Kevin Dowd
13 August 2018
Most UK Equity Release mortgages1 involve a no-negative equity guarantee (NNEG) by which the lender guarantees that the borrower (or their estate, if they have passed away by then) will never need to pay back more than the value of their house when the loan is repaid.
The valuation of these NNEGs has suddenly become a hot issue in light of this week’s two reports – Howard Mustoe’s BBC story “Home equity release may cost pension firms billions” and my report, “Asleep at the Wheel: the Prudential Regulation Authority and the Equity Release Sector” – and a BBC Radio 4 programme “The Equity Release Trap” aired last Tuesday (8 August 2018) at 8 pm.
These reports and some of the reaction to them show that there is considerable confusion about the valuation of these NNEG guarantees.
There shouldn’t be.
The truth is that we have known how to do these valuations properly since the seminal option pricing work of Black, Scholes and Merton in the 1970s (Black and Scholes, 1973; Merton, 1973; Black, 1976).
In this posting, I set out the correct way to value these guarantees and explain why the current industry standard approach is not only indefensible, but also leads to major under-valuations of these guarantees.
Let’s start with a simple example. Suppose hypothetically that a borrower takes out an equity release loan and is confidently expected to exit the house after, say, 25 years.
The NNEG is a form of put option in which the borrower is allowed to repay the loan at the lesser of the house price and the loan value when they exit the house in 25 years’ time.
We can therefore value the NNEG using option pricing theory and the natural formula to use is the put formula proposed by Fischer Black in 1976, the Black ’76 model.2
The key input to this formula is the forward house price.3
This forward house price is the price agreed now to take possession in 25 years’ time, with the payment being made at possession. The forward house price is given by the following formula:
(1) Forward house price = current house price × exp((r-q)×25)
where r is the risk-free interest rate and 𝑞 is the annual rental rate on the property. This rental rate 𝑞 might reasonably be in the region of 2% to 3%.
In reality we do not know when the borrower will exit the house. They might exit after 1 year or 2 years or 3 years and so on. To deal with this uncertainty, we estimate the probability that they will exit after 1 year and we take the 1 year put value to be the probability of exit after 1 year times the put value calculated assuming that they exit after 1 year.4 We obtain the analogous put values for years 2, 3 and so on in a similar manner and our NNEG value is the sum of all these annual put values. These calculations are easy to do on a spreadsheet.
In my report, I proposed a simple base case. Making some plausible calibrations of the other inputs to the put option formula, I estimated the NNEG for a 70 year old borrower in a plausible example case to be 52% of the amount loaned if you assume a 𝑞 rate of 2%. If you assume a higher 𝑞 rate of 3%, then the NNEG rises to 67% of the amount loaned. So I would say that the NNEG is somewhere between 52% and 67% of the amount loaned.
Naturally, if you change the input calibrations, or the age of the borrower, you get somewhat different results, but you still get results that are in the same ballpark.
More details are provided in my report, “Asleep at the Wheel”.
It is important to note that the Black model does not include the house price inflation rate or any expected house price inflation rate.
So how do I know that companies are getting their NNEG valuations wrong? Because in every single case where I can find any information at all about their approach to NNEG valuation, they tell us that they use the house price inflation rate or the expected house price inflation rate as an input to their NNEG valuation model.
So whatever they are doing, it must be wrong because they are using an input variable that they should not be using.
What firms appear to be doing is inputting the expected house price instead of the forward house price into the Black put option formula. This practice has no scientific foundation and constitutes a major error.
In one case, a company explicitly stated that it was using an expected house price inflation rate of 4.25%. If we assume a risk-free interest rate r equal to 1.5%, then this gives an implied 𝑞 rental rate of minus 4.25% plus 1.5%, or minus 2.75%.
But a negative rental rate makes no sense! A negative rental rate implies that the rent is negative, and who pays a negative rent?
And what happens if we input a rental rate of minus 2.75% into the option formula? We get a NNEG that is only 3% of the amount loaned.
So if we do the NNEG valuation properly, we get a NNEG that is in the region of 52% to 67% of the amount loaned, and if we value NNEGs the way that firms seem to be doing, we get a NNEG value in the region of 3% of the amount loaned.
Therefore, we can conclude that firms are under-valuing their NNEGs – and to a considerable extent.
- An Equity Release mortgage is a loan made to a home-owning customer late in life that is collateralised by their home. The loan is made at some fraction of the home value and is repaid when the customer leaves the house by dying or going into a care home.
- Am I saying that one must use the Black 76 formula? Not at all. There are alternative models but Black ’76 is a natural choice. One limitation of this model however is its reliance on the thin-tailed Gaussian distribution and we might prefer to use a model based on a fatter distribution instead, which would produce a higher NNEG valuation than Black ’76. But what one cannot do is use the ‘holes’ in Black ’76 as an excuse to input some expected future house price into the put formula instead of a forward house price.
- The other inputs to the Black ’76 formula are: the risk-free interest rate r, the forward house price volatility, the maturity of the option (in this case, 25 years) and the exercise or strike price, which would be the value of the loan absent any NNEG.
- Plausible values of the exit probabilities are easily estimated. I estimated them using projections of male mortality rates based on recent Continuous Mortality Investigation data.