Derivative and Periodic yield: Difference between pages

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# Abbreviation for derivative financial instrument.
__NOTOC__
# ''Maths''.  A derivative function describes the rate of change of the underlying function, with respect to changes in one of the variables in the underlying function.
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.  


::: The first derivative describes the slope of the function curve at a given point on the curve.


::: The second derivative describes the rate of change of the slope.  In other words the degree of curvature, at a given point.
====Example 1====


== See also ==
GBP 1 million is borrowed or invested.
* [[Derivative instrument]]
 
* [[Embedded derivative]]
GBP 1.03 million is repayable at the end of the period.
* [[Greeks]]
 
 
The periodic yield (r) is:
 
r = (End amount / start amount) - 1
 
Which can also be expressed as:
 
r = (End / Start) - 1
 
''or''
 
r = <math>\frac{End}{Start}</math> - 1
 
 
= <math>\frac{1.03}{1}</math> - 1
 
= 0.03
 
= '''3%'''
 
 
====Example 2====
 
GBP  0.97 million is borrowed or invested.
 
GBP 1.00 million is repayable at the end of the period.
 
 
The periodic yield (r) is:
 
r = <math>\frac{End}{Start}</math> - 1
 
 
= <math>\frac{1.00}{0.97}</math> - 1
 
= 0.030928
 
= '''3.0928%'''
 
 
''Check:''
 
Amount at end = 0.97 x 1.030928 = 1.00, as expected.
 
 
====Example 3====
 
GBP  0.97 million is invested.
 
The periodic yield is 3.0928%.
 
Calculate the amount repayable at the end of the period.
 
 
'''''Solution'''''
 
The periodic yield (r) is defined as:
 
r = <math>\frac{End}{Start}</math> - 1
 
 
''Rearranging this relationship:''
 
1 + r = <math>\frac{End}{Start}</math>
 
 
End = Start x (1 + r)
 
 
''Substituting the given information into this relationship:''
 
End = GBP 0.97m x (1 + 0.030928)
 
= '''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 yield is 3.0928%.
 
Calculate the amount invested at the start of the period.
 
 
'''''Solution'''''
 
As before, the periodic yield (r) is defined as:
 
r = <math>\frac{End}{Start}</math> - 1
 
 
''Rearranging this relationship:''
 
1 + r = <math>\frac{End}{Start}</math>
 
 
Start = <math>\frac{End}{(1 + r)}</math>
 
 
''Substitute the given data into this relationship:''
 
Start = <math>\frac{1.00}{(1  +  0.030928)}</math>
 
 
= '''GBP 0.97m'''
 
 
''Check:''
 
Amount at start = 0.97 x 1.030928 = 1.00, as expected.
 
 
====Effective annual rate====
 
The periodic yield (r) is related to the [[effective annual rate]] (EAR), and each can be calculated from the other.
 
 
'''Conversion formulae'''
 
 
EAR = (1 + r)<sup>n</sup> - 1
 
 
 
r= (1 + EAR)<sup>(1/n)</sup> - 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
 
 
 
 
==See also==
 
*[[Effective annual rate]]
*[[Discount rate]]
*[[Nominal annual rate]]
*[[Periodic discount rate]]
*[[Yield]]

Revision as of 09:51, 1 November 2015

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.


Example 1

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

or

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


= <math>\frac{1.03}{1}</math> - 1

= 0.03

= 3%


Example 2

GBP 0.97 million is borrowed or invested.

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


The periodic yield (r) is:

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


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

= 0.030928

= 3.0928%


Check:

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


Example 3

GBP 0.97 million is invested.

The periodic yield is 3.0928%.

Calculate the amount repayable at the end of the period.


Solution

The periodic yield (r) is defined as:

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


Rearranging this relationship:

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


End = Start x (1 + r)


Substituting the given information into this relationship:

End = GBP 0.97m x (1 + 0.030928)

= 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 yield is 3.0928%.

Calculate the amount invested at the start of the period.


Solution

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

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


Rearranging this relationship:

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


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


Substitute the given data into this relationship:

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


= GBP 0.97m


Check:

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


Effective annual rate

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


Conversion formulae


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



See also