Why the forward price has an interest rate term

Another day, and another challenge to the method we have proposed for the valuation of the no negative equity guarantee. In our presentation here (slide 9, equation 8) we use the term r-q in the calculation of the forward house price at time t, so aren’t we incorporating both a house price growth assumption (r = interest rate) and a net rental rate q? Thus aren’t we – implausibly – assuming that future house price growth will be equal to the interest rate?

Not at all.

To explain what r and q are doing in the forward price calculation, let’s start with the deferment price calculation

R(t) = current house price x exp(-q x t)

where R(t) is the price settled now for deferred possession in t years, and q is the deferment rate. As Kevin and I (and the PRA) have argued before, the deferment price will be less than the price of immediate possession, because the buyer will lose out on t years of income or use.

Two points.

First, the price in no way makes any assumptions about house price future growth, for that can be anything you like. The chart above shows the Japanese housing index together with the price of a 40 year deferment contract, through to maturity, assuming deferment rate of 2.5%. Both lines converge in the 40th and final year, i.e. when the deferment contract turns into a contract for immediate possession, but the convergence could be at any price, depending on how the index performs. The pricing of the deferment in no way depends on where the final convergence will be.

Second, the interest term is absent, because the buyer is handing over the money now to take possession in the future. The difference between spot and deferment price reflects the income that the buyer loses by not taking possession immediately. Put another way, the red line grows 2.5% faster than the blue line, on the assumption that immediate possession gives a net rental yield of the same amount.

Now suppose that instead of paying now, the buyer asks to pay when he or she takes possession in 40 years time. Then the seller can no longer invest the payment at a lower but safer rate, so will want compensation, which can be achieved by selling at a premium to the deferment price, reflecting the desired safe return r. So the forward price is:

F(t) = current house price x exp( (r-q) x t)

where r is the interest rate demanded by the seller. Again, this in no way reflects house price growth assumptions. The price can grow in any way you like, and the buyer takes that risk. Rather, the interest growth r reflects the non-risky growth that the seller would have achieved if had the buyer had paid upfront.

Nothing to do with house prices, more like the growth of a building society account. In summary, the r and q terms in the forward equation represent compensation to the seller and compensation to the buyer, respectively. Forget the asset growth thing.