Discount factor and Periodic discount rate: Difference between pages

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'''1.'''
__NOTOC__
A cost of borrowing - or rate of return - expressed as:


(DF).
*The excess of the amount at the end over the amount at the start
*Divided by the amount at the end


The number less than one which we multiply a single future cash flow by, to work out its present value as:


PV = DF x future cashflow.
==Example 1==
GBP 1 million is borrowed.  


GBP 1.03 million is repayable at the end of the period.


The periodic discount factor is calculated from the periodic [[yield]] as:


DF = (1 + periodic yield)<SUP>-n</SUP>
The periodic discount rate (d) is:


d = (End amount - start amount) / End amount


Commonly abbreviated as DF(n,r) ''or'' DF<SUB>n,r</SUB>
= (1.03 - 1) / 1.03


Where:
= 0.029126


n = number of periods.
= '''2.9126%'''


r = periodic yield (or periodic cost of capital).


==Example 2==
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: Discount factor calculation'''</span>


Periodic yield or cost of capital (r) = 6%.
The periodic discount rate (d) is:


Number of periods in the total time under review (n) = 1.
(End amount - start amount) / End amount


= (1.00 - 0.97) /  1.00


Discount factor = (1 + r)<sup>-n</sup>
= 0.030000


= 1.06<sup>-1</sup>
= '''3.0000%'''


= 0.9434.


==Example 3==
GBP  0.97 million is borrowed.


The greater the time delay, the smaller the Discount Factor.
The periodic discount rate is 3.0000%.


Calculate the amount repayable at the end of the period.


<span style="color:#4B0082">'''Example 2: Increasing number of periods delay'''</span>
===Solution===
The periodic discount rate (d) is defined as:


Periodic yield or cost of capital = 6%.
d = (End amount - start amount) / End amount


The number of periods delay increases to 2.
d = 1 - (Start amount / End amount)


Discount factor = (1 + r)<sup>-n</sup>


= 1.06<sup>-2</sup>
''Rearranging this relationship:''


= 0.8890.
(Start amount / End amount) = 1 - d


''(A smaller figure than the 0.9434 we calculated previously for just one period's delay.)''
Start amount = End amount x (1 - d)


Start amount / (1 - d) = End amount


End amount = Start amount / (1 - d)


'''2.'''


The yield or cost of capital used for the purpose of calculating Discount Factors, as defined above. 
''Substituting the given information into this relationship:''


For example the 6% rate applied in definition 1. above.
End amount = GBP 0.97m / (1 - 0.030000)


= GBP 0.97m / 0.97


== See also ==
= '''GBP 1.00m'''
* [[Annuity factor]]
 
* [[Certificate in Treasury Fundamentals]]
 
* [[Certificate in Treasury]]
==Example 4==
* [[Compounding effect]]
An investment will pay out a single amount of GBP 1.00m at its final maturity after one period.
* [[Compounding factor]]
 
* [[Cumulative Discount Factor]]
The periodic discount rate is 3.0000%.
* [[Day count conventions]]
 
* [[Factors]]
Calculate the amount invested at the start of the period.
* [[Present value]]
 
===Solution===
As before, the periodic discount rate (d) is defined as:
 
d = (End amount - start amount) / End amount
 
d = 1 - (Start amount / End amount)
 
 
''Rearranging this relationship:''
 
(Start amount / End amount) = 1 - d
 
Start amount = End amount x (1 - d)
 
 
''Substitute the given data into this relationship:''
 
Start amount = GBP 1.00m x (1 - 0.030000)
 
= '''GBP 0.97m'''
 
 
 
==See also==
 
*[[Effective annual rate]]
*[[Discount rate]]
*[[Nominal annual rate]]
*[[Periodic yield]]
*[[Yield]]

Revision as of 15:04, 26 October 2015

A cost of borrowing - or rate of return - expressed as:

  • The excess of the amount at the end over the amount at the start
  • Divided by the amount at the end


Example 1

GBP 1 million is borrowed.

GBP 1.03 million is repayable at the end of the period.


The periodic discount rate (d) is:

d = (End amount - start amount) / End amount

= (1.03 - 1) / 1.03

= 0.029126

= 2.9126%


Example 2

GBP 0.97 million is borrowed or invested

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


The periodic discount rate (d) is:

(End amount - start amount) / End amount

= (1.00 - 0.97) / 1.00

= 0.030000

= 3.0000%


Example 3

GBP 0.97 million is borrowed.

The periodic discount rate is 3.0000%.

Calculate the amount repayable at the end of the period.

Solution

The periodic discount rate (d) is defined as:

d = (End amount - start amount) / End amount

d = 1 - (Start amount / End amount)


Rearranging this relationship:

(Start amount / End amount) = 1 - d

Start amount = End amount x (1 - d)

Start amount / (1 - d) = End amount

End amount = Start amount / (1 - d)


Substituting the given information into this relationship:

End amount = GBP 0.97m / (1 - 0.030000)

= GBP 0.97m / 0.97

= GBP 1.00m


Example 4

An investment will pay out a single amount of GBP 1.00m at its final maturity after one period.

The periodic discount rate is 3.0000%.

Calculate the amount invested at the start of the period.

Solution

As before, the periodic discount rate (d) is defined as:

d = (End amount - start amount) / End amount

d = 1 - (Start amount / End amount)


Rearranging this relationship:

(Start amount / End amount) = 1 - d

Start amount = End amount x (1 - d)


Substitute the given data into this relationship:

Start amount = GBP 1.00m x (1 - 0.030000)

= GBP 0.97m


See also