# Evolution of spot rates as predicted by forward rates and one-year period return

Why is it that if spot rates evolve as predicted by forward rates, bonds of all maturities will realize a one-period return equal to the one-period spot rate?

Can someone help me understand the intuition please? Thanks!

It is just maths. No great economic theory.

Get excel out.

Take a 2 year cooupon bond.
Pick a 1 year and 2 year spot rate
Price the bond

Calculate the foward rate implied from the 2 spot rates.
Re-price teh bond with 1 year remaining using this forward rate.

Calculate the return over year 1.

Do you want understanding, or do you want intuition?

Thanks @MikeyF

Does it matter whether the bond pays coupons? I get different results if I assume the bond pays coupons

Without coupons:

With coupons:

Thanks so much

Hey @S2000magician , I guess doing the Excel calculation can give a mechanical understanding. But I’m hoping to understand it from (perhaps) an economic perspective so that it is hopefully more intuitive and easier to recall.

Fair enough.

In a nutshell, when you discount a single payment at time n, you can think of it as discounting it for n periods at the n-period spot rate, or you can think of it as discounting it for n periods using n 1-period forward rates: the first from time n to time n−1, the second from time n−1 to time n−2, and so on.

If spot rates evolve according to the forward curve, then one period from now, the only change to the (compounded) discount rate is that you have eliminated discounting by the 1-period spot rate (which is equal to the 1-period forward rate starting today). Thus, the value is higher by exactly the 1-period spot rate.

In the second caclualtion, with the coupons, when calculating the return for holding the bond for 1 year you have missed off the coupon that is paid at the end of year 1.

Return = (97.1876 + 4)/96.3692 -1 = 5%

This works due to the maths.

All the excel calculations you have done tell us is that if forward rates are a good predictor of future spot rates the period return you will get will equaly what you see in the spot curve now.

But Implied (calcualtd) forward rates are not good predictors of future spot rates.

The world changes and future spot rates will reflect the markest view when we get there.

The forward rates, as calculated, converge to the value because if the diverge too much arbitrage profits can be made.

But many investors/speculators take positions contary to the view that forward rates will predict future spot rates. “Riding the yield curve” makes this assumption.

Oh, OK, I think I get it.

It follows from the Forward Rate Model:

And if we expand it:

Thanks very much, @S2000magician !

Thanks @MikeyF !

This is very helpful insight. And thanks for pointing out that I missed out the coupon.