Periodic yield and Perpetuity: Difference between pages

From ACT Wiki
(Difference between pages)
Jump to navigationJump to search
imported>Doug Williamson
(Layout.)
 
imported>Doug Williamson
(Layout.)
 
Line 1: Line 1:
__NOTOC__
1. ''Valuation.''
A rate of return - or cost of borrowing - expressed as the proportion by which the amount at the end of the period exceeds the amount at the start.  


A series of cash flows modelled to carry on for an infinite amount of time in the future.


==Example 1==
GBP 1 million is borrowed or invested.


GBP 1.03 million is repayable at the end of the period.  
2. ''Fixed perpetuity.''


A fixed perpetuity is a periodic cash flow starting one period in the future, then carrying on for ever thereafter.


The periodic yield (r) is:
Each cash flow is an equal fixed amount.


r = (End amount / start amount) - 1
The present value of a fixed perpetuity is calculated - assuming a constant periodic cost of capital (r) for all periods from now to infinity - as:


= (1.03 / 1) - 1
Present Value = A<sub>1</sub> x 1/r


= 0.03


= '''3%'''
where:


A<sub>1</sub> = Time 1 cash flow


==Example 2==
r = periodic cost of capital
GBP  0.97 million is borrowed or invested.


GBP 1.00 million is repayable at the end of the period.


<span style="color:#4B0082">'''Example 1: Fixed perpetuity valuation'''</span>


The periodic yield (r) is:
Time 1 cash flow = $10m, continuing at the same amount each period thereafter in perpetuity.


(End amount / start amount) - 1
Periodic cost of capital = 5%


= (1.00 / 0.97) - 1
The present value of the fixed perpetuity is:


= 0.030928
= $10m x (1 / 0.05)


= '''3.0928%'''
= $10m x 20


= $'''200'''m


''Check:''


0.97 x 1.030928 = 1.00.


3. ''Growing perpetuity.''


==Example 3==
A growing perpetuity is an infinite series of cash flows, modelled to grow by a constant proportionate amount every period.
GBP  0.97 million is invested.  


The periodic yield is 3.0928%.
For a growing perpetuity, the present value formula is modified to take account of the constant periodic growth rate, as follows:


Calculate the amount repayable at the end of the period.
Present Value = A<sub>1</sub> x 1 / (r - g)


===Solution===
where g = the periodic rate of growth of the cash flow.
The periodic yield (r) is defined as:


r = (End amount / start amount) - 1


<span style="color:#4B0082">'''Example 2: Growing perpetuity valuation'''</span>


''Rearranging this relationship:''
Time 1 cash flow = $10m, growing by a constant percentage amount each period thereafter in perpetuity.


End amount = Start amount x (1 + r)
Periodic cost of capital = 5%.


Periodic growth rate = 2%


''Substituting the given information into this relationship:''


End amount = GBP 0.97m x (1 + 0.030928)
The present value of the growing perpetuity is:


= '''GBP 1.00m'''
= A<sub>1</sub> x 1 / (r - g)


= $10m x (1 / (0.05 - 0.02) )


==Example 4==
= $10m x (1 / 0.03)
An investment will pay out a single amount of GBP 1.00m at its final maturity after one period.


The periodic yield is 3.0928%.
= $10m x 33.3


Calculate the amount invested at the start of the period.
= $'''333'''m


===Solution===
As before, the periodic yield (r) is defined as:


r = (End amount / start amount) - 1
The modest rate of growth in the cash flow has added substantially to the total present value.




''Rearranging this relationship:''


Start amount = End amount / (1 + r)


4. ''Declining perpetuity.''


''Substitute the given data into this relationship:''
Growth can be negative, in other words, decline.


Start amount = GBP 1.00m / (1 + 0.030928)
For a declining perpetuity, the present value formula is the same as the growing perpetuity, but the growth rate (g) is entered as a negative number as follows:


= '''GBP 0.97m'''


<span style="color:#4B0082">'''Example 3: Declining perpetuity valuation'''</span>


''Check:''
Time 1 cash flow = $10m, declining by a constant percentage amount each period thereafter in perpetuity.


0.97 x 1.030928 = 1.00, as expected.
Periodic cost of capital = 5%.


