I mentioned David Land’s bemused question to the Equity Release working party yesterday. If the working party hasn’t yet fixed the right method of calculating the forward, isn’t that a pretty major source of possible error?
No coherent answer emerged, but Land raised an interesting point. If we can’t lower the value of the no negative equity guarantee by putting in an optimistic growth forecast, perhaps we can tweak the funding rate instead? He drops a hint when he suggests that there’s a large range of possible funding rates that you could think about, and that ‘The PRA thinks that you could possibly fund a house at Libor flat, which seems remarkably difficult’.
Nice try, but there is a problem with that idea too.
Let’s try to draw out what Land may have had in mind.
(1) R = S x exp(-q x t)
(2) F = S x exp( (r+s-q) x t)
where R is the price of a reversion aka deferment contract, S is the spot price of the income producing asset, q is the discount rate aka deferment rate representing the present value of lost income over the term t years. F is the price of a forward contract, r the risk free, s the funding spread over Libor (which might well be zero, let’s see).
If we use Black 76 then we use the forward price F rather than the deferment price R. Since the NNEG for any decrement t is a put option, the higher the forward price F, the more the option is out of the money, and the cost of the guarantee is correspondingly lower. Thus we can make the guarantee cheaper by assuming a non-zero spread s.
In a certain sense of the word ‘good’, a cheap guarantee is good, as it gives firms more capital. I wonder if that was what Land was driving at? Recall that he works for Rothesay Life, a major provider of annuities, a major user of the Matching Adjustment, who just bought a whacking great portfolio of Equity Release Mortgages from the taxpayer. Are Rothesay using a higher-than-Libor funding rate to cheapen the cost of their guarantees?
Who can say. In any case, I don’t think a higher than Libor funding rate is correct, as follows.
It is true that a bank will charge a buy to let customer a higher rate than Libor to fund the purchase of the underlying property. No one doubts that. But the purchaser of a forward contract doesn’t actually own any property. The deal on the forward is to agree to purchase the property at maturity for a price agreed in advance. Unlike a deferment contract, where we put up the money now for possession later, with a forward contract we pay later, too. So the analogy with buy to let is not quite right.
Now suppose we are a firm holding a deferment contract for possession at t, and we want to hedge our risk by selling the corresponding forward contract. If we price the contract per equation (2) above, it is certainly true we may want to charge a spread over the risk free rate r to compensate for the risk that the purchaser of the forward contract will default at t. But how do we represent the value of our hedged position in our books? Assuming no risk of default, the present value of the position is
(3) R’ = R x exp(s x t)
But there clearly is a risk of default, because we just assumed one when we applied the spread s! So we must reserve against that risk. By how much? Well if s is a spread representing the cost of default, i.e. if the difference in present value attributable to s is precisely the cost of default, then that same difference is the amount we would reserve.
(4) Reserve = R x exp(s x t) – R
We must therefore subtract the reserve from our portfolio value.
(5) Portfolio value = R’ – reserve = R x exp(s x t) – (R x exp(s x t) – R) = R
So we are back where we started! You cannot cheapen the value of a deferment contract by hedging it in the derivatives market. Whatever increase in value is attributable to applying a spread over Libor represents a compensation for risk, which must be subtracted again as a reserve. You can’t make money instantaneously, and good things come only to those who wait. (Sometimes bad things too).
This is for the simpler case where there is no option, only a deferment contract hedged by a forward. I will consider the option case in a later post.