2.1 Exit probabilities

The home-exit probabilities refer to the probabilities that the borrower will exit the home (and hence terminate the loan) over each of the next 1, 2, 3, … years. We assume away complicating factors such as the possibility of early repayment of the loan and the possibility that the individual will spend time in a care home at the end of his or her life.Footnote ⁴ We assume instead that the individual will die at home, i.e., so home exit occurs at death. Under these assumptions, the exit probability for year t is equal to

(1)$$exit\;prob_t = q_t \times S_t$$

where q _t is the mortality rate for year t and S _t is the probability that an individual alive now will survive to year t. Note that S ₀ = 1 and S _t = (1 − q _t−1)S _t−1 for all t > 0.

The home-exit probabilities for males currently aged 70 are illustrated in Figure 1.

Figure 1. House-exit probabilities for males currently aged 70.

Notes: House-exit probabilities are based on M5 CBD model [Cairns et al. (Reference Cairns, Blake and Dowd2006, Reference Cairns, Blake, Dowd, Coughlan, Epstein, Ong and Balevich2009)] cohort mortality rate projections using male England and Wales deaths rate data estimated over ages 55:89 and years 1971:2017.

Source: Life & Longevity Markets Association.

The left-hand (low t) house-exit probabilities are close to the low t mortality rates and reflect the early high survival probabilities (i.e., that people aged 70 have a high probability of living at least a few years), and the later (high t) house-exit probabilities primarily reflect the fact that the probabilities of living to extreme old age are low and approach zero in the limit.

2.2 Valuation issues and the put pricing model

The present value ERM of the equity release mortgage loan can be considered to be the present value L of a risk-free loan, one which is guaranteed to be repaid in full, minus the present value NNEG of the NNEG guarantee:

(2)$$ERM = L-NNEG$$

The original loan amount grows at the loan rate (sometimes called the rollup rate) l from its current amount until the time when the loan ends. Therefore L is given by

(3)$$L = \mathop \sum \limits_t [ {exit\;prob_t \times current\;loan\;amount \times e^{( {l-r} ) t}} ] $$

where r is the risk-free interest rate.Footnote ⁵

The valuation of L is straightforward.

NNEG is the sum of the products of the exit probabilities for each future time t and the present value of the NNEG guarantee for each future time t:

(4)$$NNEG = \mathop \sum \limits_t [ {exit\;prob_t \times NNEG_t} ] $$

where NNEG _t is the present value of the NNEG guarantee for period t.

The question is then how to value each of these individual NNEG _t terms.

Recall that the NNEG gives the borrower (or the person acting for the borrower) the right to repay the loan by paying the lender the minimum of the loan value or the house price at the time of death.

The right to repay the minimum of two future values (one of which, the future house price, is uncertain) at some given future time implies a European put option granted by the lender to the borrower. Since the time of exercise is uncertain, we can think of the NNEG as involving a portfolio of such put options.

In the case of our put options the underlying variable is a residential property (“house”) or more precisely, a forward contract on a house, and we should think of a house as an asset that bears a continuous yield in the form of a net rental yield. This net rental yield reflects the use benefit of living in the house or the (net, after costs) rental income we might get by renting the house out.

A natural option pricing model to use in these circumstances is the Black ’76 model [Black (Reference Black1976)]. Black ’76 is an appropriate pricing model when the underlying is a forward contract with a maturity coterminous with that of the option itself. This model is a near-relative of the famous Black–Scholes–Merton model [Black and Scholes (Reference Black and Scholes1973), Merton (Reference Merton1973)].Footnote ⁶

The Black ’76 formula for the price p _t of a European put option with maturity t on a forward contract on a commodity bearing a continuous yield q is given by the formula:

(5)$$p_t = e^{{-}rt}[ {K_tN( {-d_2} ) -F_tN( {-d_1} ) } ] $$

where K _t is the strike or exercise price for period t, F _t is the forward house price for period t, the function N(…) is the value of the cumulative standard normal distribution at the value specified in brackets, and d ₁ and d ₂ are given by:

(6)$$d_1 = [ {\ln ( {F_t/K_t} ) + \sigma_t^2 t/2} ] /\left({\sigma \sqrt t } \right)$$

(7)$$d_2 = d_1-\sigma _t\sqrt t $$

where σ _t is the volatility of the forward house price for maturity t.

