Wu Shou Zhi propounds the determination of equated yield of freeholds using imperfect market evidence.
With the drastic changes of property investment practices in the past decades, new techniques of analysis and valuation have been developed. A common factor in contemporary models is that they consider the explicit prospective future income flow based upon a DCF format employing the target rate or equated yield. A number of researchers have shown that the contemporary approaches are more logical and appropriate to property investment appraisal than traditional ones. However, the practical applications of the equated yield model are still limited.
The major argument on which the defence of conventional techniques relies is that the contemporary models have an inherent flaw in assuming an equated yield subjectively.
Since there is no problem or error in contemporary analysis and valuation where the perfect comparable exists (Baum and Crosby 1989), the debate is therefore focused upon the use of imperfect comparables. Because a perfect comparable requires not only the right physical and locational characteristics but also the same unexpired term and rent received to rental value ratio, the valuer is often faced with a lack of good comparable evidence.
As Baum and Crosby (1989) indicated:
Surely what is needed is a model which helps the valuer to reconcile non-perfect comparables with a degree of objectivity and a logical base which, in the last resort, can utilise any property comparable which has locational and physical similarities. Contemporary techniques offer an alternative which may satisfy these requirements. However, it has already been noted that the analysis process for contemporary equated yield or real value techniques requires a subiective estimate of equated yield choice.
In this article, an approach to determining equated yield of freeholds using imperfect market evidence is proposed. For a fully-let freehold, if two comparables let on different review pattens and, for a reversionary freehold, if two comparables having different unexpired terms/rents received under the existing leases are available, the equated yields required then can be calculated by the formulae established and the subjectivity of equated yield choice might thus be avoided. The sensitivities of the formulae are discussed and the mathematical analysis confirms Baum and Crosby’s conclusion that the effect of the subjective choice of equated yield is minimal for reversionary property with under normal years unexpired.
Fully-let freeholds
For a fully-let freehold, the following formula describing the quantitative relationship between equated yield e (decimal), rental growth pa g (decimal), review period t (years) and capitalisation rate k (decimal) is well known.
Now assume two similar freeholds just let at their rental values on t1 year and t2 year review pattems recently sold at capitalisation rates of k1 (decimal) and k2 (decimal) respectively.
Since the two comparables are similar in location and physical characteristics, they should have the same equated yield e and the same rental growth g. For the first freehold, by formula (ii), one has
By valuation tables and linear interpolation technique the equated yield could be objectively determined now.
Example 1
Value the freehold interest in a property just let at its rental value of £20,000 pa on a seven-year review pattern. One similar property just let on a five-year review pattern recently sold for a price based on a 6% capitalisation rate. Another comparable let on a three-year review pattern recently sold at a capitalisation rate of 5.6%.
Analysis
Calculation for equated yield by formula(1) Trial rate 12%
Reversionary freeholds
The analysis and valuation of reversionary freeholds are more complex than that of fully let freeholds. According to the equated yield model, the formula to calculate the capital value of a reversionary freehold is as follows:
In order to determine the equated yield one needs two market transactions of similar properties with different lease structures. Assume that one property is let on a lease with n1 years unexpired at a current net rent passing of £r1 pa. ERV based on t1 year reviews is £R1 pa and rack rented capitalisation rate for this type of property, also let on t1 year reviews, is k1 (decimal). The property has just been sold for £C1. The other property is let on lease with n2 years unexpired at £r2 pa. ERV based on t2 year reviews is £R2 pa and rack rented capitalisation rate is k2 (t2 year reviews). It has just been sold for a price of £C2.
By formula (v), one has:
As formula (1), equated yield of reversionary freeholds can now be calculated by formula (2) using valuation tables and linear interpolation.
Example 2
Value a reversionary freehold property let on a lease with three years unexpired at £12,000 pa, with ERV based on a letting on seven-year rent reviews of £20,000 pa. Assume the following information available from comparable transactions:
A similar property is let at a current rent of £15,000 pa with two years unexpired. The estimated rental value is £20,000 pa and the rack rented capitalisation rate is 5% (three-year reviews) . The property has just been sold for £393,000. Another similar property is let on a lease with seven years unexpired at £10, 000 pa, with an ERV based on five-year reviews of £20,000 pa and capitalisation rate of 5.5%. The property has recently been sold for £315,000.
Some special cases of the second formula
Formula (2) is more complex than formula (1). However, there are some special cases in which the formula will become much simpler.
Example 4
Two similar properties both let on a lease with four years unexpired, but one at £8,000 pa, the other at £6,000 pa. They will have the same five-year reviews after reversion and have recently been sold for £165,000 and £159,500 respectively.
It can be seen from the discussion above that the more similar the two comparables, the less market information needed and so the simpler the calculation. Conversely, if the market can offer only evidence with widely different lease structures, more calculations have to be done to offset the imperfections.
Sensitivity of equated yield
Since formulae (1) and (2) are derived from reasonable prerequisites and a strictly mathematical inference, they should be theoretically logical. For the purpose of practical applications, the sensitivities of the formulae have yet to be considered. By “sensitivity” I mean the effect of the choice of trial rates on the values of the functions in the left side of the formulae.
The two formulae resemble each other somewhat in appearance. There is, however, a great difference between the two in sensitivity. Comparing the two functions in the left side of the formulae, one can easily conclude that formula (2) is much less sensitive than formula (1). In formula (2), in fact, the value of term
(r x YPn years @e)
is extremely small relative to the whole capital value C when the unexpired term is not very long. Thus the chance of trial rate e affects the value of (C – r x YPn years @ e) and, in turn, the whole value of function
little. In other words, the solution to formula (2) is sensitive to the comparables’ capital values, C1 and C2 , while formula (1) performs otherwise.
This sensitivity of equated yield choice reconciles completely Baum and Crosby’s conclusions. They (Baum and Crosby 1989, Crosby 1983) demonstrated that a realistic range of equated yield would result in very little difference in the valuation of a reversionary property with normal years unexpired and that, for the valuation of a fully let freehold, the wider the review pattern the more significant the effect of equated yield choice. The mathematical analyses of the formulae (1) and (2) support this view strongly.
The non-sensitivity of trial rate choices may sometimes produce difficulty in finding the practical solution of formula (2). As is known, interpolation technique requires the practical solution of formula (2). As is known, interpolation technique requires the selection of two trial rates, one giving a positive function value, the other a negative value, and then interpolating between the two. Since the chance of trial rate affects the function value little, it is not impossible that either a positive function value or a negative value can not be got in a reasonable range of trial rates. It happens especially when the data from market evidence are not so reliable. It is thus more appropriate to use formula (2) for a longer reversionary comparable. As for a property with normal years unexpired, the reader is recommended to attach great importance to Baum and Crosby’s research mentioned above.
Conclusion
The subjective choice of equated yield in DCF based models is not unavoidable. What is required is only one more comparable, even though imperfect. The formulae established in this article are logical in theory and useful in practice. However, one must bear in mind that formula (2) is sensitive to data and its use is thus subject to the reliability of market evidence.
Wu Shou Zhi is a mathematician from Suzhou Institute of Urban Construction and Environmental Protection in China and is currently based at the Centre for Research in the Built Environment at the University of Glamorgan.
