Converting from forward rates and Depreciation: Difference between pages

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The forward rate is the rate of return - or cost of borrowing - contracted in the market today for a notional or actual deposit or borrowing:
1. ''Financial reporting - accounting practices.''
#Starting at a fixed future date; and
#Ending on a later fixed future date.


Accounting depreciation spreads the cost of a long-term tangible asset over its total life.


The forward rate is also known as the [[forward yield]].
The depreciation accounting charge reflects:
* the estimated periodic cost to a business
* of a physical capital asset
* spread over its estimated useful economic life.  




'''Conversion'''
Accounting depreciation seeks to ensure that the total accounting cost of a capitalised asset is appropriately spread and matched to the economic benefits of using the asset. 


If we know the forward yield, we can calculate both the [[zero coupon yield]] and the [[par yield]] for the same maturities and risk class.
Accounting depreciation is applying the accruals accounting principle to spread the total cost of tangible long term assets over their expected useful life.


The conversion process and calculation stems from the '[[no-arbitrage]]' relationship between the related yield curves. This means - for example - that the cash flows from a two-year '[[outright]]' deposit must be identical to the cash flows from a '[[synthetic]]' two-year deposit, built from a combination of forward deals.


Methods of spreading the total accounting cost include Straight line, Reducing balance and Sum of the digits.


<span style="color:#4B0082">'''Example 1: Forward to zero coupon rates'''</span>
Financial reporting standards generally permit the use of any systematic basis of allocating the total cost over the useful life of the asset.


Periodic forward yields ('''f''') are:


f<sub>0-1</sub> = 0.02 per period (2%)
It's important to be clear about the distinction between the:
*Depreciation charge for the period, reflected in the income statement; and
*Cumulative depreciation provision at the end of the period, reflected in the balance sheet.


f<sub>1-2</sub> = 0.04 per period (4%)


The depreciation charge is an in-period accounting expense, charged against profits for the period.


The total accumulated cash at Time 2 periods hence, from investing £1m at Time 0 is:
The cumulative provision for depreciation is a liability in the balance sheet. It's offset against the cost of the assets, to calculate their accounting net book value.


£1m x 1.02 x 1.04


= £'''1.0608'''m
Some accounting jurisdictions use the term ''amortisation'' both for this aspect of accounting both for tangible and intangible assets.




Under no-arbitrage pricing conditions, the identical terminal cash flow of £1.0608m results from investing in an outright zero coupon investment of two periods maturity, at the market yield of '''z<sub>0-2</sub>''' per period, as follows:
2. ''Foreign exchange''.


£1m x (1 + z<sub>0-2</sub>)<sup>2</sup> = £'''1.0608'''m
A decrease in the value of a currency.




Using this information, we can now calculate the zero coupon yield for two periods' maturity.
3. ''Other contexts.''


 
More generally, any decrease in the value of an asset resulting from the passing of time.
(1 + z<sub>0-2</sub>)<sup>2</sup> = 1.0608
 
1 + z<sub>0-2</sub> = 1.0608<sup>(1/2)</sup>
 
z<sub>0-2</sub> = 1.0608<sup>(1/2)</sup> - 1
 
= '''0.029951''' per period (= 2.9951%)
 
 
This is the market zero coupon rate which we would enjoy if we were to make a two-year deposit, putting our money into the deposit today.
 
 
The no-arbitrage relationship says that making such a deposit should produce the identical terminal cash flow of £1.0608m. Let's see if that's borne out by our calculations.
 
 
Investing the same £1m in a two-periods maturity zero coupon instrument at a rate of 2.9951% per period would return:
 
£1m x (1.029951)<sup>2</sup>
 
= £'''1.0608'''m
 
 
''This is the same result as enjoyed from the forward investments, as expected.''
 
 
<span style="color:#4B0082">'''Example 2: Forward to par rates'''</span>
 
Now using the zero coupon rates ('''z'''), the par rates ('''p''') can also be calculated in turn.
 
 
The periodic zero coupon yields ('''z''') are:
 
z<sub>0-1</sub> = 0.02 per period (2%)
 
z<sub>0-2</sub> = 0.029951 per period (2.9951%)
 
 
The no-arbitrage relationship between par rates and zero coupon rates is summarised in the formula:
 
p<sub>0-n</sub> = (1 - DF<sub>n</sub>) / CumDF<sub>n</sub>
 
 
''Where:''
 
p<sub>0-n</sub> = the par rate for maturity n periods, starting now
 
DF<sub>n</sub> = the discount factor for 'n' periods maturity, calculated from the zero coupon rate (z<sub>n</sub>)
 
CumDF<sub>n</sub> = the total of the discount factors for maturities 1 to 'n' periods maturity, again calculated from the zero coupon rates (z<sub>1</sub> to z<sub>n</sub>)
 
 
''Applying the formula:''
 
p<sub>0-2</sub> = (1 - DF<sub>2</sub>) / CumDF<sub>2</sub>
 
p<sub>0-2</sub> = (1 - 1.029951<sup>-2</sup>) / (1.02<sup>-1</sup> + 1.029951<sup>-2</sup>)
 
= 0.029803 (= 2.9803% per period)
 
 
This is the theoretical fair (no-arbitrage) market price for the par instrument.
 