The strike price K _t is then the rolled up or accumulated loan amount by period t:

(8)$$K_t = current\;loan\;amount\;\times e^{lt}\;$$

and the forward price F _t, the price agreed now to be paid on possession in period t, is:

(9)$$F_t = current\;house\;price\;\times e^{( {r-\tilde{q}} ) t}\;$$

where $\tilde{q}$ is the deferment rate, namely the discount rate applied to the current house price to give the deferment price, the price we would agree to pay today to take possession of the house in t years' time. Thus, the deferment house price R _t is given by:

(10)$$R_t = current\;house\;price\;\times e^{-\tilde{q}t}\;$$

The difference between the forward house contract and the deferment house contract is that with the forward we settle when we take possession in t years' time, but with the deferment contract we settle today.Footnote ⁷ Therefore, the deferment house price R _t is the present value of the forward price, where the present value is obtained by discounting at the risk-free rate r.

It is important to note that the deferment house price will be less than the current house price S ₀ because the deferment rate $\tilde{q} > 0$.

The forward house price F _t should not be confused with future house prices or expected future house prices:

• • Forward prices for future period t are known (or can be approximated) now and we need to be able to price options using information available now.

• • Options cannot be priced using future house prices because future house prices are currently unknown.

• • Expectations of future prices do not appear in the Black ’76 option pricing formula.

We should also keep in mind that although the original Black ’76 article discussed options on futures, futures prices are the prices of futures contracts, a form of forward contract, not actual or expected future prices of any sort.

A mistake to be particularly avoided—the one common among UK ERM actuaries—is to confuse forward and expected future prices. This mistake typically manifests itself in the inputting of an assumed expected house price inflation rate into (9) instead of the forward rate $r-\tilde{q}$.

To repeat: it is not the future or expected future price of a contract for immediate possession that we use in the option pricing equation, but rather the current price of a contract for future possession.

Finally, we make the point that the most important of these parameters in terms of their impacts on results are $\tilde{q}$ and σ _t.

3.1 Calibrating the net rental yield

We now provide a more precise calibration of the net rental yield. Define the maintenance cost m as the rate of expenditure (as a percentage of gross rental) required to keep the property in perfect condition (i.e., such as to achieve the best sales price for a property of that size in the same area), and define the tenant maintenance share (s) as the proportion of m that the tenant is likely to spend on maintenance. s will typically vary between 0% and 100%. For a short rental, s will be close to zero, and for a long let we would expect s to be close to 100% in the early years of tenancy, falling over time. In the final years it might fall to zero, even for a standard tenancy, given the lack of incentive to keep in full order for the landlord's benefit.Footnote ¹⁰ For an ERM, it would seem unlikely that the “tenant” at the end of life, perhaps in the situation where the NNEG had bitten, would have any incentive to keep the property in good condition, so we would expect s to fall toward zero in that case too.

We now use the following calibrations:

• • Void: we use the standard “1 month in 12” rule of thumb, i.e., v = (1/12) × y.Footnote ¹¹

• • Management cost: following Tunaru and Quaye (Reference Tunaru and Quaye2019, p. 32), we assume management cost c = 10% × y.

• • Maintenance cost: again following Buckner and Dowd (Reference Buckner and Dowd2020a, Reference Buckner and Dowd2020b, p. 37), we assume maintenance cost m = 15% × y.

Thus, the maintenance cost borne by the landlord and to be subtracted from the gross rental yield is m = 15% × 50% × y = 7.5% × y.

We then have

(13)$$net\;rental\;yield = y \times ( {11/12-0.1-0.075} ) = y \times 0.7417.$$

Thus, the net is 74.17% of the gross.

