The standard approach used by UK ERM actuaries to value NNEGs and ERMs is the Discounted Projection (DP) approach, which its adherents like to call the ‘real world’ approach.
This approach is based on the use of a projection of future house price growth to value the NNEG. In particular, it replaces the forward house price as the underlying in the Market Consistent (i.e., correct) approach with some expected future house price ‘forecast’ that a cynic (not us!) might say was indistinguishable from a convenient guess.
Equivalently, the DP approach replaces the forward rate f in the MC approach with some assumed rate of future house price growth hpi.
Since f=r-q, replacing f by hpi gives
(1) hpi=r-q.
Given also that hpi has been specified and r can be easily calibrated from the spot rate curve, (1) implies that we can back out the following implied q:
(2) q=r-hpi.
To give an example, if we set r=1.5% and use the 4.25% hpi assumed recently used by Just Group1 then we can back out q as
(3) q = 1.5% – 4.25% = -2.75%.
The negative sign in front of the ‘2.75%’ on the right-hand side of (3) is not a typo. For this calibration, and indeed for any calibration in which hpi exceeds the risk-free rate, the DP approach produces a negative q net rental rate.
One can implement this DP approach by taking an otherwise sound MC (e.g., Black’ 76) calibration and replacing the forward rate with an assumed future house price growth rate hpi. So one replaces the q rate one would otherwise use (e.g., a sensible q rate of around 3% or more) with an implied q rate equal to r-hpi (e.g., a q rate of about -2.75% as in the Just example).
To illustrate the impact of this approach, Table 1 shows the NNEG and related valuations for our baseline calibration, obtained using each approach:
Table 1: Baseline ERM/NNEG Valuations: Market Consistent vs. Discounted Projection Approaches
| Approach | L | NNEG | ERM |
| Market consistent | £74.84 | £32.19 | £42.66 |
| Discounted projection | £74.84 | £4.37 | £70.47 |
Notes: L is the present value of the loan component of the Equity Release Mortgage, NNEG is the is the present value of the NNEG guarantee, and ERM is the present value of the Equity Release Mortgage. Based on the baseline assumptions: male aged 70, LTV=40%, r=1.5%, l=5.25%, q=4.2% for the MC approach and q=-2.75% for the DP approach, and σ=14.8%. Exit probabilities are based on M5-CBD model projections using England & Wales male deaths rate data spanning years 1971:2017 and ages 55:89.
In this case, which is not untypical of others we have looked at, the DP approach gives NNEG valuations that are close to an order of magnitude lower than those produced by the MC approach. The result is a considerable overvaluation of the ERM, in this case by 57.3%.
But ask yourself: do the DP valuations even look right? If you believe them, then you have to believe that the ‘true’ NNEG is only 4.37/74.84 = 5.8% of L. This NNEG/L ratio looks awfully low when you consider the spread between the loan rate and the risk-free rate, which is 5.25% – 1.5% = 3.75%. If the loan has so little risk, then why is the spread so high?
Here we see in a nutshell the valuation problems entailed by the use of the DP approach.
The underlying error is that the DP approach confuses the future price of spot possession with the current price of deferred possession. This error is hugely material, because the one variable (the future price of spot possession) usually goes up over time, whilst the other (the current price of deferred possession) falls with maturity.
To illustrate the magnitude of this difference, consider the plots in Figure 1:
Figure 1: Future vs Deferment House Prices
Notes: Based on current house price = £100, hpi=4.25% and q=4.25%.
The blue plot gives the correct price to use in the option pricing formula, i.e., the spot price of deferred possession. For a maturity of, say, 15 years, and our recommended deferment rate of 4.2%, the deferment house price (the price of a spot contract for possession in 15 years) is £53.3. The red plot gives the incorrect price that the Hosty et alia argument implied should be used, i.e., the future house price. This particular plot is based on an assumed 4.25% hpi rate. The expected future price in 15 years is £189.3.
So based on these calibrations, the Hosty et alia argument implies that we use an underlying value of £189.3 in the option pricing equation when the correct value is £53.3. If you believe that the Hosty approach is right, then you are believing that an asset, a forward worth £53.3, is actually worth £189.3, in which case let’s do a trade.
