Spot rates and forward rate

When you buy a four year zero coupon bond , you calculate your purchase price by discounting by the 4 year spot rate. And when you sell that 4 year zero coupon after 2 years, how do you calculate the future price, to use it to arrive on returns?

Do you calculate the price (2 years remaining for that 4 year zero coupon) by discounting with 2 year spot rates or you calculate f(2,2) to arrive at that price?

You use the 2-year spot rate at the time you sell it.

The market doesn’t care about what it thought when you bought it.

Why do we use the 2 year spor rate ? And why not the f (2,2).

f(2,2) would be pretty much irrelevant at t=2 in any case as the 2 year rate beginning in 2 years would be to the period expiring at t=6 i.e. beyond the maturity of your ZCB. Why would you want to discount something with this rate rather than the current spot rate at time t=2?

I think you have misread.

So its a 4 year zero coupon bond

I am supposed to sell it after 2 years. So if I have to calculate my return, I need to derive the buying price ans the selling price.

The discussion is on selling price, which I want to know why can we not calculate that from f(2,2) which is the 2 year rate 2 years from now to discount, as I want to sell after 2 years.

Because in 2 years rates will almost certainly have changed.

As I wrote above, at the time you sell it, the market doesn’t care what it thought when you bought it. It cares only about what it thinks when you sell it.