Again following Buckner and Dowd (Reference Buckner and Dowd2020a, Reference Buckner and Dowd2020b, p. 37), we take

(14)$$\;y = 5.6\% .$$

Therefore

(15)$$\tilde{q} = net\;rental\;yield = 74.17{\rm \% } \times 5.6\% = 4.15\% \;\approx 4.2\% .$$

So we use $\tilde{q} = 4.2\percnt$ as our “best estimate” of the deferment rate.Footnote ^12, Footnote ¹³

4.1 Volatility estimation: a first pass

We begin by defining the volatility σ of the return on a forward contract as the square root of the variance of that return:

(16)$$\sigma ^2 = \displaystyle{1 \over n}\mathop \sum \limits_1^n ( {FR_t-\overline {FR} } ) ^2.$$

where FR _t is the forward return over the period to time t and $\overline {FR}$ is the mean forward rate.

More specifically, we seek to obtain the variance of the return on a forward with maturity T, F _t,T say, based on observations of a spot price S _t and we would typically take S _t to be some HPI. We already know (see (9)) that

(17)$$F_{t, T} = S_te^{( {r-q} ) T}$$

where the variables have their usual interpretations.

The return on a forward with maturity T is

(18)$$FR_{t, T} = {\rm ln}\,F_{t, T}-{\rm ln}\,F_{t-1, T}$$

where variables have their obvious interpretations. Substituting (17) into (18)

(19)$$\eqalign{& {FR_{t, T} = {\rm ln}\big[{S_te^{( {r-q} ) T}\big]{-{\rm ln}} \big[S_{t-1}e^{( {r-q} ) T}} ] }\cr & \qquad= \ln S_t + ( {r-q} ) T-\ln S_{t-1}-( {r-q} ) T \cr & \qquad= \ln S_t-\ln S_{t-1}} $$

which is the return on the spot house price.

Therefore, the variance of FR _t,T must be equal to the variance of the spot return, regardless of the maturity of the forward.

So what should we take the value of the spot return variance to be?

We would suggest to go with the PRA's central estimate of 13%.

4.2 Volatility around the index

The estimated 13% volatility only refers to the volatility of the index, but there is also the volatility around the index. This additional volatility would include the impact of regional variation around the index, but there are further contributory factors as well. These include, e.g., the impact of changes in consumer-relative demand for different types of property, expansions of nearby roads, the impact of new housing estates, yuppification, middle class flight, the opening or closing of a good nearby school, and so forth.

Figure 2 shows scatterplot of the “achievement rates” of ERMed properties, i.e., the amounts that the lender was able to realize after the borrower exited, expressed as a percentage of the indexed value, based on the Shared Appreciation Mortgage Securities (SAMS) originated by HBOS in the late 1990s.

Figure 2. Indexed vs. achieved house prices.

Source: SAMS.

The red dots are the scattershot of the individual achievement rates in the sample. The blue random looking line is a simulated HPI.

We immediately see that the achieved values are much more volatile than the index. Above all, when seeking to calibrate the volatility, we need to keep in mind that it is that dispersion that matters, not the volatility of the index itself.

It would then behoove us to revise our earlier index-based volatility estimates to take account of this additional source of volatility. We could start with our earlier index volatility. Let us label this volatility as σ _INDEX. We now obtain σ _AR, the volatility of the achievement rate, as follows: take a rolling standard deviation of the achievement rate and divide by root time to get an annualized value. We found that the annualized volatility values vary from 7% to 10%. Let's work with the middle value of 8.5%. We then have to assume a plausible correlation between the index vol and the achievement rate vol. Assuming zero correlation, which is not unreasonable, we obtain the results reported in Table 1.

Table 1. Illustrative volatility including impact of achievement rate volatility

Note: The term in the rightmost cell is obtained by Pythagoras.

4.3 Interest rate risk as a further contributor to volatility

We have hitherto assumed (as per Black–Scholes/Black ’76) that the interest rate is constant, i.e., that there is no interest rate risk. In fact, interest rate risk not only exists, but arises from two different sources. Recall the following components of the Black model, reproduced here in slightly simplified form:

(20)$$p_t = e^{{-}rt}[ {K_TN( {-d_2} ) -F_TN( {-d_1} ) } ] $$

(21)$$d_1 = [ {\ln ( {F_T/K_T} ) + \sigma_T^2 T/2} ] /( {\sigma_tT} ) $$

(22)$$d_2 = d_1-\sigma _T\sqrt T $$

(23)$$F_T = S\;e^{( {r-q} ) T}\;$$

where p _t is the value of the t decrement put, K _T the strike price, F _T the forward price, σ _T the annualized input volatility, T the time to maturity in years, r the interest rate, S the price of “spot” possession of the property, and N(…) is the cumulative normal distribution function.

