Dean Buckner and Kevin Dowd 27 February 2019
Professor Radu Tunaru’s new report on NNEG valuation is attracting a lot of discussion and rightly so. Its key result – that the net rental yield used in NNEG valuation should be 20% of the observed yield because only 20% of properties are rented out – was discussed in our blog posting on Monday.
Today we address a second issue, his assumptions about volatility.
Tunaru states in the introduction (p.1) that the Nationwide historical index data suggest a range of volatility values between 3.85% to 6.5%, where volatility is taken to be the standard deviation. This is a good starting point.
Tunaru selects his baseline scenario inputs (see pp. 27-28) “based on discussions with experts working on ERMs and using public available tables from Legal & General, Just Group and Equity Release Council, as of November/December 2018.” These include a net rental rate of 1% (which would be 0.66% rounded up), a rollup (or lending rate) of 4.15% and a volatility of 3.9% (p. 35). These calibrations are very low and coincidentally all tend towards lower NNEG valuations.
Figure 1 shows our estimates of corresponding range of NNEG valuations for an illustrative case involving a just turned 70 year old, a 30% LTV, a house price of £100 and other specified parameters, including those from his baseline case.
Figure 1: Illustrative NNEGs for Volatility Range 3.85% to 6.5%
Notes: Results based on the Black’ 76 put option model and the M5 version of the Cairns-Blake-Dowd mortality model (Cairns et alia, 2006, 2009). The mortality model is calibrated using Life & Longevity Markets Association data for England & Wales males spanning years 1971:2017 and ages 55:89. Source: llma.org. The borrower is assumed to have just turned 70 and to have taken out the ERM loan on their 70th birthday. The Loan-to-Value ratio is 30% and the house price is £100, so the initial loan is £30. The risk-free interest rate is 1.5% and the ERM loan (or rollup) rate is 4.15%.
One is immediately struck by the fact that his baseline volatility is just next to the minimum of the volatility range: one usually puts baseline parameters somewhere in the middle, because otherwise the range becomes redundant. But then look at the NNEG valuations. These vary from £0.13 to £0.55. For a loan of house price times LTV = £100 times 30% = £30, the ratios of NNEGs to the amount loaned vary from 0.4% to 1.8%. These ratios are extremely low.
It is brave to choose such a low baseline volatility based on discussions with firms which have a vested interest in low NNEG valuations, all the more so when the PRA recommends a much higher minimum volatility of 13% and a plausible volatility range of 10% to 15% (CP 13/18).
One doubts that the PRA would be happy with these calibrations and rightly so.
Basis risk
Then there is the issue of basis risk, the risk of losses from discrepancies between the prices of individual properties and the property price index. Tunaru acknowledges (p. 33) that while “volatility may be 5% for the index” (what – not 3.9%?) “it may be adjusted to 7% or even 10%, to reflect the basis risk.” That is correct. The chart below shows how different regions of the UK performed from 2000 until now:
Figure 2: House Prices by Region 2000 to 2017
Source: Nationwide.
There is considerable volatility around the index. So why then does Professor Tunaru suggest 3.9% as a baseline? And why does he say on p. 1: “Based on the Nationwide index data, a range of values between 3.85% to 6.5% seems representative for [the] GBM [Geometric Brownian Motion] volatility parameter.” It’s all a bit of a puzzle.
The basis risk issue is more involved than the regional variations suggest, however. The next Figure shows the ‘achieved’ value of a property, i.e., the amount that the lender was able to realise after the borrower exited, expressed as a percentage of the indexed value.
Figure 3: Indexed vs. Achieved House Prices aka Stochastic Dilapidation
Source: SAMS. The achievement rates are real data; the indexed value is a simulation.
The darker blue random-looking line is a simulated house price index. The lighter blue upward sloping line is the standardised loan value, whose upward slope is driven by the loan (or rollup) rate. The red dots are a scattershot of the individual achievement rates in the sample.
What jumps out is that the achieved values are all over the place relative to the index.
If all property values exactly followed the index, then there would have been no NNEGs expiring in the money and hence no NNEG-related losses. But on average the achieved value is close to 95% of the index and there is a huge dispersion around the index. The volatility of the index does not capture the volatility of the dispersion around the index! We are particularly interested in those red dots below the blue loan value line: these are the properties with put options expiring in the money, every one of which represents a guarantee that had to be paid, i.e., a NNEG realised loss to the lender.
