Real Estate Yield Math in the UK

The Business Functions Approach

Introduction/Synopsis

Yield is used all the time to establish the value of a rental stream, and is intrinsic to the formal valuations of commercial and residential property. Yield has two distinct faces: in everyday terms, yield is analogous to the Profit/Equity (P/E) ratio used to value companies, in that it converts a current annual income into a capital value using a more or less observable market value indicator (yield). In a more theoretical sense, yield is a shortcut to determining the Net Present Value of a rental stream.

It is common practise today to use full cashflow DCF methods in tandem with conventional property yields to establish a property’s value. Controversy lingers on about the relative merits of yield methods versus DCF methods. I’m not a valuer, but strongly recommend doing both – a DCF valuation is highly transparent and consistent with how other asset classes are valued, and the yield calculation calibrates your result to the market through your knowledge of market yields.

What this article does is introduce the basic yield concept, then present the conventional mathematics. Then, because the conventional math makes some approximations, we go on to present what is called here DCF math. The DCF Math derives the same types of equations as the conventional math but by making fewer assumptions. The DCF math versions of the equations are not in general use, probably because they are more complicated. They are presented here and offered as an option in Business Functions because they are easily programmable and, to me, more correct.

Business Functions (BF) has functions for both conventional and DCF math, so you can use both methodologies. In addition, BF has functions for calculating the full-blown DCF present value of its many rental functions, thereby providing a highly accurate double-check.

Basic Principles

We are all used to calculating a value V using the standard yield equation:

Strictly speaking, this only applies for a scenario we call here the Basic rack-rented case, meaning that the market rent is equal to the passing rent, so you don’t need to worry about the extent to which the passing rent is different from the market rent (‘over’ and ‘under’ renting). This case is only really valid either (i) when the rental growth is zero between reviews; (ii) you’ve just had a rent review; or (iii) you are at the start of a lease with no rent-free period. However, it’s a convenient method of getting a handle on value before progressing to more accurate methods.

Where does the yield come from? It comes from observations in the market and has a strong empirical element – in particular, yield embodies a lot more than just the cost of money, it has rent review pattern, market and tenant qualities to it as well. Because of its all inclusive nature, yield is sometimes known as an ‘all-risks’ yield.

Notice that the yield in equation was written as y_N, meaning the Nominal Yield. This means that it is the yield applied to the aggregate annual rent as is, regardless of rental frequency.

Much of the yield math has historically been presented on the assumption that rents are annually in arrears, because the formulae and arithmetic were much simpler to manage in the days before computers. This has caused some confusion that hopefully this article will help resolve. Clearly rent receivable quarterly in advance is worth more than rent received annually in arrear, and I will go on to derive how the math deals with both situations.

The key concept is the difference between the Nominal Yield and the True Yield. The difference between the two is analogous to the distinction between a Simple interest rate and an AER (Annual Equivalent, or Annual Effective Rate ).

The first thing to say is that if you in a rack-rented situation, Equation works fine. It does not assume rents are annually in arrear, it just requires that you use the nominal yield that has been derived from properties with a similar rental frequency i.e. if you derived the yield from a quarterly rented property, make sure you use it for valuing a quarterly rented property.

Secondly, if the rents are receivable annually in arrear then the nominal yield equals the true yield. This rather academic result is intrinsic in the definition of an AER, which says it’s the rate which would apply if the rents (or interest) were annually in arrear instead of the actual payment frequency. If rents are payable annually in arrear, then:

That gives us enough to go and derive the conventional yield math that assumes rent is paid annually in arrear, and once we’ve got that, we can go on to derive the much more useful situation where rents are paid at a specified frequency, such as quarterly in advance.

A. Conventional Yield Math assuming rents annually in arrear

A.1. Simple Yield Valuation – Basic Rack-Rented Case ^CONV

Both equations and apply in this situation because the nominal yield is the same as the true yield for rents annually in arrear. For the rest of this section (A) the equations will be derived in terms of the true yield y_T but remember that for rents paid annually in arrears y_T=y_N.

A.2. Yield Valuation Where the Passing Rent is different to the Market Rent ^CONV

This is often the situation in reality, as the lease being valued is usually inbetween reviews. Here there is more to consider. There is the Passing Rent P_,the Market Rent (ERV) R , and the Next Market Review r years away (or ‘deferred’).

