Leverage and Periodic discount rate: Difference between pages

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Periodic discount rate is 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


==Leverage calculation==


Leverage is most commonly defined as debt divided by Debt plus Equity
==Calculating periodic discount rate from start and end cash==


= D / (D + E).
Given the cash amounts at the start and end of an investment or borrowing period, we can calculate the periodic discount rate.




<span style="color:#4B0082">'''''Example 1: Leverage calculation'''''</span>
<span style="color:#4B0082">'''Example 1: Discount rate of 2.91%'''</span>


If the amounts of debt and equity were equal then leverage under this definition would be calculated as:<br />
GBP 1 million is borrowed.  
1 / (1 + 1) = 50%.


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


==Broader definitions==


The term 'leverage' is also used in a broader sense to refer to the amount of debt in a firm's financial structure.<br />
The periodic discount rate (d) is:
Used in this broader sense, 'leverage' means very much the same as 'gearing'. <br />
However, leverage and gearing are normally quantified by different calculations.


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


==Leveraging up==
Which can also be expressed as:


To 'leverage up' means to increase the level of gearing in an operational or financial structure.  The intention of leveraging up is to improve expected net results.  <br />
d = (End - Start) / End
A consequence of leveraging up is normally to increase financial risk.<br />
Many financial disasters have been a consequence of leveraging up excessively in this way in earlier periods.


= (1.03 - 1) / 1.03


<span style="color:#4B0082">'''''Example 2: Virgin's loan notes secured on Heathrow landing slots'''''</span>
= 0.029126


:"Virgin Atlantic Airways secured an impressive £220m senior secured note transaction using the airline's [rights to use] take-off and landing slots at London Heathrow Airport. It is the first time in European air travel history that airport slots have been leveraged in this way."
= '''2.9126%'''


:''The Treasurer magazine, February 2017 p25 - Deals of the Year - Bonds below £500m winner.''


<span style="color:#4B0082">'''Example 2: Discount rate of 3%'''</span>


==Leverage in banking==
GBP 0.97 million is borrowed or invested


Banks tend to have very high levels of leverage, compared with non-financial corporates.
GBP 1.00 million is repayable at the end of the period.  


Maximum levels of leverage are established by prudential regulation, including regulatory leverage ratios.


The periodic discount rate (d) is:


Leverage ratios in banking are usually defined as the ratio of total balance sheet assets to equity.
= (End - Start) / End




==Leverage in derivatives trading==
= (1.00 - 0.97) / 1.00


Leverage is also the ratio of the total value of a derivatives contract relative to the size of the required margin or collateral. <br />
= 0.030000


= '''3.0000%'''


<span style="color:#4B0082">'''''Example 3: Leverage in derivatives trading'''''</span>


10:1 leverage means that an investor needs to provide GBP 10,000 in order to control a position of a GBP 100,000 value futures contract while taking responsibility for any losses or gains their investments incur. <br />As a result if the value of the contract rose by 10% to GBP 110,000, there will be a potential profit of 100% (= 10 x 10%) relative to the amount of GBP 10,000 invested.<br /><br />
==Calculating end cash from periodic discount rate==


Similarly if the value were to fall by 10% to GBP 90,000, there would be a loss of the all the initial investment.<br />
We can also work this relationship in the other direction.
Again the change in the value of the total position is 10 x the 10% movement in the value of the contract.<br />
In this case, a loss of 10 x 10% = 100%.<br />
<br />
It is also possible to lose more than the entire value of the initial investment.<br />
This is why derivatives trading can be so dangerous for the investor.


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


== See also ==
* [[CRD IV]]
* [[Debt]]
* [[Deleverage]]
* [[Equity]]
* [[Gearing]]
* [[Leverage Ratio]]
* [[Liquidity]]
* [[Liquidity risk]]
* [[Prudential Regulation Authority]]
* [[Stability]]


==Other links==
<span style="color:#4B0082">'''Example 3: End amount from periodic discount rate'''</span>
[http://www.treasurers.org/node/8012 Masterclass: Measuring financial risk, The Treasurer, July 2012]


[[Category:Corporate_finance]]
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:
 
d = (End - Start) / End
 
 
d = (End / End) - (Start / End)
 
 
d = 1 - (Start / End)
 
 
''Rearranging this relationship:''
 
1 - d = (Start / End)
 
 
End = Start / (1 - d)
 
 
''Substituting the given information into this relationship:''
 
End = 0.97 / (1 - 0.030000)
 
 
= 0.97 / 0.97
 
 
= '''GBP 1.00m'''
 
 
==Calculating start cash from periodic discount rate==
 
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 discount rate, we can calculate the start amount.
 
 
<span style="color:#4B0082">'''Example 4: Start amount from periodic discount rate'''</span>
 
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 - Start) / End
 
 
d = 1 - (Start / End)
 
 
''Rearranging this relationship:''
 
(Start / End) = 1 - d
 
 
Start = End x (1 - d)
 
 
''Substitute the given data into this relationship:''
 
Start = 1.00 x (1 - 0.030000)
 
= '''GBP 0.97m'''
 
 
==Periodic yield==
 
The periodic discount rate (d) is also related to the [[periodic yield]] (r), and each can be calculated from the other.
 
 
====Conversion formulae (d to r and r to d)====
 
r = d / (1 - d)
 
d = r / (1 + r)
 
 
''Where:''
 
r = periodic interest rate or yield
 
d = periodic discount rate
 
 
 
==See also==
 
*[[Effective annual rate]]
*[[Certificate in Treasury Fundamentals]]
*[[Certificate in Treasury]]
*[[Discount rate]]
*[[Nominal annual rate]]
*[[Periodic yield]]
*[[Yield]]

Revision as of 12:27, 14 December 2016

Periodic discount rate is 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


Calculating periodic discount rate 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 discount rate.


Example 1: Discount rate of 2.91%

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

Which can also be expressed as:

d = (End - Start) / End

= (1.03 - 1) / 1.03

= 0.029126

= 2.9126%


Example 2: Discount rate of 3%

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 - Start) / End


= (1.00 - 0.97) / 1.00

= 0.030000

= 3.0000%


Calculating end cash from periodic discount rate

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 discount rate, we can calculate the end amount.


Example 3: End amount from periodic discount rate

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:

d = (End - Start) / End


d = (End / End) - (Start / End)


d = 1 - (Start / End)


Rearranging this relationship:

1 - d = (Start / End)


End = Start / (1 - d)


Substituting the given information into this relationship:

End = 0.97 / (1 - 0.030000)


= 0.97 / 0.97


= GBP 1.00m


Calculating start cash from periodic discount rate

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 discount rate, we can calculate the start amount.


Example 4: Start amount from periodic discount rate

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 - Start) / End


d = 1 - (Start / End)


Rearranging this relationship:

(Start / End) = 1 - d


Start = End x (1 - d)


Substitute the given data into this relationship:

Start = 1.00 x (1 - 0.030000)

= GBP 0.97m


Periodic yield

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


Conversion formulae (d to r and r to d)

r = d / (1 - d)

d = r / (1 + r)


Where:

r = periodic interest rate or yield

d = periodic discount rate


See also