Some further light on these issues comes from our correspondents and many thanks to you all: please keep them coming! Dean addressed some of these issues in his post last week on Brownian dilapidation and the remarkable Aviva ERM dataset. It turns out that the exercise of the put option appears to be due to the underperformance, often a dramatic underperformance, of the properties used as collateral.
In his post, he gave an example of a property that was originally valued at £1.2m when it was ERMed. The loan amount was about £520k. Over the period to exit, the Halifax went up 70%, so the indexed value of the property at exit would have been over £2m. The indexed value should have been more than enough to cover the £1.4m exit value of the loan, except that the house fell in value to £625,172, i.e., the house price fell by over 50%, and the lender was left with a NNEG loss of £763,225. The lender had lost more than the amount of the original loan despite the index rising by 70%.
We were puzzled by this example and others like it. Didn’t ERM contracts have clauses about maintenance? Yes, they do. But the reality is that once a property is ERMed there is no incentive for the property owner to make long-term investments in it (e.g., as opposed to need-to-do repairs to keep it functioning) and the people who are most likely to take out ERM loans are asset-rich, cash-poor types who can’t make long-term investments in the upkeep of their property, even if they wanted to. One of Kevin’s daughters recently bought a house and the old dear who had previously owned it still had a 70’s style washing hanger hanging over her kitchen and a massive and potentially lethal hole in her kitchen floor. No work seemed to have been done on it for 50 years.
Opinion on the appropriate way to address these issues was divided. Suggestions included adjustments to property value, property volatility, HPI and a margin for stochastic dilapidation. Some felt that systematic underperformance risk due to adverse selection should be allowed for in the valuation.
There is also a further factor. As one of our correspondents explained:
The achievement rates are like betas in stocks. Every house is individual. But the dispersion of results around the average and the volatility of that dispersion are both enormous. The effect you identify in the Aviva data regarding old people allowing their properties to generally subside in later life is the first reaction of most people, but it is in fact a small part.
The problem is that once properties revert, the properties are just dumped on the market:
In theory, the difference in price of the cheapest house on a street compared to the most expensive house on a street – assuming they’re all the same size and design – should be no more than about 20% at current average house price levels. That’s from our practical experience of taking ownership of properties arising from securitisations and managing the sale process ourselves (rather than relying on bank servicers). That’s what it costs to fix a new bathroom and kitchen, take control of the gardens and generally paint the place before sale. So the maximum spread of a series of spruce houses and neglected houses shouldn’t be more than 20% – as any buyer knows that’s the cost of returning to top quality. None of this process occurs in bank managed sales – all of which happen “as is” once the property falls into the servicer’s ownership. No one in the chain of sales set up by general servicers has any incentive to take control of the value at sale in the interests of the actual owner (in this case the bondholders). Everyone has an interest in the quickest possible sale and the removal of “in-hand” properties from the balance sheet. General servicing is passive and thus entirely accepting of any price as long as the property is sold.
Thus the lenders experience greater losses than would be the case if the sales were managed in (for want of a better term) a more professional manner.
In short, we are already seeing a number of exercises where the achieved value is significantly lower than the indexed value. Last week, the underlying distribution was not clear to us, given that the achieved/indexed ratio in the Aviva dataset is only reported when the guarantee is exercised, i.e., there is sample selection bias. However the SAMS data is for all properties and our preliminary analysis shows the distribution around the index is close to normal. Thus, the achievement rates are amenable to modelling.
The implication is that there could be some nasty surprises in store for equity release lenders.
Autocorrelation
Another problem is that for any market with strong autocorrelation, the risk attached to shorting the market via a naked put is much much greater than if there was no autocorrelation. We have discussed the Japan market before. Now market risk is a capital requirement question, not one for option pricing (although the two questions are frequently confused). For pricing purposes, we should take whatever volatility corresponds to our imputed hedging frequency and input that volatility to our favourite option pricing model. It does not matter whether the market is autocorrelated or not, for it is the volatility at the hedging frequency that determines the cost of hedging. If we hedge every quarter, use quarterly sampled volatility, if every 5 years, use returns from a 5 year period, and so on. But if the market is strongly autocorrelated and we are short a put, then we are effectively taking a long position in a market that may go down for a very very long period, as the Japan index showed. Even worse, the gamma effect, i.e., the tendency of delta to increase as you approach the money, means you are getting longer and longer in a falling market, which is where no trader wants to be. So there is considerable risk and it would be prudent to reflect that risk, somewhere.