What we have to do is adjust the basic Yield equation, and here we see some nice shortcut maths.

The method adjusts by horizontally slicing the rental income into ‘core’ (the passing rent) and ‘layer’ (the excess of the market rent over the passing rent). Two rather strong assumptions are made in the process.

The ERV is the same at the valuation date as it is at the review date ie no growth between these two dates. (g=0 until next review).
The discount rate is equal to the true yield over this period (y_T=d).

So you can say:

Then if, as is a common, we use the same yield for core and layer, and substituting y_T for d you get:

The effect of the assumptions does in practise cancel out to a degree, but it does introduce a small inaccuracy I shall look at later.

Incidentally, if you don’t like the ‘core’ and ‘layer’ approach, you can arrive at the same result with this derivation:

A.3. Unexpired Rent Free Periods – The Standard Case ^CONV

The existence of a rent-free period of f years just means that the passing rent term in equation is deferred, and again we assume y_T=d:

Like before, if you don’t like the ‘deferred layers’ method of derivation, there is another way:

As an aside, it has been said that you could use equation with the appropriate discount rate and all is well. I think that this may make matters worse because the g=0 assumption is still there, and the d=y_Tassumption acts to counteract the inaccuracy introduced.

A.4. The General Result ^CONV

It is possible to extend the yield math to a series of rental steps. Essentially the logic is the same, in that each layer can be valued separately. At this point we have a generalised equation for valuing deferred perpetuities by layers:

Whilst R_n is the market rent, R_1..n-1arefixed rents in this formula. It is assumed that d=y_T and g=0 (inbetween reviews). One way of putting the above equation into practise is to use the ‘Years Purchase of a Perpetuity Deferred’ term,

to multiply each ‘layer’ of deferred rent. Remember you can also use y_T=y_N in any of the equations in this section (A) because it is for payments annually in arrear.

A.5. Costs

Finally there are the usual costs to deduct from a valuation, intended to represent selling fees:

where V_{BEFORE_FEES}is any of the valuation equations.

A.6. Equivalent Yield

Equivalent Yield is just the reverse of determining the value: in other words it is the yield that gives rise to the value V. Unfortunately it’s not possible to re-arrange most valuation equations to solve directly for y, so a goalseek is usually performed, iteratively recalcing with different values of y until V reaches the desired result.

It’s just worth mentioning here that an equivalent yield to some people means the rate of return (IRR) of the property. This is the case in North America and even sometimes in the UK for investment purposes. Business Functions future work in this particular area will refer to this latter type of equivalent yield as the Equivalent Investment Yield.

B. Conventional Yield Math for actual Rental Frequency

All of the equations in section (A) assume rents are paid annually in arrear.

The standard way to convert a Present Value assuming payments annually in arrear to payments m times per year in advance, is to multiply by the formula used in the Business Functions CorrectionM function:

where m is the payments per year (+ve for advance, -ve for in arrears). By the way, this equation has an elegant and handy characteristic, which is often overlooked, that just by changing the sign of m you can specify payments in advance (eg rent) or arrears (eg interest).

We are going now to derive the same relationships as previously for rental payments m times per year. Of course you can simply assume payments in arrear and multiply resultant value by CorrectionM , but the derivations will be useful to solve for other parameters such as equivalent yield.

B.1. Application of Rental Frequency to the Basic Rack-Rented Case ^CONV

Now it is time to adapt the math derived above for rents quarterly in advance, starting with the Basic rack-rented case.

B.1.1 Value

For calculating value from the Nominal Yield we have the original Equation :

This equation works just fine with rents paid quarterly, the only restriction being that is only applicable for the rack-rented situation. Going to express in terms of True Yield, y_T, and using equation :

Equation useful if you like to use True AER Yields to calculate values in the basic Rack-Rented Case, but most people use the Nominal Yield and equation .

B.1.2 Converting the yield between True and Nominal

In the Basic Rack-Rented Case, if you have a value already calculated using a Nominal Yield you can convert it to a True Yield by saying:

assume d=y_Ttherefore

TrueToNom Function

and the converse,

NomToTrue Function

B.2. Application of Rental Frequency to the Standard Case of Passing Rent different to Market Rent ^CONV

B.2.1 Value

Applying the same logic to the Standard Case as in the Basic rack-rented case, you can say the same thing as before:

This simplifies slightly to:

CapValue Function

This is a useful standard formula for a situation that has an unexpired rent-free period, a passing rent different to the market, and a review to market after t years. If you don’t have a rent-free period, you can just make f=0.

