# Deferred perpetuity / Delayed Perpetuity

###### What is a deferred perpetuity ?

A Perpetuity is an infinite stream of cash flows. A deferred perpetuity is a stream of cash flows that start a point in the future and continue till infinity. It is also known as delayed perpetuity.

Retirement accounts pay a set stream of cash flows after the retirement account holder has attained the age of retirement. Thus is one of the reasons most people in the US invest their savings in 401(k) that will pay them a set amount of money once they retire. The amount of payment is set or calculated based on the plan chosen.

The mutual fund market is another place where delayed perpetuities are common. Some mutual funds have terms which requires investors to lock their investment in the fund for a specific period. It is after the expiry of this period, that dividends are paid to the investor.

###### Who should invest in a deferred perpetuity ?

An investment that has delayed perpetuity as a feature is made with an intention of ensuring future financial security. It is not an option for investors who are looking for immediate returns on their investments.

###### What is the Formula for calculating the present value of a deferred perpetuity ?

The formula for calculating the present value of a deferred perpetuity is

= ( CF / r ) * ( 1 / ( 1 + r ) ( n  –  1 ) )

Where

CF = Annual cash flow  ;

r = discount rate  ;

n = Number of periods of deferral  ;

Example :

Shawn has made an investment that will pay him amount of \$ 1000 every year starting 7 years from now. Calculate the present value of these payments assuming a discount rate of 7 %. ( Round the solution to two decimal places )

Solution :

The formula for calculating the present value of a deferred perpetuity is

= ( CF / r ) * ( 1 / ( 1 + r ) ( n  –  1) )

As per the information given in the question we have

CF = \$ 1,000  ;  r = 7 % = 0.07  ;  n = perpetuity beginning 7 years from now i.e., for seven years from Year 0 to Year 6 = 7 periods

Applying the above information in the formula we have the present value of a deferred perpetuity which is due from today as

= ( \$ 1,000 / 0.07 ) * ( 1 / ( 1 + 0.07 ) ( 7 – 1 ) )

= ( \$ 1,000 / 0.07 ) * ( 1 / ( 1 + 0.07 ) ( 6 ) )

= ( \$ 1,000 / 0.07 ) * ( 1 / ( 1.07 ) ( 6 ) )

= ( \$ 1,000 / 0.07 ) * ( 1 / 1.500730 )

= \$ 14,285.714286   * 0.666342

= \$ 9,519.174626

= \$ 9,519.17 ( when rounded off to two decimal places )

The present value of delayed payments is = \$ 9,519.17

Note : The value of ( 1.07 ) 6   has been calculated using the excel function =POWER(Number,Power). Thus  =POWER(1.07,6) = 1.500730

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