Periodic growth rate = -(2)% negative = -0.02


==See also==


*[[Effective annual rate]]
The present value of the declining perpetuity is:
*[[Discount rate]]
 
*[[Nominal annual rate]]
= A<sub>1</sub> x 1 / (r - g)
*[[Periodic discount rate]]
 
*[[Yield]]
= $10m x (1 / (0.05 - -0.02) )
 
= $10m x (1 / 0.07)
 
= $10m x 14.3
 
= $'''143'''m
 
 
The small negative rate of growth in the cash flow has reduced the total present value very substantially.
 
 
 
The growing / declining perpetuity concept is applied in many contexts.
 
For example, the Dividend growth model for share valuation.
 
 
== See also ==
* [[Annuity]]
* [[Discounted cash flow]]
* [[Dividend growth model]]
* [[Growing annuity]]
* [[Growing perpetuity]]
* [[Growing perpetuity factor]]
* [[Irredeemable]]
* [[Perpetuity due]]
* [[Perpetuity factor]]
* [[Simple annuity]]
 
 
==The Treasurer articles==
[[Media:2013_10_Oct_-_The_real_deal.pdf| The real deal, The Treasurer]]
 
''Real rates of corporate decline often lead to miscalculation, overpaying for acquisitions and disastrous losses.''
 
''Read this article to discover how to avoid the most common errors, and add value for your organisation.''
 
[[Category:Corporate_finance]]
[[Category:Long_term_funding]]

Revision as of 12:45, 12 June 2021

1. Valuation.

A series of cash flows modelled to carry on for an infinite amount of time in the future.


2. Fixed perpetuity.

A fixed perpetuity is a periodic cash flow starting one period in the future, then carrying on for ever thereafter.

Each cash flow is an equal fixed amount.

The present value of a fixed perpetuity is calculated - assuming a constant periodic cost of capital (r) for all periods from now to infinity - as:

Present Value = A1 x 1/r


where:

A1 = Time 1 cash flow

r = periodic cost of capital


Example 1: Fixed perpetuity valuation

Time 1 cash flow = $10m, continuing at the same amount each period thereafter in perpetuity.

Periodic cost of capital = 5%

The present value of the fixed perpetuity is:

= $10m x (1 / 0.05)

= $10m x 20

= $200m


3. Growing perpetuity.

A growing perpetuity is an infinite series of cash flows, modelled to grow by a constant proportionate amount every period.

For a growing perpetuity, the present value formula is modified to take account of the constant periodic growth rate, as follows:

Present Value = A1 x 1 / (r - g)

where g = the periodic rate of growth of the cash flow.


Example 2: Growing perpetuity valuation

Time 1 cash flow = $10m, growing by a constant percentage amount each period thereafter in perpetuity.

Periodic cost of capital = 5%.

Periodic growth rate = 2%


The present value of the growing perpetuity is:

= A1 x 1 / (r - g)

= $10m x (1 / (0.05 - 0.02) )

= $10m x (1 / 0.03)

= $10m x 33.3

= $333m


The modest rate of growth in the cash flow has added substantially to the total present value.



4. Declining perpetuity.

Growth can be negative, in other words, decline.

For a declining perpetuity, the present value formula is the same as the growing perpetuity, but the growth rate (g) is entered as a negative number as follows:


Example 3: Declining perpetuity valuation

Time 1 cash flow = $10m, declining by a constant percentage amount each period thereafter in perpetuity.

Periodic cost of capital = 5%.

Periodic growth rate = -(2)% negative = -0.02


The present value of the declining perpetuity is:

= A1 x 1 / (r - g)

= $10m x (1 / (0.05 - -0.02) )

= $10m x (1 / 0.07)

= $10m x 14.3

= $143m


The small negative rate of growth in the cash flow has reduced the total present value very substantially.


The growing / declining perpetuity concept is applied in many contexts.

For example, the Dividend growth model for share valuation.


See also


The Treasurer articles

The real deal, The Treasurer

Real rates of corporate decline often lead to miscalculation, overpaying for acquisitions and disastrous losses.

Read this article to discover how to avoid the most common errors, and add value for your organisation.

Retrieved from ‘https://wiki.treasurers.org/w/index.php?title=Periodic_yield&oldid=37395’