To find the True Equivalent Yield, y_T, you need to goalseek using equation .

If you want Equation expressed using the Nominal Yield y_N, then by substituting equation you get:

CapValue Function

This is, to me, an important formula, because it’s the one that ought to be used routinely to value. You should either use this or the generalised formula in equations and . Unfortunately, for me anyway and as I go on to describe later, people usually use the annual-in-arrears equations of section A (eg eqn ) and, well, basically get it wrong.

B.2.2 Converting the Yield between True and Nominal

Using the early forms of equation and equation expressed in terms of y_N:

therefore

TrueToNom Function

NomToTrue Function

which is the same result as with the Basic-Rack Rented case equations and .

Note also that:

B.3. Application of Rental Frequency to the General Case ^CONV

Equation enables you to deduce the multi-payment version of the general result in equation :

The term

YPPDef Function

is the ‘Years Purchase in Perpetuity Deferred’ term adjusted for rental periodicity, and can be used as before to calculate the value of each ‘layer’ of rent, or, using Equation you can use the exact same in terms of y_T:

YPPDef Function

Notice that if m=-1 (ie payments annually in arrear) then both Equations and are the same as the ‘annually in arrears’ equation .

B.4 Solving for Equivalent Yield ^CONV

You can use equations , or and goalseek to find the yield. Or you can use Business Functions EqYield function, which does a iterative goalseek internally.

B.5 Wrong Practises?

B.5.1 Wrong 1: Annual-in Arrears Valuations used for True Equivalent Yields

There is one practise that is very widely adopted, that seems to us theoretically flawed. What is commonly done is that a Yield is used with annually-in-arrears formulae (equations or ) to calculate the property’s value. Then, with this value and using the rental frequency math (equation ), a True Equivalent Yield is derived based on payments quarterly in advance. In other words, the value is derived using:

and then the True yield is calculated by iteratively solving equation :

What is the aim? The argument is that, although it is wrong not to take into account rental frequency in calculating a value, the market has been doing it so long that it adjusts to imperfections in the technique. As much as I have gone over and over the math, I don’t think this is right. The traditional method of V=P/y_N is and always was fine, because it was simply the value divided by the rent. Where the methodology went wrong was when, many years ago, V=P/y_N was extrapolated to a non-rack rented situation and the ‘layer’ formulae of Section A were created without regard for rental frequency. Instead of using formulae that accepted a quarterly nominal yield (equations / or ) which would have been consistent with the rack-rent equation, annual-in-arrears formulae (equations / or equation ) have historically been widely adopted. These are inconsistent with the rack-rent equation because, unlike that equation, they require payments to be annually in arrears, and the result is a method that it’s clear many are uncomfortable with, but is seldom called wrong. We are now seemingly in the process of trying to backtrack and encourage the use of yields on a more consistent basis, whilst at the same time trying to retain the traditional annual-in-arrears methodology and this is a fudge that is not really sustainable.

So having calculated a slightly dubious value, what relevance has the equivalent yield? It’s a bit confusing, isn’t it? Sure, the combination of valuation and rent gives rise to True and Nominal Yields, but they have their origins in an incorrect valuation. So is the True Yield really equivalent to the yield used for the valuation (which now has the dodgy status of being Nominal, annually in arrears)? Well, no, they are not equivalent, because, to labour the point, one yield has been applied incorrectly. It is because of this inconsistency that it is sometimes stated that Nominal to True yield conversion formulae mysteriously don’t work in other than rack-rented scenarios. They would work if they were applied correctly. Well, we know from the math in this paper that the conversion formulae do work, so what’s gone wrong?

What happened is that by assuming that the annually in arrears value was correct, it has been implicitly assumed that d=y_N, where that y_N was a quarterly nominal yield. Then in the goalseek for the equivalent yield formulae are used that assume d=y_T. That inconsistency is why the circle won’t square, and the conversion formulae don’t appear work with these yields, and their values will never agree with DCF, etc. If you assume d=y_N, you can easily derive the value that is called the ‘annual in arrears’ value, but you’ve broken the laws of finance by discounting annually at a quarterly nominal rate.

