# Spot and Forward rate

Per the CFAI book
“A spot interest rate (in this reading, “spot rate”) is a rate of interest on security that makes a single payment at a future point in time.
The forward rate is the rate of interest set today for single-payment security to be issued at a future date”

Both of them are future payments so, what is the difference between both in simple terms?

that was a really poor explanation for the spot and forward rate.
the fact that CFAI does such a poor job explaning this fundamental theory is disappointing.

below is a paraphrase from a financial math book i used:

spot rate of interest is the annualized effective rate of interest from time 0 to time t, where time 0 is the present time. you can have a spot rate of interest over a year (from time 0 to time 1), over 5 years (from time 0 to time 5), etc. The spot rate is determined at time 0.

Forward rate of interest is the rate of interest for a single period in the future (from time t - 1 to time t). For example, the rate of interest from time 0 to time 1, from 1 to time 2, from time 2 to time 3, from time 3 to time 4, etc. this rate of interest is determined at time 0. Unlike spot rates, forward rates only apply for a single period.

also note that when t = 1, spot rate and forward rate are identical.

for example, you could have a spot rate of interest over 2 years to equal 4%, while the forward rate from time 0 to time 1 is 5% and the forward rate from time 1 to time 2 equal 3% (the value 4% for the spot rate is only approximate. see below)

one important relation between spot rate of interest and forward rate of interest is the following:

let F(T - 1, T) be the forward rate of interest from time T - 1 to T.
and S(0, T) be the spot rate of interest, that is, the annualized effective rate of interest over T periods. then:
(1 + S(0, T) )^T = (1 + F(0, 1) ) x (1 + F(1, 2) ) x … x (1 + F(T - 1, T) )
also,
1 + F(T - 1 , T) = {(1 + S(0, T) ) ^T } / { (1 + S(0, T - 1)^ (T - 1) }

maybe it is easier to see the difference with a time diagram (that is how the book showed the difference)

draw a time diagram and draw a line on top from time 0 to time T. that is your spot rate. now draw a line from time 0 to time 1, and another line from time 1 to time 2, etc. these are your forward rates.

1 Like

A spot rate discounts a single future payment back to today.

A forward rate discounts a single future payment back to a nearer time, but not necessarily today.

All spot rates are, in fact, forward rates.

Not all forward rates are spot rates. For example, there is a forward rate that will discount a payment three years from today back to one year from today (known as the 2-year forward rate starting one year from today); that one’s not a spot rate.

it is worth nothg that on my first post, i was referring to a single-period forward rate (usually just called forward rate), while magician above is talking about multi-period forward rate. the logic for multi-period forward rate is the same as for a single-period forward
(instead of drawing a line for 1 period, draw line for multiple periods. for example, draw a line from time 2 to time 5 - that is your multi-period forward rate from time 2 to time 5. again, this rate is determined at time 0)
the equations relating the multi-period forward and spot rate is slightly different from the ones i gave above.

Actually, what I wrote applies to single-period and multi-period forward rates. The example I gave is a multi-period forward rate.

The 1-year spot rate is a one-period forward rate (specifically, the 1-year forward rate starting today).

you’re right of course. i was referring to your example, but was lazy to be specific in my previous post.