The PRA chose to reflect the risk by bumping up the input volatility, hence the 13% minimum expected in PS 31/18. Now that adjustment is a fudge. To include such a risk adjustment in a valuation model is theoretically wrong, but it roughly works in practice. From the perspective of a short put, a market with 13% volatility and with no autocorrelation is unlikely to be wildly different from a market with 7% volatility (sampled annually) but strongly autocorrelated. All good risk management is about suitably applied – and prudent – fudges. Forget the clever mathematics and remember where that got us in 2008.
Professor Tunaru does not agree:
Some market practitioners believe that for shorter durations, the presence of serial correlation might mean that the BS volatility needed to be higher than historic volatility in order to fit an ARMA result that did pick up that serial correlation. It is not clear to me how increasing volatility for a process that does not have serial correlation, such as the GBM, will induce or recover values as if serial correlation existed. This is an ad-hoc procedure that does not have, to the best of my knowledge, any grounding into statistical modelling. (Tunaru, 2019, p. 48).
But it is not clear to us why he is not clear.
Plausible estimates for vol and NNEG
We have still to come up with our own plausible volatility values, however. We would start with the PRA’s minimum of 13%, but how to adjust for the dispersion of achievement ratios? Our suggestion: take a rolling standard deviation of the achievement ratio and divide by root time to get an annualised value. This annualised value varies from 5% to 9%. We then take the maximum (9%) on prudential grounds, i.e., where volatility goes all over the place, assume the worst. We then have to work out a plausible correlation between the index vol and the achievement vol. Assuming zero correlation (which is not unreasonable) then the root sum of squares gives 16%. (13^2+9^2=250 and square root of 250 = 15.8.) Alternatively, we could be prudent and add the two vols together, which would give us a vol = 13%+9% = 22%. We are then looking at a plausible range of something like 16% to 22%, and the plausible vols would be even higher (18% to 24%) if we used the upper end of the PRA’s volatility range (i.e., 15% instead of 13%) as our starting point. So let’s go for 16% to 24% as a plausible range.
There are also other calibration differences to consider. Tunaru has a rollup rate of 4.15% and a net rental rate of 0.66%. We would recommend a rollup rate of 6% (see our report “Equity Release: Another Equitable in the Making,” p. 13) but would accept his original net rental rate of 3.62%, i.e., the rental rate that Tunaru obtained before he applied his deus ex machina 20% multiplier.
The resulting NNEGs from these alternative calibrations are shown in Figure 4.
Figure 4: Our NNEG Valuations vs. Tunaru NNEG Valuations
Notes: Results based on the Black’ 76 put option model and the M5 version of the Cairns-Blake-Dowd mortality model (Cairns et alia, 2006, 2009). The mortality model is calibrated using Life & Longevity Markets Association data for England & Wales males spanning years 1971:2017 and ages 55:89. Source: llma.org. The borrower is assumed to have just turned 70 and to have taken out the ERM loan on their 70th birthday. The Loan-to-Value ratio is 30%. The house price is £100, so the initial loan is £30. The risk-free interest rate is 1.5%. The Tunaru calibrations are a net rental rate of 0.66% and rollup rate of 4.15%. Our calibrations are a net rental rate of 3.62% and rollup rate of 6%.
The blue line in the lower part of the Figure shows the NNEG valuations using the Tunaru calibrations but across the entire range of volatilities. The blue * point shows the NNEG valuations using the Tunaru calibrations including his baseline vol of 3.9%. The red line in the upper part of the Figure shows the NNEG valuations using our calibrations across the entire range of volatilities. The red * plot shows our NNEG valuations across the range of volatilities that we consider plausible.
So it would appear that the Tunaru calibrations lead to a large (and by large we mean, very large) NNEG undervaluation or our calibrations lead to a large NNEG overvaluation.