Reading 48 - Forwards

[kshitijharve]

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

[S2000magician]

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 p_resent 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.

[kshitijharve]

Thanks certainly made a lot of things clear!

[S2000magician]

@kshitijharve

Thanks

You’re quite welcome.

@kshitijharve

certainly made a lot of things clear!

I’d hoped that it would.

[S2000magician]

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,

V__t_ = PV(S__t_ − PV(FT))

= PV(S__t_ − FT × (((1 + r_DC)^{−(T − t)}) / (1 + r_FC)^{−(T − t)}))

= PV(S__t_ − FT × (((1 + r_FC)^{T − t}) / (1 + r_DC)^{T − t})))

= (S__t_ − FT × (((1 + r_FC)^{T − t}) / ((1 + r_DC)^{T − t})))) / (1 + r_FC)^{T − t}

= [S__t_ / (1 + _r__FC)^{T − t}] − [FT × ((1 + r_FC)^{T − t} / (1 + r_DC)^{T − t})) / (1 + r_FC)^{T − t}]

= [S__t_ / (1 + _r__FC)^{T − t}] − [FT / (1 + r_DC)^{T − t}]

Whew!

[kshitijharve]

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?

[S2000magician]

@kshitijharve

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.

[Michael_V]

@S2000magician

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,

V__t_ = PV(S__t_ − PV(FT))

= PV(S__t_ − FT × (((1 + r_DC)^{−(T − t)}) / (1 + r_FC)^{−(T − t)}))

= PV(S__t_ − FT × (((1 + r_FC)^{T − t}) / (1 + r_DC)^{T − t})))

= (S__t_ − FT × (((1 + r_FC)^{T − t}) / ((1 + r_DC)^{T − t})))) / (1 + r_FC)^{T − t}

= [S__t_ / (1 + _r__FC)^{T − t}] − [FT × ((1 + r_FC)^{T − t} / (1 + r_DC)^{T − t})) / (1 + r_FC)^{T − t}]

= [S__t_ / (1 + _r__FC)^{T − t}] − [FT / (1 + r_DC)^{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.

[S2000magician]

@Michael_V

@S2000magician

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,

V__t_ = PV(S__t_ − PV(FT))

= PV(S__t_ − FT × (((1 + r_DC)^{−(T − t)}) / (1 + r_FC)^{−(T − t)}))

= PV(S__t_ − FT × (((1 + r_FC)^{T − t}) / (1 + r_DC)^{T − t})))

= (S__t_ − FT × (((1 + r_FC)^{T − t}) / ((1 + r_DC)^{T − t})))) / (1 + r_FC)^{T − t}

= [S__t_ / (1 + _r__FC)^{T − t}] − [FT × ((1 + r_FC)^{T − t} / (1 + r_DC)^{T − t})) / (1 + r_FC)^{T − t}]

= [S__t_ / (1 + _r__FC)^{T − t}] − [FT / (1 + r_DC)^{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 + r_DC)^{T − t} / (1 + r_FC)^{T – t}

which is the same as multiplying by:

(1 + r_DC)^{−(T − t)} / (1 + r_FC)^{−(T − t)}

[Pyng]

@S2000magician

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,

V__t_ = PV(S__t_ − PV(FT))

= PV(S__t_ − FT × (((1 + r_DC)^{−(T − t)}) / (1 + r_FC)^{−(T − t)}))

= PV(S__t_ − FT × (((1 + r_FC)^{T − t}) / (1 + r_DC)^{T − t})))

= (S__t_ − FT × (((1 + r_FC)^{T − t}) / ((1 + r_DC)^{T − t})))) / (1 + r_FC)^{T − t}

= [S__t_ / (1 + _r__FC)^{T − t}] − [FT × ((1 + r_FC)^{T − t} / (1 + r_DC)^{T − t})) / (1 + r_FC)^{T − t}]

= [S__t_ / (1 + _r__FC)^{T − t}] − [FT / (1 + r_DC)^{T − t}]

Whew!

Such a Masterpiece Sir!!

[breadmaker]