Currency forward valuation at time t

Can someone help me understand how currency forward is valued at time t after initiation?

I am having a hard time to recall how this works from Level II. The formula is:

Value of currency forward at time t = Spot FX rate at time t / (1+Foreign interest rate)^(T-t) - FX Forward rate set when contract initiated / (1+domestic rate)^ (T-t)

using example 8 in reading 34 (page 238), question 1

US investor long a forward contract at $0.90 per euro in 2 years. Interest rate in 6% in the US, 5% in the Europe. After 6 months, the spot rate is $0.862 per euro, interest rates in US and EU not changed, the book says the value of the contract at that time is 0.862/(1.05)^1.5 - $0.90 / (1.06)^1.5

My question is,

based on my understanding, value of forward at any given time t, is the difference between market value of underlying asset at time t, and the PV of forward price. So in the first part of the formula, why the spot fx rate needs to be discounted at foreign interest rate? shouldn’t this just be the spot rate, which is the market price of underlying FX?

Thanks for your help. I must be missing something here…

The short answer is that if you settle a currency forward early, you won’t settle for the nominal value; you’ll settle for the present value of the nominal value. Hence, everything is discounted at the foreign currency risk-free rate.

Got it. Thanks man! Would love to see your notes though (its removed by the forum)

Can you send me a copy?

hey my friend, in order to get this straight I always think about it like this. Let’s say the quote is USD/AUD, or AUDUSD. In this case AUD is the base currency and USD is the quote currency.

Lets say that you are long an AUD$50M 1-yr forward at 1.05. Spot is 1.00 and the US interest rate is 1% and AUD interest rate is 3%.

Now lets mark it to market.

In this case if you took delivery, in 1 year, you would receive 50$M AUD into your bank account and for it, you would pay 50*1.05 = $52.5 USD

Therefore here are the cash flows:

Pay 52.5M USD

Receive 50M AUD

In order to mark this to market, what we need to do exactly (as you already know) is find the expected future cash flow payoff in 1 yr, if we were to take an offsetting position, and then discount this payoff until today.

So, if we were going to take a position at current market rates to offset this forward exactly, what we would need to do is go short AUD$50M at the prevailing forward rate.

So first lets solve for the prevailing forward rate:

Spot * (1+US Rate) / (1+AUD rate) = 1.00*1.01/1.03 = 0.981 = ~0.98

So in order to lock in the position, we would need to sell AUD forward for 1 year at 0.98.

Ok, so lets do it and here is our new portfolio cash flows in 1 year:

Pay 52.5M USD

Receive 50M AUD

Pay 50M AUD

Receive 49M (USD) (50M * 0.98)

So as you can see, we are net 0 AUD (+50-50)

But we are negative -3.5M USD (+49-52.5)

So from these two transactions, I have locked in a cash outflow of $3.5m USD.

Obviously time value of money prevails here and as I don’t need to make this USD cash flow for 1 year, I can discount my loss at the same rate as the cash flow currency - USD

-3.5 / 1.01 = -3.47M

This is the current value of the forward contract.

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