Periodic yield: Difference between revisions

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__NOTOC__
Periodic yield is 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.  
Periodic yield is 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.  


It is often denoted by a lower case (r).


<span style="color:#4B0082">'''Example 1'''</span>
 
==Calculating periodic yield from start and end cash==
Given the cash amounts at the start and end of an investment or borrowing period, we can calculate the periodic yield.
 
 
<span style="color:#4B0082">'''Example 1: Periodic yield (r) of 3%'''</span>


GBP 1 million is borrowed or invested.  
GBP 1 million is borrowed or invested.  
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r = (End amount / Start amount) - 1
r = (End amount / Start amount) - 1


Which can also be expressed as:
''Which can also be expressed as:''


r = (End / Start) - 1
r = (End / Start) - 1


''or''
= (1.03 / 1.00) - 1
 
r = <math>\frac{End}{Start}</math> - 1
 
 
= <math>\frac{1.03}{1}</math> - 1


= 0.03
= 0.03
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<span style="color:#4B0082">'''Example 2'''</span>
<span style="color:#4B0082">'''Example 2: Periodic yield of 3.09%'''</span>


GBP  0.97 million is borrowed or invested.  
GBP  0.97 million is borrowed or invested.  
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The periodic yield (r) is:
As before, the periodic yield (r) is:
 
r = <math>\frac{End}{Start}</math> - 1


r = (End / Start) - 1


= <math>\frac{1.00}{0.97}</math> - 1
= (1.00 / 0.97) - 1


= 0.030928
= 0.030928
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''Check:''
''Check:''


Amount at end = 0.97 x 1.030928 = 1.00, as expected.
Amount at end = 0.97 x 1.030928 = GBP 1.00m, as expected.
 


==Calculating end cash from periodic yield==
We can also work this relationship in the other direction.


<span style="color:#4B0082">'''Example 3'''</span>
Given the cash amount at the start of an investment or borrowing period, together with the periodic yield, we can calculate the end amount.
 
 
<span style="color:#4B0082">'''Example 3: End amount from periodic yield'''</span>


GBP  0.97 million is invested.  
GBP  0.97 million is invested.  
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'''''Solution'''''
'''''Solution'''''


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


r = <math>\frac{End}{Start}</math> - 1
r = (End / Start) - 1




''Rearranging this relationship:''
''Rearranging this relationship:''


1 + r = <math>\frac{End}{Start}</math>
1 + r = (End / Start)




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<span style="color:#4B0082">'''Example 4'''</span>
==Calculating start cash from periodic yield==
We can also work the same relationship reversing the direction of time travel.
 
Given the cash amount at the end of an investment or borrowing period, again together with the periodic yield, we can calculate the start amount.
 
 
<span style="color:#4B0082">'''Example 4: Start amount from periodic yield'''</span>


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


As before, the periodic yield (r) is defined as:
As before, the periodic yield (r) is:
 
r = (End / Start) - 1


r = <math>\frac{End}{Start}</math> - 1




''Rearranging this relationship:''
''Rearranging this relationship:''


1 + r = <math>\frac{End}{Start}</math>
1 + r = (End / Start)




Start = <math>\frac{End}{(1 + r)}</math>
Start = End / (1 + r)




''Substitute the given data into this relationship:''
''Substitute the given data into this relationship:''


Start = <math>\frac{1.00}{(1 + 0.030928)}</math>
Start = End / (1 + 0.030928)




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''Check:''
''Check:''


Amount at start = 0.97 x 1.030928 = 1.00, as expected.
Amount at end = 0.97 x 1.030928 = GBP 1.00m, as expected.




====Effective annual rate====
==Effective annual rate (EAR)==


The periodic yield (r) is related to the [[effective annual rate]] (EAR), and each can be calculated from the other.
The periodic yield (r) is related to the [[effective annual rate]] (EAR), and each can be calculated from the other.




'''''Conversion formulae (r to EAR and EAR to r):'''''
====Conversion formulae (r to EAR and EAR to r)====


EAR = (1 + r)<sup>n</sup> - 1
EAR = (1 + r)<sup>n</sup> - 1
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Where:
''Where:''


EAR = effective annual rate or yield
EAR = effective annual rate or yield
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====Periodic discount rate====
==Periodic discount rate (d)==


The periodic yield (r) is also related to the [[periodic discount rate]] (d), and each can be calculated from the other.
The periodic yield (r) is also related to the [[periodic discount rate]] (d), and each can be calculated from the other.