The interest rate term r appears first [see (20)] as a discount term wrapped around the terms representing the future value of the put option, which brings the future value (i.e., [K _TN( − d ₂) − F _TN( − d ₁)]) back to present value. Here r plays the role of an outer discount factor.

The parameter r then appears again [see (23)] as a projection term or inner discount factor taking us from the spot price S to the forward price F _T.

Each appearance gives rise to interest rate risk, but in different ways.

4.4 Discount rate risk

The first can be called discount interest rate risk. This risk can be hedged relatively easily, the gist of it being to swap floating into fixed.

A more detailed explanation goes as follows. When a trading desk sells an option, it places the premium in an account called the “hedging account.” This account earns interest from the firm's central funding desk and the interest earned will typically be close to the firm's overall funding rate. To hedge the risk arising from changes in this outer discount factor, the desk should make an internal or external IR swap into a fixed rate with maturity at the option expiry date.

It can then be shown that this swap guarantees that, with no other change taking place in the market, the hedge account will earn the fixed rate r in the outer discount factor e ^−rt. The demonstration goes as follows. Let

(24)$$P = FV \times e^{{-}rT}$$

where P is the option premium paid, r here is the long-term rate earned on the option account, and FV is the future value of the option given by the undiscounted Black formula.

Now suppose the long-term interest rate r changes, but there is no change in the forward price F. Such a circumstance would occur where the spot rate S changed by an amount ΔS in such a way that F remained constant under the formula connecting S with F, i.e.,

(25)$${\rm \Delta }S = S( {e^{-{\rm \Delta }rT}-1} ) $$

where Δr is the change in discount rate. In this situation F will remain the same and hence the future value FV of the put option will also remain the same. The change ΔP in the value of the option premium will then be a simple discount function:

(26)$$P + {\rm \Delta }P = FV[ {e^{-( {r + {\rm \Delta }r} ) T}-e^{{-}rT}} ] .$$

Assuming the amount P is currently held in the hedging account, we could replicate the change ΔP if P were invested in a long-dated zero-coupon bond with maturity T. In practice the same effect can be achieved by investing P at the firm's short-term funding rate, but swapping the short-term floating payments into a zero-coupon swap.

4.5 Projection rate risk

In the previous example we assumed that the forward price remains constant while the spot price changes, i.e., a rise in long-term interest rates will force the spot price lower, while a fall in the interest rate forces the spot price higher. This effect might be explained by a market expectation of unchanged future nominal rental cashflows, whose discounted present value would fall or rise according to the long-term interest rate operating as a discount factor.

The opposite case can also occur, i.e., we could have a situation where the spot remains steady, but the forward changes due to the way in which the interest rate operates as a projection factor.

Using the standard formula for the forward house price [i.e., (30)], and assuming constant q, the return on the forward is calculated as follows:

(27)$${\rm forward\ return}\approx \Delta HP + ( {\Delta IR-\Delta q} ) \times T$$

where ΔHP = ln((S + ΔS)/S) and T is the maturity of the forward at any point in the historical time series for the given combination of interest rate (IR), deferment rate (q), and house price index (HPI).

A proof of (27) is provided in Appendix A.

Three key points follow from (27).

4.6 Four risk factors

The first is that we now have four risk factors (index, achievement rate, interest rate, and deferment rate) impacting the forward price.

4.7 Correlations

The second is that we need to consider their correlations. We are now interested in particularly (a) the correlation between the HPI and the interest rate, and (b) the correlation between the HPI and the deferment rate.

Let's consider each of these in turn.

4.7.1 Correlation between HPI and interest rate

We have already assumed that the correlation between the index and the achievement rate is zero. Table 2 shows the correlation between the 10 year interest rate (which we take as a benchmark for the whole term structure) and the housing index, for 10 representative countries.

