Perpetuity Present Value and Intra Year Compounding and Discounting - Time Value of Money

8/13/2019 Perpetuity Present Value and Intra Year Compounding and Discounting - Time Value of Money

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Perpetuity is an annuity of limitless period. For instance, the Britishgovernment has issued bonds called ‘CONSOLS’ which pay annual

interest without end.Present Value of Perpetuity:

The present value of perpetuity may be expressed as follows:

P∞ = A x PVIFA



Where P∞ is the present value of perpetuity and A is the continuousannual payment. The value of PVIFAr∞

∞ Σ 1 / (1+r) ^t = 1/rt=1



The formula for PVIFAr, ∞ is derived as follows:

PVIFAr, ∞ = 1(1+r) ^-1 + 1(1+r) ^-2 +...+ 1(1+r) ^-∞ (A)

Multiplying both the sides of (A) by (1+r) provides:

PVIFAr, ∞ (1+r) =1 + 1(1+r) ^-1 +...+ 1(1+r) ^-∞+1 (B)

Subtracting (A) from (B) yields:



The formula for PVIFAr, ∞ is derived as follows:

PVIFAr, ∞ x r = 1-1(1+r) ^-∞ (C)

In view of the fact that the 2nd term on the right hand side of (C)vanishes, we get:

PVIFAr, ∞ x r = 1 (D)The outcome will be:PVIFAr∞ =1/r (E)



Put in phrase, it implies that the present value interest factor ofperpetuity is simply 1 divided by the interest rate articulated indecimal form. Therefore the present value of perpetuity is just equal

to the continuous yearly payment divided by the interest rate. Forinstance, the present value of perpetuity of $5,000 if the interestrate is 10% is equivalent to $5,000/0.10 = $50,000. Instinctively this

is somewhat persuasive as an original sum of $50,000 would, ifinvested at a rate of interest of 10%, present continuous yearlytakings of $5,000 everlastingly devoid any mutilation of the capitalvalue.



Growing Perpetuity explained with IllustrationAn office complex is anticipated to fetch a net rental of $1.50 millionthe subsequent year, which is anticipated to augment by 5% every

year. If we assume that the increase will continue indefinitely, therental stream is a growing perpetuity. If the discount rate is 5percent, the present value of the rental stream is:



Intra Year Compounding and Discounting :

We are going to discuss when compounding is done more often.

Illustration :

$500 is deposited in a finance company that offers 12%

interest semi-annually. Calculate principal at the end.



Solution :First 6 months: Principal at the start $500Interest for 6 months $30$500 x 0.12/2Principal at the end $530

Second 6 months: Principal at the start $530

Interest for 6 months $31.80$530 x 0.12/2Principal at the end $561.80



Students should note that if compounding is done annually, theprincipal at the end of one year would be $500 x (1.12) = $560. Thedifference is 1.8 (between 561.80 under semiannual compoundingand $560 under annual compounding) denotes interest on interestfor the second half year.



The universal formula for the future value of a single cash flowafter n years when compounding is done m times a year is asbelow:

FVn = PV x (1 + (r / m)) ^ m x n

Perpetuity Present Value and Intra Year Compounding and Discounting - Time Value of Money

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