'''''Conversion formulae (r to d and d to r):'''''
====Conversion formulae (r to d and d to r)====


d = r / (1 + r)
d = r / (1 + r)
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Where:
''Where:''


d = periodic discount rate
d = periodic discount rate
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==See also==
==See also==


*[[Discount rate]]
*[[Effective annual rate]]
*[[Effective annual rate]]
*[[Discount rate]]
*[[Forward yield]]
*[[Nominal annual rate]]
*[[Nominal annual rate]]
*[[Nominal annual yield]]
*[[Nominal annual yield]]
*[[Par yield]]
*[[Periodic discount rate]]
*[[Periodic discount rate]]
*[[Yield]]
*[[Yield]]
*[[Forward yield]]
*[[Zero coupon yield]]
*[[Zero coupon yield]]
*[[Par yield]]
 
 
== Other resources ==
[[Media:2016_02_Feb_-_Many_happy_returns.pdf| Many happy returns - calculating and applying interest rates and yields, The Treasurer]]
 
[[Media:2013_09_Sept_-_Simple_solutions.pdf| Simple solutions - converting between yields, The Treasurer]]
 
[[Category:Corporate_financial_management]]
[[Category:Cash_management]]

Latest revision as of 21:07, 15 May 2020

Periodic yield is 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.

It is often denoted by a lower case (r).


Calculating periodic yield from start and end cash

Given the cash amounts at the start and end of an investment or borrowing period, we can calculate the periodic yield.


Example 1: Periodic yield (r) of 3%

GBP 1 million is borrowed or invested.

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


The periodic yield (r) is:

r = (End amount / Start amount) - 1

Which can also be expressed as:

r = (End / Start) - 1

= (1.03 / 1.00) - 1

= 0.03

= 3%


Example 2: Periodic yield of 3.09%

GBP 0.97 million is borrowed or invested.

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


As before, the periodic yield (r) is:

r = (End / Start) - 1

= (1.00 / 0.97) - 1

= 0.030928

= 3.0928%


Check:

Amount at end = 0.97 x 1.030928 = GBP 1.00m, as expected.


Calculating end cash from periodic yield

We can also work this relationship in the other direction.

Given the cash amount at the start of an investment or borrowing period, together with the periodic yield, we can calculate the end amount.


Example 3: End amount from periodic yield

GBP 0.97 million is invested.

The periodic yield is 3.0928%.

Calculate the amount repayable at the end of the period.


Solution

As before, the periodic yield (r) is:

r = (End / Start) - 1


Rearranging this relationship:

1 + r = (End / Start)


End = Start x (1 + r)


Substituting the given information into this relationship:

End = GBP 0.97m x (1 + 0.030928)

= GBP 1.00m


Calculating start cash from periodic yield

We can also work the same relationship reversing the direction of time travel.

Given the cash amount at the end of an investment or borrowing period, again together with the periodic yield, we can calculate the start amount.


Example 4: Start amount from periodic yield

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%.

Calculate the amount invested at the start of the period.


Solution

As before, the periodic yield (r) is:

r = (End / Start) - 1


Rearranging this relationship:

1 + r = (End / Start)


Start = End / (1 + r)


Substitute the given data into this relationship:

Start = End / (1 + 0.030928)


= GBP 0.97m


Check:

Amount at end = 0.97 x 1.030928 = GBP 1.00m, as expected.


Effective annual rate (EAR)

The periodic yield (r) is related to the effective annual rate (EAR), and each can be calculated from the other.


Conversion formulae (r to EAR and EAR to r)

EAR = (1 + r)n - 1

r = (1 + EAR)(1/n) - 1


Where:

EAR = effective annual rate or yield

r = periodic interest rate or yield, as before

n = number of times the period fits into a calendar year


Periodic discount rate (d)

The periodic yield (r) is also related to the periodic discount rate (d), and each can be calculated from the other.


Conversion formulae (r to d and d to r)

d = r / (1 + r)

r = d / (1 - d)


Where:

d = periodic discount rate

r = periodic interest rate or yield


See also


Other resources

Many happy returns - calculating and applying interest rates and yields, The Treasurer

Simple solutions - converting between yields, The Treasurer

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