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Reading 48 - Forwards

hi guys,

i didnt really understand why in the formula for valuing a currency forward they divide the spot exchange rate (St) by the foreign interest rate?

Can someone please explain the logic behind that…schweser just states the formula. doesnt explain it.

Thanks

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I had promised myself that I wouldn’t do this, but  as there is a specific question on this, I will renege.

Here’s a write-up I did on this very question:

[link removed by admin]

(Note to mods: if this reference is inappropriate, please remove it.)

The synopsis is that there are two discounts happening in a currency forward.  The first is that the exchange rate is discounted back to today using interest rate parity.  The second is that the notional amount is discounted back to today because if the forward were settled today it wouldn’t be for the notional amount, but for the present value of the notional amount.

Not only does Schweser not explain this formula, nobody does.  I had to ponder it for quite a while before I figured out why it was correct.

Simplify the complicated side; don't complify the simplicated side.

Financial Exam Help 123: The place to get help for the CFA® exams
http://financialexamhelp123.com/

Thanks :) certainly made a lot of things clear!

k****ijharve wrote:
Thanks :)

You’re quite welcome.

k****ijharve wrote:
certainly made a lot of things clear!

I’d hoped that it would.

Simplify the complicated side; don't complify the simplicated side.

Financial Exam Help 123: The place to get help for the CFA® exams
http://financialexamhelp123.com/

OK, the link got removed (no grousing: I told the admins about it and agree fully with their reasoning), so I’ll try to write something intelligible here.  There’s no equation editor, so you’ll have to be patient with the notation.

  • Vt = value of the currency forward (to the DC payer / FC receiver) at time t
  • T = expiration of the forward contract
  • St = spot exchange rate at time t (in DC/FC)
  • FT = forward exchange rate at time T (in DC/FC)
  • rDC = domestic risk-free rate
  • rFC = foreign risk-free rate

For all other forwards, we have:

Vt = St – PV(FT)

The subtlety of currency forwards (as I mentioned in post 2, above) is that if we settle a currency forward early, we don’t settle it for the notional amount of the foreign currency (which is what the DC payer is buying); we settle it for the present value of the notional amount of the foreign currency.  So, for a currency forward, the formula is:

Vt = PV(St – PV(FT))

We get PV(FT) by discounting using interest rate parity, and we get PV(St – PV(FT)) by discounting by discounting at the foreign risk-free rate.  So,

Vt = PV(St − PV(FT))

= PV(St − FT × (((1 + rDC)−(T − t)) / (1 + rFC)−(T − t)))

= PV(St − FT × (((1 + rFC)T − t) / (1 + rDC)T − t)))

= (St − FT × (((1 + rFC)T − t) / ((1 + rDC)T − t)))) / (1 + rFC)T − t

= [St / (1 + rFC)T − t] − [FT × ((1 + rFC)T − t / (1 + rDC)T − t)) / (1 + rFC)T − t]

= [St / (1 + rFC)T − t] − [FT / (1 + rDC)T − t]

Whew!

Simplify the complicated side; don't complify the simplicated side.

Financial Exam Help 123: The place to get help for the CFA® exams
http://financialexamhelp123.com/

just one small query…if we settle a forward contract prior to expiration…we get the PV of the notional because of the implicit assumption that currency does not lie idle and will be earning a risk free rate? or is there another reason behind it?

k****ijharve wrote:
just one small query…if we settle a forward contract prior to expiration…we get the PV of the notional because of the implicit assumption that currency does not lie idle and will be earning a risk free rate? or is there another reason behind it?

That’s exactly the reason.

Simplify the complicated side; don't complify the simplicated side.

Financial Exam Help 123: The place to get help for the CFA® exams
http://financialexamhelp123.com/

S2000magician wrote:

OK, the link got removed (no grousing: I told the admins about it and agree fully with their reasoning), so I’ll try to write something intelligible here.  There’s no equation editor, so you’ll have to be patient with the notation.

  • Vt = value of the currency forward (to the DC payer / FC receiver) at time t
  • T = expiration of the forward contract
  • St = spot exchange rate at time t (in DC/FC)
  • FT = forward exchange rate at time T (in DC/FC)
  • rDC = domestic risk-free rate
  • rFC = foreign risk-free rate

For all other forwards, we have:

Vt = St – PV(FT)

The subtlety of currency forwards (as I mentioned in post 2, above) is that if we settle a currency forward early, we don’t settle it for the notional amount of the foreign currency (which is what the DC payer is buying); we settle it for the present value of the notional amount of the foreign currency.  So, for a currency forward, the formula is:

Vt = PV(St – PV(FT))

We get PV(FT) by discounting using interest rate parity, and we get PV(St – PV(FT)) by discounting by discounting at the foreign risk-free rate.  So,

Vt = PV(St − PV(FT))

= PV(St − FT × (((1 + rDC)−(T − t)) / (1 + rFC)−(T − t)))

= PV(St − FT × (((1 + rFC)T − t) / (1 + rDC)T − t)))

= (St − FT × (((1 + rFC)T − t) / ((1 + rDC)T − t)))) / (1 + rFC)T − t

= [St / (1 + rFC)T − t] − [FT × ((1 + rFC)T − t / (1 + rDC)T − t)) / (1 + rFC)T − t]

= [St / (1 + rFC)T − t] − [FT / (1 + rDC)T − t]

Whew!

