Guy Thomas has an interesting post (‘No-negative-equity guarantees: Black-Scholes and its discontents’, guythomas.org.uk, Thursday 06 September 2018), arguing that the use of the Black-Scholes formula in the context of valuing the no-negative-equity guarantee (NNEG) in equity release mortgages, is flawed in ways that are more fundamental than the PRA blandly suggests.
You can read his article for yourself, but his key points are (1) that the Black-Scholes argument depends crucially on the idea of dynamic hedging and arbitrage, which is not met in the case of housing assets, and ‘is simply not possible in any shape or form’; (2) that Black-Scholes assumes when constructing the dynamic hedge that the underlying asset follows a geometric Brownian motion; (3) that there is no meaningful market in deferment prices [sic] over the periods of 20-40 years most relevant to NNEGs, and furthermore a deferred interest might well be more attractive, particularly if in the form of cash-settled financial contracts, so that all the problems of current interests (nasty tenants, management costs, legal risk etc) are permanently avoided.
Let’s look at these arguments carefully.
Guy’s article has a lot of words, and talks about a lot of things including ‘Black-Scholes’, ‘geometric Brownian motion’, ‘dynamic hedging’ ‘no-arbitrage’ etc., but these are irrelevant to what I call the ‘upper bound’ principle of ERM valuation.
The argument for the upper bound, which is the rationale for Principle II of CP 13/18, is as follows. (1) A rational investor would prefer a loan without a NNEG, to a loan with a NNEG. This preference is because the NNEG limits the maximum amount paid by the borrower to the value of the house at exit, which might be less than the contractual value of the loan without such a limit. (2) A rational investor would prefer deferred possession of the property to an ERM. This preference is because the NNEG limits the exit value of the ERM to the property value, so the present value of the ERM can never be greater than the present value of deferred possession.
This result means we can draw the following chart.
The green line is the present value of the loan for any maturity up to 40 years. The red line is the present value of a contract for deferred possession for any maturity up to 40 years. Note that the chart is not saying what the value of that contract will be in, say, 40 years. It is saying what the value of that contract is now, where the contract is some legal document saying that I will get possession of the property in 40 years. Imagine 40 separate contracts, the first to receive possession in 1 year, the second to receive in 2 years, up to 40 years. The red line on the chart specifies how much each contract will be worth now. (Apologies for labouring this point, but in my experience people find it terribly confusing).
You see the red line slopes downwards. Why? Won’t property go up through time? Yes it will, but as explained above, we are interested in the value of each deferment contract now, not at some future time. So the value of the 1 year deferment will be less now than the value of a contract for immediate possession, the value of a the 2 year deferment will be less then the value of the 1 year contract, and so on up to 40 years. The curve will fall monotonically, i.e. you always lose value by prolonging the deferment period.
The argument so far is essentially the rational for principles II and III of the CP. You see that nowhere do there occur expressions like ‘Black-Scholes’, ‘geometric Brownian motion’, ‘dynamic hedging’ ‘no-arbitrage’ etc. The upper bound does not depend on the shape of the distribution, nor on dynamic hedging arguments.
Has the no-arbitrage principle been smuggled in somewhere? No, because it has been replaced by the fiduciary principle. The principle dictates that if a rational investor would price an instrument at X, then the accountant should value the instrument at X. Thus, if a rational investor prefers possession in year 1 to possession in year 2 etc., that is how the accountant should represent that preference in the books of the company. Anything else would be a breach of trust.
Now Guy suggests that an investor might prefer deferred to immediate possession. Here he challenges principle III of the CP. Kevin has already addressed this argument in a post on August 28, so I won’t repeat his reasoning here. Read his post, but his argument is essentially that most properties most of the time generate a positive net rental stream. ‘Therefore, when looking for a general rule to assess deferment value, the only sensible rule is to assume a positive rental stream – and a positive rental stream implies that the deferment price will be less than the current property price’.
In summary, most of the arguments provided by Guy for ERM valuation are irrelevant to establishing an upper bound for the value at each maturity. Of course, the actual value will be lower than the bound, and will depend on dynamic hedging arguments. But set that aside for now.The question is whether the arguments for principles II and III of the CP are valid.
All objections happily considered, the contact form is there at the top on the right.