates, using the formula:
The formula is used to find e, where the price paid is known and although the solution could be found using calculus, the solution for the tables was found using iterative methods. The internal rate of return is found when the present value of the income flow equates to the value or purchase price. Thus the true return of an investment can be calculated where the value of the investment, the rent and the full rental value are known. Growth can be included by compounding the full rental value at the appropriate rate.
It should be noted that in this formula, g, the rate of growth is taken into account only to the next review and, it is conceded:
If growth were built in for further periods, alternative rent review patterns would need to be covered. This would require far more space to be devoted to this table, perhaps at the expense of other tables… If this type of approach is contemplated, more complex models more suited to computer analysis should be used.
Bowcock’s discounted cash flow tables take account of growth or inflation at the effective annual rate and these allow for the calculation of equated rents, the capitalisation of rising incomes and perpetual incomes with regular rent reviews. Other tables give the years’ purchase for perpetual incomes using the inflation-risk-free yield (real value method), where the real value is assumed to be constant but where, even so, the income will be inflation-prone during the period between reviews.
Rose’s tables include calculations of the effective risk rate of an investment for perpetual freehold interests. For leasehold calculations the number of variables is such that only the formulae are provided.
No sets of tables could be published that would be other than fortuitously useful in circumventing the need for such a calculation, so many are the variables involved and so wide the range over which their variations would need to be covered.
Miscellaneous tables
Life tables are included in all the main tables and are taken from the tables of the Office of Population Censuses and Surveys based on the mortality experience of the population in England and Wales. These tables show a continuing improvement of life expectancy, but note the contemporary warning in the latest edition of Parry’s Tables concerning the possible effect of AIDS. The life tables are used as the basis for YP tables for single and joint lives to assist with valuation problems concerning life interests.
Other tables included in the various publications include metric conversion tables, mortgage redemption tables, logarithms of the compound interest functions, tables showing the accumulation of simple interest from date to date as well as calendars.
Summary of notation
One of the problems with valuation formulae is that there is no standard form of notation and, as can be seen right, the three main volumes adhere to different forms which overlap only in part.
Tables remain a useful, and in some cases, an essential tool for the valuer despite certain limitations. Students of valuation should be competent in their use and would be well advised to acquaint themselves with the commentaries accompanying each of the tables which, while avoiding issues of valuation practice, do help to further a greater understanding of the relationship between different types of financial asset, an area often glossed over in the standard valuation texts.
References
Bowcock P, Property Valuation Tables, 1978, Macmillan.
Rose JJ, Property Valuation Tables, Constituting the 34th Edition of Inwood’s Tables 1975, Freeland Press.
Davidson A W, Parry’s Valuation and Investment Tables, 11th edition, 1989, Estates Gazette.
