Internal rate of return and Put-call parity theory: Difference between pages

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(IRR).  
Put-call parity theory links put and call option values via ‘no arbitrage’ market pricing assumptions and the related:
#underlying asset price
#option strike price
#time to maturity and
#theoretically risk-free rate of return.




=== Definition of IRR ===
So for example if the put option value, underlying asset price, strike price, time to maturity, and risk-free rate of return are known, then the call option value can be calculated using the put-call parity relationship:


The internal rate of return of a set of cash flows is the [[cost of capital]] which, when applied to discount all of the cash flows (including any initial investment outflow at Time 0) results in a [[net present value]] (NPV) of NIL.
Underlying asset price + Put value ''less'' Call value = Present Value of option strike price


For an investor, the IRR of an investment proposal therefore represents their expected rate of [[return]] on their investment in the project.
Call value = Underlying asset price + Put value ''less'' Present Value of option strike price


In the special case where the strike price of the options is equal to the forward price of the underlying asset, the Put value and the Call value are exactly equal.


'''Example 1'''


A project requires an investment today of $100m, with $110m being receivable one year from now.
== Theoretically risk-free portfolios ==
The no-arbitrage pricing relationship is based on the theory that combinations of market assets and liabilities with the same terminal cash flows, must also have the same present values (i.e. the same theoretical current market prices).


The IRR of this project is 10%, because that is the cost of capital which results in an NPV of $0, as follows:
For example both the left side and the right side of the put-call parity formula represent portfolios with the same terminal value:


Underlying asset + Put ''less'' Call = Present Value of option strike price


[[PV]] of Time 0 outflow $100m


= $(100m)
The left side portfolio is built by buying the underlying asset, buying a put option, and selling a call option with the same strike price.


The theoretically risk free terminal value of this portfolio is the equal strike price of the two options.


PV of Time 1 inflow $110m
The present value of this left side portfolio is the present value of the strike price.


= $110m x 1.10<sup>-1</sup>


= $100m
The right side portfolio is a deposit of cash.


This cash portfolio also produces a theoretically risk free terminal value, equal to the strike price of the options.


NPV = - $100m + $100m
The current market pricing of these two portfolios must in theory be exactly the same.


= '''$0'''.
If this relationship did not hold, there would be an arbitrage opportunity to buy the cheaper portfolio and sell the more expensive one, to earn an immediate risk free profit.


 
Therefore market supply and demand pressures will act to quickly re-establish the no arbitrage pricing relationship, following any temporary pricing mis-alignments.
If the project had been funded by borrowing all the required money at the IRR of 10%, there would have been exactly the right amount of surplus from the project to repay the borrowing and interest, with neither a deficit nor a surplus.
 
This is another way to define the IRR.
 
 
 
=== Determining IRR ===
 
 
Unless the pattern of cash flows is very simple, it is normally only possible to determine IRR by trial and error (iterative) methods.
 
 
'''Example 2'''
 
Using straight line interpolation and the following data:
 
First estimated rate of return 5%, positive NPV = $+4m.
 
Second estimated rate of return 6%, negative NPV = $-4m.
 
The straight-line-interpolated estimated IRR is the mid-point between 5% and 6%.
 
This is '''5.5%'''.
 
 
Using iteration, the straight-line estimation process could then be repeated, using the value of 5.5% to recalculate the NPV, and so on.
 
The IRR function in Excel uses a similar trial and error method.
 
 
 
=== IRR project analysis decision making ===
 
 
Target or required IRRs are set based on the investor's [[weighted average cost of capital]], appropriately adjusted for the risk of the proposal under review.
 
In very simple IRR project analysis the decision rule would be that:
 
(1) All opportunities with above the required IRR should be accepted.
 
(2) All other opportunities should be rejected.
 
 
However this assumes the unlimited availability of further capital with no increase in the cost of capital.
 
 
A more refined decision rule is that:
 
(1) All opportunities with IRRs BELOW the required IRR should still be REJECTED; while
 
(2) All other opportunities remain eligible for further consideration (rather than automatically being accepted).




== See also ==
== See also ==
* [[CertFMM]]
* [[Arbitrage]]
* [[Effective interest rate]]
* [[Interest rate parity]]
* [[Hurdle rate]]
* [[Option]]
* [[Implied rate of interest]]
* [[Parity]]
* [[Interpolation]]
* [[Put option]]
* [[Iteration]]
* [[Risk-free rate of return]]
* [[Linear interpolation]]
* [[Market yield]]
* [[Net present value]]
* [[Present value]]
* [[Shareholder value]]
* [[Weighted average cost of capital]]
* [[Yield to maturity]]

Revision as of 21:06, 5 February 2018

Put-call parity theory links put and call option values via ‘no arbitrage’ market pricing assumptions and the related:

  1. underlying asset price
  2. option strike price
  3. time to maturity and
  4. theoretically risk-free rate of return.


So for example if the put option value, underlying asset price, strike price, time to maturity, and risk-free rate of return are known, then the call option value can be calculated using the put-call parity relationship:

Underlying asset price + Put value less Call value = Present Value of option strike price

Call value = Underlying asset price + Put value less Present Value of option strike price

In the special case where the strike price of the options is equal to the forward price of the underlying asset, the Put value and the Call value are exactly equal.


Theoretically risk-free portfolios

The no-arbitrage pricing relationship is based on the theory that combinations of market assets and liabilities with the same terminal cash flows, must also have the same present values (i.e. the same theoretical current market prices).

For example both the left side and the right side of the put-call parity formula represent portfolios with the same terminal value:

Underlying asset + Put less Call = Present Value of option strike price


The left side portfolio is built by buying the underlying asset, buying a put option, and selling a call option with the same strike price.

The theoretically risk free terminal value of this portfolio is the equal strike price of the two options.

The present value of this left side portfolio is the present value of the strike price.


The right side portfolio is a deposit of cash.

This cash portfolio also produces a theoretically risk free terminal value, equal to the strike price of the options.

The current market pricing of these two portfolios must in theory be exactly the same.

If this relationship did not hold, there would be an arbitrage opportunity to buy the cheaper portfolio and sell the more expensive one, to earn an immediate risk free profit.

Therefore market supply and demand pressures will act to quickly re-establish the no arbitrage pricing relationship, following any temporary pricing mis-alignments.


See also

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