The leasehold extension formula

In the United Kingdom, the amount of capital that must be paid by the leaseholder (the ‘Lessee’) to the landlord (the ‘Lessor’) is determined, or at least guided, by Act of Parliament. (The relevant legislation is the Leasehold Reform, Housing and Urban Development Act 1993 and the Commonhold and Leasehold Reform Act 2002.)

The relevant parameters are as follows,
 G the ground rent (pounds per annum) T the time left on the lease (in years) v the current value of the property (pounds) V the value of the property after lease extension (pounds) ε the yield rate (so ε = 0.05 is a 5% yield rate)
G and T are given in the title deeds of the property. ε, in the current application of the law, is taken to be 0.05. v and V depend on the local property market, and their values need to be judged by an experienced chartered surveyor.

D is the diminution in value of the landlord's interest, and is defined as the sum of,

(a) the loss of income of ground rent for the remainder of the term,
(b) the loss due to the additional wait for reversion.

If the ground rent is reduced to a peppercorn rent by the lease extension, the landlord will lose an income of G pounds a year for T years. This sum can be provided by an investment of

 (a)           D1 = G ε { 1 − (1 + ε)−T }

pounds. At the end of each year this sum increases by a factor (1 + ε) with the accrual of interest, but decreases as the fixed sum G is withdrawn. After just T years it is reduced to zero.

The loss due to the additional wait for reversion means that the landlord, instead of acquiring the property at value V after a wait of T years, must wait longer. An indefinite wait is assumed, which means that the time by which the lease is actually extended does not form part of the final formula. A sum V to be realised in T years time can be achieved by an investment now of,

 (b)           D2 = V (1 + ε)−T

This grows by a factor (1 + ε) each year, so after T years will equal V.

The landlord’s diminution of interest, D, is then given by,

 (d)           D = D1 + D2

The marriage value, M, is defined as the increase in value of the property plus the landlord’s diminution of interest, and is therefore,

 (m)           M = V − v + D

And L, the cost of extending the lease, is determined to be the landlord’s diminution of interest plus half the marriage value,

 (l)           L = D + M 2 = 1 2 (V − v + D)

And substituting (a) and (a) in (l) we get,

 (L)           L = 1 2 { G ε (1 − (1 + ε)−T) + V (1 + (1 + ε)−T) − v }

For example, suppose we have the following typical values,
G = 10.5, T = 58, v = 75,000, V = 85,000 and ε = 0.05
Then (1 + ε)−T = 0.0590228, and L is,
L = 0.5 (197.61 + 90,016.93 − 75,000) = 7,607
Note the correct behaviour of the formula when T sinks to zero. The expiry of the lease makes v = 0. Then L = V, so the cost of extending the lease becomes equal to the full value of the property.

The derivative dL/dT is an approximate measure of the extra cost (negative) of delaying the lease extension by one year. We have,

 dL dT = 1 2 (V − G ε ) d dT (1 + ε)−T

 = − 1 2 (V − G ε ) (1 + ε)−T loge (1 + ε)

If, in the example above, the lease is delayed until T = 57, the extra cost is 129. If the lease had been extended when T = 59, the saving would have been 114. These compare with the value of dL/dT, which is about 122.