Question

A perpetuity makes payments at the end of each year at an annual effective rate of
3.4%. The payment pattern is 3,2,1 (at the end of years 1, 2 and 3) and this pattern repeats for the
balance of the perpetuity. Calculate the present value of the perpetuity at the beginning of the rst
year. Please show all work.

Answer 1

Value of perpetuity = Value ofperpetuity($3 at yr 1,4,7...) + Value of perpetuity($2 at yr 2,5,8...) + Value of perpetuity($1 at yr 3,6,9...)

Value of perpetuity($3 at yr 1,4,7...) = V₁ = 3/1.034 + 3/1.034⁴ + 3/1.034⁷ + 3/1.034¹⁰+...

V₁ = 1/1.034 * {3 + 3/1.034³ + 3/1.034⁶ + 3/1.034⁹+...}

V₁ = 1/1.034 * {3 + [3/1.034³ + 3/(1.034³)² + 3/(1.034³)³+...]}

Note: [3/1.034³ + 3/(1.034³)² + 3/(1.034³)³+...] can be applied annuity formula with rate r = (1.034³-1)

V₁ = 1/1.034 * {3 + [3/(1.034³-1)]}

V₁ = 30.4004 ...(1)

Similarly

V₂ = 2/1.034² + 2/1.034⁵ + 2/1.034⁸ + 2/1.034¹¹+...

V₂ = 1/1.034² * {2 + 2/1.034³ + 2/1.034⁶ + 2/1.034⁹+...}

V₂ = 1/1.034² * {2 + [2/1.034³ + 2/(1.034³)² + 2/(1.034³)³+...]}

Note: [2/1.034³ + 2/(1.034³)² + 2/(1.034³)³+...] can be applied annuity formula with rate r = (1.034³-1)

V₂ = 1/1.034² * {2 + [2/(1.034³-1)]}

V₂ = 19.6005 ...(2)

Similarly

V₃ = 1/1.034³ + 2/1.034⁶ + 2/1.034⁹ + 2/1.034¹²+...

V₃ = 1/1.034³ + 1/(1.034³)² + 1/(1.034³)³+...

V₃ = 1/(1.034³-1) = 9.4780 ...(3)

Value of perpetuity = Value ofperpetuity($3 at yr 1,4,7...) + Value of perpetuity($2 at yr 2,5,8...) + Value of perpetuity($1 at yr 3,6,9...)

= V₁ + V₂ + V₃

= 30.4004 + 19.6005 + 9.4780

= 59.4789