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Saturday, July 2, 2022

### Methodology

The net present value today of the pension you will receive over your lifetime is given by the following equation:

$v(0)=\int _{ T }^{ D }{ p(1+bT){ e }^{ -rt } } dt$

where

p is the current value of the State pension

D is the number of years until death

T is the number of years until the State pension commences (chosen start date)

b is the bonus for waiting an extra year

r is the real real rate of return on savings (assuming they are continuously compounded)

In the situation where r=0, by setting $\frac { \delta v }{ \delta T } =0$ it can be shown that optimum time to defer the State Pension (T*) is given by:

${ T }^{ * }=\frac { 1 }{ 2 } (D-\frac { 1 }{ b } )$

In the general case, where r is non-zero, by setting $\frac { \delta v }{ \delta T } =0$ it can be shown that:

$1+b{ T }^{ * }= \frac { b }{ r } (1-{ e }^{ -r(D-{ T }^{ * }) })$

This equation cannot be solved analytically in terms of T*, but since |F”(T*)/F'(T*)|<1 – where F(T*) is the above equation set to zero – the calculations use the Newton-Raphson numerical method to find an iterative solution. With a suitably chosen initial estimate of T*(we use 1) this rapidly converges on a solution (the calculation uses only three steps).