In practise, the debate never gets this far. You either run with the conventional math, as is, or you give up and go for a full-blown DCF, forever baffled. In fact, the muddy application of yield math is the main reason I advocate for adopting a full-blown DCF approach.

My recommendation, if you are using the conventional yield math, is to use equations / or , which are simply the ‘layered’ equivalent of the rack-rent equation . In using these equations you have simply used a nominal quarterly yield in the way it was meant to be used, and therefore the method is highly defendable. You still have the debatable assumptions of y=d and g=0, but the methodology is consistent, the yield conversion formulae work, and you can work much more directly knowing the math is correct.

B.5.2 Wrong 2: Comparing Bond Yields to Property Yields

It’s common to compare property yields to bond yields, sometimes with an emphasis on demonstrating how, if you compare on a like for like basis, using a True Property Yield as against an AER effective bond yield, the comparison is more favourable for property as an asset class than if you simply compared the nominal yields. The comparison clearly is better (for property) if you adjust for the in-advance rental payments as against the in-arrears interest payments, and to compare apples with apples in this way is no less than what is required. But to make the comparison at all is fundamentally flawed.

What is happening is that the property yield is being used as proxy of rate of return. With a bond, it’s yield to redemption is its rate of return. Not so property (unless you are in North America where by definition a property yield is the property’s rate of return). What is the real relationship between yield and rate of return? If you look at the Implied Discount Rate section in the appendix, you will see that the discount rate (ie rate of return) depends on the rental growth rate. And that’s the crux of the issue – a property grows in value, a bond doesn’t, so there really isn’t a comparison (unless perhaps you are looking index linked bonds). If you do get property to compare well against bonds, then you have done very well, because the yield ought to be a lot lower. I think this is one reason why commercial property tends to be if anything undervalued in the UK, and why North American DCF valuation techniques are accused of over-valuing properties. I suppose you can have a working rule of thumb that if your property yield exceeds a bond yield, then your property growth is an added bonus, and intuitively it looks a good deal. But it’s not very scientific.

C. DCF Math

The conventional yield math has certain problems and is, in my view, inaccurate. In particular it is hard not to feel deeply uncomfortable with the two assumptions which are made

The growth rate between the Valuation Date and the next review is zero.
The Discount Rate applicable until next review is equal to the True Yield.

These assumptions are, in practise, not disastrous, because:

The period of time between valuation and next review is not that great, 5 years at the most.
To some extent they cancel out.

However, no amount of algebra seems to be able to make them right, and the problem is that the generalised result of equation offers the promise of being able to value stepped rents far out into the future, at which point the defective assumptions really start to bite, and errors become significant.

DCF Math approaches the yield calculation afresh by saying that rents are valued by quarterly nominal yields, which are empirical PE ratio measures observed in the market. However, where you are inbetween reviews and/or have rent-free periods, DCF Math says that you can adjust correctly over this time-period if you use an analytical relationship, coined here as the Theoretical Yield, which is a function of discount rate, rental growth and time inbetween reviews (review periodicity).

C.1 Theoretical Yield ^DCF

A regular stepped rental profile with growth rate g per annum and reviewing every t years has a useful algebraic formulation, derived in detail in the Appendix. From this you find:

Nominal Theoretical Yield for rents annually in arrear:

TheorYield Function

Nominal Theoretical Yield for rents paid m times per year:

TheorYield Function

True (AER) Yield:

TheorYield Function

These equations are based on the PV of a growing rental stream, and assume that a review has just occurred. When they are used, you should check that this assumption is more or less valid.

Equations , and are useful, because they show you what yields ought to be (in a perfect world) and because it helps us simplify DCF Math by substituting away any (1+g) terms that come up in subsequent derivations.

It’s an open question as to whether these theoretical yields concur with the market, because you only ever know y_Nby market observation and can therefore only deduce g and d. Also note that the equations don’t cater for any breaks, so the yield if anything could be on the low side. However, for the narrow use we are putting them to, that of adjusting values inbetween reviews, they should be a good basis of estimation.

C.2 Application of DCF Math to the Standard Case
(inbetween Reviews with unexpired Rent-Free Period) ^DCF

Using the Theoretical Yield and the derivations detailed in the Appendix, you can derive equivalent DCF equations that correspond to the standard cases in equations and .