It is the calculated rate of interest payable on a two-year investment on par rate terms. This means that 2.9803% interest will be paid on the amount of the original investment, at Times 1 and 2 periods. In addition, the original investment will be repaid at Time 2.
 
 
In theory, such an investment - again of £1m - should produce the same terminal cash flow. Let's see.
 
 
Cash flows from the two period par instrument, paying periodic interest of 2.9803% per period, assuming an initial investment of £1m:
 
 
Interest coupon at Time 1 period = £1m x 0.029803 = £<u>0.029803</u>m
 
Principal + interest at Time 2 periods = £1m + 0.029803m = £'''1.029803'''m
 
 
The coupon receivable at Time 1 period is reinvested at the pre-agreed forward rate of 4% (0.04) for the maturity 1-2 periods.
 
So the Time 2 proceeds from the reinvested coupon received at Time 1 are:
 
£0.029803 x 1.04
 
= £'''0.030995'''m at Time 2
 
 
The total terminal value at Time 2 periods is:
 
0.030995 + 1.029803
 
= £'''1.0608'''m (as before)
 
 
The par rate we have calculated is indeed consistent with the no-arbitrage pricing relationship.




== See also ==
== See also ==
* [[Forward yield]]
* [[Accruals accounting]]
* [[Yield curve]]
* [[Accumulated depreciation]]
* [[Zero coupon yield]]
* [[Amortisation]]
* [[Par yield]]
* [[Appreciation]]
* [[Forward rate agreement]]
* [[Assets]]
* [[Periodic yield]]
* [[Capital allowances]]
* [[Discount factor]]
* [[Capitalisation]]
* [[Coupon]]
* [[Carry trade]]
* [[Flat yield curve]]
*[[Cash flow]]
* [[Rising yield curve]]
* [[CertICM]]
* [[Falling yield curve]]
* [[Cost]]
* [[Positive yield curve]]
* [[EBITDA]]
* [[Negative yield curve]]
* [[Impairment]]
* [[Converting from zero coupon rates]]
* [[Intangible assets]]
* [[Converting from par rates]]
* [[International Fisher Effect]]
 
* [[Net book value]]
* [[Property, plant and equipment]]
* [[Provision]]
* [[Reducing balance]]
* [[Straight line]]
* [[Sum of the digits]]
* [[Tangible asset]]
* [[Tax depreciation]]
* [[Writing down allowance]]


===Other resources===
[[Category:Accounting,_tax_and_regulation]]
[[Media:2013_09_Sept_-_Simple_solutions.pdf| The Treasurer students, Simple solutions]]
[[Category:Corporate_finance]]

Revision as of 22:53, 18 July 2022

1. Financial reporting - accounting practices.

Accounting depreciation spreads the cost of a long-term tangible asset over its total life.

The depreciation accounting charge reflects:

  • the estimated periodic cost to a business
  • of a physical capital asset
  • spread over its estimated useful economic life.


Accounting depreciation seeks to ensure that the total accounting cost of a capitalised asset is appropriately spread and matched to the economic benefits of using the asset.

Accounting depreciation is applying the accruals accounting principle to spread the total cost of tangible long term assets over their expected useful life.


Methods of spreading the total accounting cost include Straight line, Reducing balance and Sum of the digits.

Financial reporting standards generally permit the use of any systematic basis of allocating the total cost over the useful life of the asset.


It's important to be clear about the distinction between the:

  • Depreciation charge for the period, reflected in the income statement; and
  • Cumulative depreciation provision at the end of the period, reflected in the balance sheet.


The depreciation charge is an in-period accounting expense, charged against profits for the period.

The cumulative provision for depreciation is a liability in the balance sheet. It's offset against the cost of the assets, to calculate their accounting net book value.


Some accounting jurisdictions use the term amortisation both for this aspect of accounting both for tangible and intangible assets.


2. Foreign exchange.

A decrease in the value of a currency.


3. Other contexts.

More generally, any decrease in the value of an asset resulting from the passing of time.


See also

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