Table 2. Correlation between 10 year interest rate and index

Source: OECD (10 year interest rate) and Dallas Fed (HPIs). Data are quarterly from January 1980 to June 2017.

The takeaway points from this table are that the correlations between interest rates and HPIs are generally low and that a reasonable correlation for the UK would be zero.Footnote ¹⁴

4.7.2 Correlation between HPI and q

Equation (27) indicates that the deferment rate q is a further source of volatility. Take equation (11), which shows that the deferment rate is equal to the rental yield divided by the house price, then replace the house price by the HPI and add a time t subscript. We then obtain:

(28)$$q_t = d_t/HP_t$$

where d _t is the aggregate nominal rental. We have no time series data on aggregate nominal rentals, but we can estimate their change using rental and HPIs. Data from OECD suggest that the deferment rate q is not constant (the annual volatility of q for the UK is of the order of 0.3%) and that changes in q are negatively correlated with changes in the index. These effects are shown in Figure 3.

Figure 3. UK nominal house and rental indices and implied deferment rate. (a) UK nominal house prices and nominal rentals (b) Implied UK deferment rate.

Source: OECD.

As one of us commented in early 2019Footnote ¹⁵:

When I worked at the PRA on the paper that became CP 13/18, I had assumed that the deferment rate stays roughly constant. The rationale is that if rentals are expected to increase, this would increase the market value of properties, all other things being equal. But all other things aren't equal: there is strong evidence that nominal rentals track price inflation, and also strong evidence that interest rates anticipate inflation.Footnote ¹⁶ So an increase in expected nominal rentals should correlate strongly with an increase in the interest rate used to discount the same rentals, and the rental yield, hence the deferment rate, should remain roughly constant.Footnote ¹⁷ I assumed this, and I imagine the PRA assumed this too.

A possible explanation for the volatile $\tilde{q}$ rate and the negative correlation with house prices might then go as follows. Nominal house prices, which in theory should reflect the net present value of all future nominal (net) rental cashflows, tend quickly to anticipate—perhaps to over-anticipate—future upward or downward changes in rentals. Nominal rentals are sticky however and respond slowly.Footnote ¹⁸ Thus the large fall in house prices which occurred in the housing recession of the early 1990s was not reflected in rents, which continued to rise slowly, and so $\tilde{q}$ rose in that period. Conversely, the significant rise in house prices from the late 1990s to 2007 was notably higher than the rise in rentals, so $\tilde{q}$ fell over this later period.

As an aside, this combination of a volatile q that is negatively correlated with house prices has an interesting policy implication. If house prices go up, the loan-to-value of an existing ERM will fall, which will decrease the cost of the NNEG. At the same time, Figure 3 suggests the deferment rate will also fall, which will make the NNEG even cheaper, given that the deferment rate is the main driver of NNEG cost. Conversely, a fall in house prices will make the NNEG more expensive because of the fall itself, and will then make the NNEG even more expensive because of the implied rise in the $\tilde{q}$ rate. The cost of the embedded guarantee is thus doubly geared to the state of the housing market. The PRA would appear to be still unaware of this double exposure, which has implications for how it should design its capital requirement regime for equity release. But we digress.

Nor is this negative correlation effect unique to the UK. Table 3 shows evidence for a strong and consistent negative correlation between the deferment rate and the HPI of our 10 countries.

Table 3. Correlation between q and index

Source: OECD (10 year interest rate) and Dallas Fed (HPIs).

4.8 Volatility term structure

The third important corollary of equation (34) is that it implies that there is a volatility term structure. For example, other things being equal, the effect of a 10 bp change on a contract with 3 months to maturity will be 10 × 3/12 = 2.5 bp, which is trivial. But the impact of the same bp change on a 30 year forward will be 10 × 30 = 300 bp,Footnote ¹⁹ which is a whole lot more. This changing sensitivity throughout the life of the contract means that the impact on volatility caused by changes in the interest rate is not constant, and the same applied to changes in the deferment rate. Instead, mapped against time, the volatility starts high when the maturity is far from maturity and falls toward zero as the contract approaches expiry. Mapped against maturity, the volatility starts from zero and increases as the maturity gets larger.

