In Part one we addressed the Merton extension to the Black Scholes Option Pricing Model (“BSOPM”) which adds a function for continuously compounding dividends.
As a reminder, the No Negative Equity Guarantee ('NNEG') included in equity release mortgages is the equivalent of a long-term put option: the homeowner can require the funding mortgagee to accept the property in settlement of the mortgage sum outstanding. In cases of extreme longevity this means that the estate of the deceased is not hit with the mortgage shortfall.
The expectation is that the put option will only be exercised if the value of the property is less than the mortgage and rolled up interest.
Modelling future house prices
The PRA has stated four principles which it has applied in measuring the value of the NNEG liability. One of these principles is that the present value of deferred possession of property (the deferment price) should be less than the value of immediate possession. At its simplest, the value of immediate possession includes the ability to rent out or occupy the property, whereas deferred possession does not.
The deferment price is the price that is settled today to take ownership of the property at some date in the future. This means that the deferment rate should be positive, reflecting the foregone rights to income or to use of the property in the deferment period. The PRA does not permit the use of a negative deferment rate, based on the unarguable logic above.
The PRA has required the use of certain standard inputs into the Black Scholes Merton model:
- The volatility of house prices (sigma σ) has been retained in the latest iteration of the guidance at 13%.
- The deferment rate (q) has tracked longer-term market interest rates. It has been increased to 4% from March 2025.
- The risk-free rate (r) is the published Solvency II basic risk-free interest rate for maturity t, stated on a continuously-compounded basis.
- The current value of the underlying (S) is the current estimate at the relevant date of the value of the property concerned.
- The strike price of the put option (K) is the loan principal and expected accrued interest at time t.
The “t” function
The PRA anticipates that the liability will be modelled using a range of values for t, with the various probabilities being based on life tables.
This can be applied to individual mortgages or on a portfolio basis.
The impact of the “q” function
We can all recognise that the value of a put option increases as the level of the expected dividend payouts increase: the higher the payout ratio, the more likely it is that the future price will be lower than the current price.
For valuing the NNEG the PRA has used the “q” function to denote the value of either occupancy or rental income of the property. It has tethered this function to long-term interest rates. In December 2018 “q” was set at 1% and this reduced to 0.5% at December 2019. Since that low point the rate has increased in steps to the current value of 4%.
The overall impact is that the “r-q” function in the model is close to nil, as both are linked to interest rates, albeit r is the risk-free rate and q is the market rate.
The absence of dynamic hedging
The major problem of applying BSOPM, with or without the Merton extension, remains: there is no liquid observable market in the underlying, that is the potential portfolio of residential properties. There is therefore no means of arranging a perfect hedge of the open positions.
Lessons for business valuers
In my view the application of BSOPM Merton in this way by the PRA adds credibility to the work that we do as business valuers. Despite the challenges of applying BSOPM outside liquid markets, an important regulator, far removed from business valuation, requires its use for the valuation of a liability.
That liability is the guarantee that the recipient of funds from an equity release mortgage will not have to pay more than the eventual sales price of the property to redeem the mortgage.
That regulator has also provided helpful guiderails as to some of the inputs, including minimum values for both the income potential of the property (q) and house price volatility (sigma σ).
There can be all sorts of arguments that house prices can be expected to increase at a higher rate than the risk-free rates that are used in the model. The simple truth is that any such expectation is baked into the current values, as is the case with all asset classes.
Andrew Strickland
Consultant