The Hosty approach then produces a NNEG of £3.00 if we make our other baseline assumptions when Black ’76 correctly applied would give us a NNEG of £31.42. The Hosty approach thus leads to a NNEG valuation that is 9.5 % of the Black ’76 valuation in this case.
Consider also that the Discounted Projection approach:
- has never been convincingly justified by those who advocate it (more on that in our report);
- is being promoted by practitioners with a vested commercial interest who are promoting it for openly commercial reasons and are dismissive of the only approach that is scientifically respectable because they do not like the valuations it produces;
- has not been endorsed by a recognised independent expert, including Tunaru;
- does not appear in the corpus of recognised scientific research journals that are subject to rigorous peer-review;2 and
- is contradicted by alternative approaches such as Black ’76 that are used and taught all over the world and have been published in top tier academic journals, albeit that their applications are sometimes still controversial.
There are two root problems with the DP approach. The first is that it is based on a forecast, in this case a forecast of future house price growth. However, it is a serious mistake to base an option valuation on a forecast, because basic option pricing theory (e.g., Black-Scholes or Black’ 76) tells us that the value of an option is not dependent on any forecast, let alone some guess value pulled out of thin air. Instead, the option should be priced using current variables only – admittedly subject to some judgements about calibration but experts know how to handle these calibration issues. It follows that any approach that does depend on a forecast must be invalid and when a firm says that it bases its NNEG valuations on a forecast – any forecast – then we know that the firm must be getting it wrong.
The second root problem is that it is wrong on principle to replace the forward house price in the put pricing equation with the expected future house price (or equivalently, to replace the forward rate with the expected future house price inflation rate). This error can produce results that are known to be impossible. As Table 2 shows, the DP approach can produce results that violate two different sets of impossibility bounds: the Principle II bounds and Principle III bounds examined in Chapter 19 of our report:
Table 2: Baseline ERM and NNEG Valuations: Discounted Projection Valuations vs Bounds
| Approach | NNEG | ERM |
| Discounted projection | £4.37 | £70.47 |
| NNEG lower bound | ERM upper bound | |
| PRA Principle II bounds | £28.09 | £46.75 |
| PRA Principle III bounds | £13.15 | £61.69 |
Notes: NNEG is the is the present value of the NNEG guarantee, and ERM is the present value of the Equity Release Mortgage. Based on the baseline assumptions: male aged 70, LTV=40%, r=1.5%, l=5.25%, q=-2.75% for the DP approach and q=4.2% for the bounds, and σ=14.8%. Exit probabilities are based on M5-CBD model projections using England & Wales male deaths rate data spanning years 1971:2017 and ages 55:89.
The DP valuations violate all these bounds. The DP NNEG valuations fall below the lower bounds, and the DP ERM valuations exceed the upper bounds. Since these DP valuations are known to be impossible, then no auditor can sign off on them because fair value principles do not allow impossible values.
So if we wanted a one sentence assessment of the validity of the DP approach, all we need to know is that it produces valuations that violate bounds that cannot be violated.
It is hard to see how any approach can get much more wrong than that.
In sum, the correct approach is to start with the forward price = Se(r-q)t, which then gives us the discounted forward price or deferment price = Se-qt.
The DP approach incorrectly treats the ‘forward price’ as the projection price = Sehpi×t, which then gives the ‘discounted forward price’ or ‘discounted projection’ price = Se(hpi-r)t.
The DP is approach is wrong on principle and confuses the future price of spot possession with the current price of future (=deferred) possession.
It will give the wrong answers in general except in the special case where hpi=r-q.
All of which would just be a matter of academic debate among the pointy headed brigade except for the fact that the DP approach is the one used by the UK equity release industry.
- We did not make this number up. The firm reported using this number in both its 2016 and 2017 Annual Reports (see pp. 163 and 110 respectively). The same number also appears in its 2018H1 results (p. 18).
- Admittedly, Hosty et alia (2007) was later published in the British Actuarial Journal, but it is not clear whether BAJ articles (or for that matter, any articles and reports published by the IFoA, e.g., such as the Tunaru report) are subject to “rigorous peer review” and the only thing that is clear about the review process, whatever that might be, is that it is unclear.