Discount rate/Tutorials

The present value of future costs and benefits

The present value V of a cost (or benefit) $c_{t}$ occuring after an interval of t years at a dicount rate of r is given by:

$V={\frac {c_{t}}{(1+r)^{t}}}$

The net present expected value of a future cost (or benefit) that has z possible values is given by calculating the value of $c_{t}$ in the above equation as:

${\mbox{c}}=\sum _{x=1}^{z}p_{x}c_{x}$

where $p_{x}$ is the probability of occurrence of the value $c_{x}$

The present value of a series of annual costs and benefits, ocurring after annual intervals 0 to n is given by:

${\mbox{V}}=\sum _{t=0}^{n}{\frac {c_{t}}{(1+r)^{t}}}$.

The social time preference rate

The social time preference rate, s, is given by:-

s = δ + ηg

where:

δ is the pure time preference rate (otherwise known as the utility discount rate);

η is the elasticity of marginal utility with respect to consumption; and,

g is the expected future growth rate of consumption.

Evidence based upon the structure of personal income tax rates in OECD countries suggests that the value of η for most developed countries is close to 1.4 . Estimates for the United Kingdom have ranged from 0.7 t0 1.5. .

The UK Treasury Green Book uses

δ=1.5%, η=1.0, g=2%, yielding s = 3.5%

The Stern review uses

δ=0.1%, η=1.0, g=2%, yielding s= 2.1%