Replicating portfolio Equation, Risk Free Return = Risky Asset + Derivative

CFA Level I
#1

I am extremely confuse in the replicating portfolio equations which states that:

Risky Asset + Derivative = Risk Free Return

Risky Asset - Risk Free Return = - Derivative

Derivative - Risk Free Return = - Risky Asset

in first equation why derivative is long. My ques is if you want to hedge your position than in 1st Eq. you LONG asset SHORT derivative of that asset = Risk free asset. How is this possible that you long asset and you long derivative of that asset too… ?

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#2

The signs here are misleading, indeed. Just ignore them. From the logic:

In the first equation you are long the asset and short the derivative, hence at expiration under any scenarios you’ll only earn the risk-free rate (in any absence of transaction costs).

Regards, Oscar

#3

bro thanks for your reply.

I am little worry about what to follow from exam perpective since the same thing is mentioned in curriculum. If I follow the curriculum it creates no logic and complete confusion and if we make our own equation i.e. SHORT derivative then may be in exam we make mistakes by doing so…

Extreme confusion… I would request seniors to pls commnt on this.

#4

If you understand the “sign convention defaults” behind the first equation it all makes sense.

+ Risk Free Rate = Investing/Lending @ risk-free rate = Buy/Long risk-free bonds

(- means borrowing @ risk free rate or selling risk-free bonds)

+ Risky Asset = Buy/Long Risky Asset

(- means Short sell Risky Asset)

+ Derivative on Risky Asset = Short Derivative on Risky Asset

( - means Long Derivative on Risky Asset)

So Equation 1 basically says: Risky Asset + Derivative = Risk Free Return

=>[Buy/Long Risky Asset , Short Derivative] ‘equals’ [Buy/Long risk-free bonds (Invest/Lend @ risk-free rate)]

Equation 2: Risky Asset - Risk Free Return = - Derivative

=> [Buy/Long Risky Asset , Borrow @ risk-free rate] ‘equals’ [Long Derivative]

Equation 3 : Derivative - Risk Free Return = - Risky Asset

=> [Short Derivative, Borrow @ risk-free rate] ‘equals’ [Short sell Risky Asset]

Alternatively, a much simpler way to remember is :

Risk Free Return = Risky Asset - Derivative where + sign indicates Long and - sign indicates Short

(So for example, - Risk Free Return = Short risk-free bonds = Borrow @ risk free rate and -Derivative is actually defined as Short Derivative)

I suppose this is the convention you were alluding to.

At the end of the day, all this is just one equation such that no arbitrage exists.

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