We now wish to determine the average lifetime volatility of the contract. If X is the series of maturities and Y is the series of forward returns, it can then be shown that the volatility of the product σ(XY) is the following simple function:

(29)$$\sigma ( {XY} ) = \sigma ( Y ) \times T/\sqrt 3 $$

where σ(XY) is the volatility of a time series of returns of the forward contract, and σ(Y) is the volatility of the interest rate. A proof is given in Appendix B. Then the volatility of the returns on the forward contract is directly proportional to T (but see Figure 4).

Figure 4. Term structure of total forward volatility.

4.9 Total forward volatility

Finally, we might consider the effect of all four risk factors (index, interest rate, q, and achievement rate) in the forward rate (27) to give what we might call the total forward volatility. We can do so by estimating a correlation matrix between the four risk factors as shown in Table 4.

Table 4. Correlation matrix for the four main risk factors

Table 5 shows the volatilities for the component risk factors.

Table 5. Volatilities of component risk factors

We next combine the correlations in Table 4 with the component volatilities in Table 5 and then apply (29) to obtain the term structure for the total forward volatility shown in Table 6.

Table 6. Term structure of total forward volatility

If we worked with these results, we would apply a 15.66% volatility to the put for decrement t = 1, a 16.39% volatility to the put for decrement 5, and so on, and a 26.05% volatility to the put for decrement 30.Footnote ²⁰

Figure 4 shows a plot of the total forward volatility over a horizon of up to 40 years.

The blue plot shows total forward vol, which starts at a little over 15% for a maturity of T = 1 and rises to a little over 31% for T = 40. The red line shows the contribution to that total forward vol made by the q and interest rate risk factors. These contributions start (for T = 1) at close to zero, but rise linearly with T, reaching over 24% for T = 40. Thus, as T increases, the q and interest rate risk factors become increasingly important drivers of total forward volatility.

4.10 Obtaining a single volatility estimate for use in all put decrements

Having established that, in principle, each put decrement requires its own (potentially) different vol estimate, it is still possible to use a single volatility for all puts, provided one uses a single volatility calibration that is appropriate to the borrower's age and gender. To obtain such a calibration, one could use the expected volatility obtained by weighting each volatility by its exit prob or decrement probability. The formula for the expected volatility σ ^e is then

(30)$$\sigma ^e = \mathop \sum \limits_t [ {exit\;prob_t \times \sigma_t} ] .$$

Table 7 shows these expected volatilities against borrower age: for males 26.1% for age 55, 27.3% for age 70 (and hence our earlier baseline single volatility recommendation for males aged 70), and so on. These expected volatilities give results that are very close to the results that one would have obtained had one used the full volatility term structure.

Table 7. Expected volatilities for different ages

Note: Exit probabilities as per Figure 1.

5.1 Sensitivities of valuations to input parameters

Table 9 shows the sensitivities of L, NNEG, and ERM to changes in key parameter inputs. These are expressed in elasticity form, i.e., where the elasticity of the relevant output with respect to a change in an input is the % change in the output divided by the % change in the input.

Table 9. Sensitivities of valuations in elasticity form: male aged 70

Note: As per Table 8.

These results indicate that NNEG valuations are highly sensitive to changes in the l and LTV input parameter calibrations. It is also interesting to note that the ERM valuations are much less so, because of the offsetting impacts on the loan value and NNEG valuations.

Table 10 shows the same valuations for age 70 as percentages of the initial loan amount, £40.

Table 10. ERM and NNEG valuations as percentages of loan amount: male aged 70

Note: Based on baseline case calibrations.

Figures 5 and 6 show plots of NNEG/loan values against borrower age and ERM/loan values against borrower age.

Figure 5. NNEG/loan ratios vs. borrower age.

Note: As per Table 10.

Figure 6. ERM/loan ratios vs. borrower age.

Note: As per Tables 8 and 10.

One notices that the ratios of NNEG to loan amount fall sharply with age. However, what is most significant are the low ratios of ERM to loan amount, which suggest that ERMs generate low (and in some cases, negative) profits to lenders, especially loans to younger borrowers.