Assuming rent paid annually in arrear

CapValueDCF Function

Assuming rent paid m times per year (in terms of true yield y_T)

CapValueDCF Function

and in terms of nominal yield y_N

CapValueDCF Function

They are pretty frightening, but they are correct. Crucially you can now assume a discount rate when adjusting for rent-free periods and being inbetween reviews, rather than assuming discount rate=yield. Also implicit in the formula derivation, but not obvious in its final form, is that growth from the valuation date until the next review is taken into account in the formula. If you use these formulae you will get the same results as a full-blown DCF, which is, in my opinion, the acid test.

C.3 Application of DCF Math to the General Result ^DCF

DCF Math can also be used to get a general result for stepped rent scenarios. You can’t however, use any of the ‘layering’ logic of deferred perpetuities used behind the conventional math equation , because that logic just doesn’t work with a DCF of growing rents – in fact, as far as DCF Math is concerned, that neat layering logic is the reason conventional Math is wrong.

What DCF Math can do though is have two ‘Years Purchase’ factors. One, the YPAD (Years Purchase of a deferred annuity) can be used for each of the rental steps, and YPPD (Years Purchase of a deferred perpetuity) can be used for the final reversion to market.

In terms of the True Yield y_T:

In terms of the Nominal Yield y_N:

These formulae are used in Business Functions’ YPPDefDCF , YPADefDCF , CapValueDCFG and EqYieldDCFG functions.

C.4. Equivalent Yield ^DCF

You can use equations , or and goalseek to find the yield. Or you can use Business Functions EqYieldDCF function, which does an iterative goalseek internally.

D. Business Functions Approach

Business Functions implements a number functions in the Valuation Family and Valuation DCF Family.

Most of them accept or return either a True or Nominal Yield, depending on what you specify in the YieldOpt variable.

For Conventional Yield Math:

CapValue calculates a traditional yield-based valuation, allowing for proper rental frequency.
EqYield does the inverse of CapValue, solving iteratively for the required yield.
CapValueG and EqYieldG are the general versions that allow for stepped rents.
YPPDef calculates the ‘Years Purchase of a Deferred Perpetuity’, allowing for rental frequency.
NomToTrue converts a Nominal Yield (ie one based on actual rental frequency) to a True Yield (ie an AER that would have been the yield if payments had been annually in arrear)
TrueToNom converts a True Yield to a Nominal Yield.

For DCF Math:

CapValueDCF does the same as CapValue but accepts the input of a discount rate and does not require the assumption that the yield equals the discount rate.
EqYieldDCF does the inverse of CapValueDCF, solving for the Equivalent Yield.
CapValueDCFG and EqYieldDCFG are the general versions that allow for fixed stepped rents correctly.
NomToTrueDCF converts a Nominal to a True yield, accepting in addition to the yield and the rental frequency, a discount rate.
TrueToNomDCF converts a True to a Nominal yield.
ImpGrowth deduces the implied annual rental growth rate, given a yield and a discount rate.
ImpDiscRate, like ImpGrowth, deduces, in this case, the implied discount rate, given a growth rate and a yield.
PVRent calculates the Present Value of a rental stream based on discount rate and rental growth.
TheorYield , related to PVRent, calculates the Theoretical Yield given a value, a discount rate and a growth rate.

E. Recommendations

These are my personal recommendations.

For valuing a property, whether rack-rented or stepped rents, do a full DCF cashflow. Then work out the equivalent yield (DCF Math, EqYieldDCF or EqYieldDCFG) to validate against market data. Use functions like PVRentGrow to do the full DCF in a single function call, either as a double-check or instead of the full DCF on the spreadsheet.
For calculating an Exit Cap at the end of the cashflow, or a development appraisal use CapValueDCF.
If you are wedded to using conventional (as opposed to DCF) yield math, use CapValue to calculate value and EqYield to get the equivalent yield. When calculating the value, use a quarterly nominal yield (ie simply rent/value) with the correct quarterly (m=4) rental frequency – don’t use the annual-in-arrears machinations of section B.5.1 unless you really have a reason to. Be very wary of using the generalised versions (CapValueG and EqYieldG) where the steps over 5 years in the future.