# Single rate and dual rate in investment valuation

Suppose I want to buy a property investment, and I want a return of 5.00% on it. If it produces an income of £100 a year, I would pay £2,000 for such an investment. (Note that I ignore acquisition costs and other complications for the purpose of this explanation).

If it is a freehold, I will still have my £2,000 ten years later – either in the value of the property or, if I sell it, in cash. (Again, I am ignoring changes in yields and property values generally). For the purpose of the discussion of single and dual rates, it is useful to call this 5.00% the “remunerative rate”. In other words, by laying out £2,000, I get a 5.00% remunerative rate on my outlay of money and, in ten years’ time, I will still have that money or its equivalent in property value.

Compare that with the acquisition of a leasehold interest where the lease I buy as an investor only lasts for ten years. I still want my remunerative rate of 5.00%, but I must take into account that, in ten years’ time, whatever I have paid for the investment will be gone: the lease will come to an end, and I will therefore have nothing – no property interest and nothing to sell to recover my money. Even the income will now be received by the freeholder, not me. To deal with this problem, property valuers long ago developed the idea of the sinking fund. Instead of re-selling (or keeping) my investment at the end of ten years, as I can with a freehold, I put aside an annual sum out of whatever income I get, so as to reconstitute my capital. The only source for such contributions to my sinking fund is the income I get from the property during my lease. If that is still £100 a year, then each year, out of that £100, I must put aside a sum sufficient to get me back whatever price I paid for the investment – in the case of our ten-year lease, one-tenth of the price I pay to buy this leasehold investment.

The attached worksheet gives the results of calculations along these lines. The text and figures outside the thick-bordered, yellow box show the factors and some intermediate calculations, which can be ignored for this purpose. The first of the lines inside the box shows the result of what is set out in the paragraph above. The line “Disregarding the accumulative rate and tax” shows that the price I can pay and still achieve my 5.00% remunerative rate is £666.67. I will set aside £66.67 each year out of the £100 a year income as a sinking fund to reconstitute my capital – the price I paid – in ten years’ time. The balance of £33.33 a year is 5.00% on my outlay of £666.67, so I have achieved the remunerative rate I want.

Most investors have to pay tax on income. Let’s say my rate of tax is 40%, to illustrate. HMRC do not regard sinking fund contributions as a cost (although possibly they should), hence the line “Disregarding the accumulative rate only”. As my sinking fund contributions will have to pass through a tax sieve, I will have to reserve a larger part of the £100 a year income for those contributions. Now, I realise, I can only afford to pay £461.54. I will have to make annual sinking fund contributions of £76.92. Of that, £30.77 goes to HMRC, £46.15 to my sinking fund. That leaves £23.08 for me, which is 5.00% on my outlay of £461.54, again achieving my desired remunerative rate.

We have not yet come to dual rates. The only rate used so far is the single, remunerative rate of 5.00%. However, what I have set out above would result in a slight understatement of the price I should pay for the investment because, while I am putting aside money for the sinking fund, it will itself attract some interest. I can place that money on deposit, and a small interest accumulation will result. I say a “small” rate, because the rate on the sinking fund will be different from the remunerative rate. It is known as an “accumulative rate”, because it is the rate that is progressively added to the accumulating sinking fund. This must be a “risk-free rate” – not, in other words, a rate as high as the remunerative rate, which can expected to vary with market conditions, but one which gives me a certain return on my sinking fund to make sure that I definitely can reconstitute my capital when the lease comes to an end in ten years’ time.

This is why a calculation of this kind is called a “dual rate years’ purchase (or YP)” calculation. There are two different rates at work, doing different things: the remunerative rate giving me my true return; and the accumulative rate enhancing my sinking fund. As regards the accumulative rate, the Lands Tribunal (as it then was – now the Upper Tribunal (Lands Chamber)) decided convincingly in *Sportelli* that the risk-free rate is 2.25%.

To illustrate, see the “Dual rate without tax” and “Dual rate with tax” lines here. They show that the two calculations using single rates mentioned above slightly understate the price I can afford to pay while still getting my 5.00% remunerative rate. In fact, if I can disregard tax, I pay £712.82 and set aside £64.36 annually. If, like most people, I cannot disregard tax, I pay £498.80 and set aside £75.06 annually.

As Harry Hill might say, dual rates in a nutshell!

PB