S2000 can you explain why the exponent is negative in the first line of your derivation? This is the only thing that is tripping me up.

Michael V wrote:
S2000magician wrote:
OK, the link got removed (no grousing: I told the admins about it and agree fully with their reasoning), so I’ll try to write something intelligible here.  There’s no equation editor, so you’ll have to be patient with the notation.
  • Vt = value of the currency forward (to the DC payer / FC receiver) at time t
  • T = expiration of the forward contract
  • St = spot exchange rate at time t (in DC/FC)
  • FT = forward exchange rate at time T (in DC/FC)
  • rDC = domestic risk-free rate
  • rFC = foreign risk-free rate

For all other forwards, we have:

Vt = St – PV(FT)

The subtlety of currency forwards (as I mentioned in post 2, above) is that if we settle a currency forward early, we don’t settle it for the notional amount of the foreign currency (which is what the DC payer is buying); we settle it for the present value of the notional amount of the foreign currency.  So, for a currency forward, the formula is:

Vt = PV(St – PV(FT))

We get PV(FT) by discounting using interest rate parity, and we get PV(St – PV(FT)) by discounting by discounting at the foreign risk-free rate.  So,

Vt = PV(St − PV(FT))

= PV(St − FT × (((1 + rDC)−(T − t)) / (1 + rFC)−(T − t)))

= PV(St − FT × (((1 + rFC)T − t) / (1 + rDC)T − t)))

= (St − FT × (((1 + rFC)T − t) / ((1 + rDC)T − t)))) / (1 + rFC)T − t

= [St / (1 + rFC)T − t] − [FT × ((1 + rFC)T − t / (1 + rDC)T − t)) / (1 + rFC)T − t]

= [St / (1 + rFC)T − t] − [FT / (1 + rDC)T − t]

Whew!

S2000 can you explain why the exponent is negative in the first line of your derivation? This is the only thing that is tripping me up.

It’s negative because we’re discounting FT from the maturity date (time T) back to today (time t).  We’re dividing by:

(1 + rDC)T − t / (1 + rFC)T – t

which is the same as multiplying by:

(1 + rDC)−(T − t) / (1 + rFC)−(T − t)

Simplify the complicated side; don't complify the simplicated side.

Financial Exam Help 123: The place to get help for the CFA® exams
http://financialexamhelp123.com/

S2000magician wrote:

OK, the link got removed (no grousing: I told the admins about it and agree fully with their reasoning), so I’ll try to write something intelligible here.  There’s no equation editor, so you’ll have to be patient with the notation.

  • Vt = value of the currency forward (to the DC payer / FC receiver) at time t
  • T = expiration of the forward contract
  • St = spot exchange rate at time t (in DC/FC)
  • FT = forward exchange rate at time T (in DC/FC)
  • rDC = domestic risk-free rate
  • rFC = foreign risk-free rate

For all other forwards, we have:

Vt = St – PV(FT)

The subtlety of currency forwards (as I mentioned in post 2, above) is that if we settle a currency forward early, we don’t settle it for the notional amount of the foreign currency (which is what the DC payer is buying); we settle it for the present value of the notional amount of the foreign currency.  So, for a currency forward, the formula is:

Vt = PV(St – PV(FT))

We get PV(FT) by discounting using interest rate parity, and we get PV(St – PV(FT)) by discounting by discounting at the foreign risk-free rate.  So,

Vt = PV(St − PV(FT))

= PV(St − FT × (((1 + rDC)−(T − t)) / (1 + rFC)−(T − t)))

= PV(St − FT × (((1 + rFC)T − t) / (1 + rDC)T − t)))

= (St − FT × (((1 + rFC)T − t) / ((1 + rDC)T − t)))) / (1 + rFC)T − t

= [St / (1 + rFC)T − t] − [FT × ((1 + rFC)T − t / (1 + rDC)T − t)) / (1 + rFC)T − t]

= [St / (1 + rFC)T − t] − [FT / (1 + rDC)T − t]

Whew!


Such a Masterpiece Sir!! 

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