# Spot rate investment horizon

I'm learning about spot rates from Financial Mathematics for Actuaries and something about the definition is confusing me.

Spot rates are used (in the textbook) to specify the interest rate on a payment made at time 0, for some fixed but arbitrary period into the future. But elsewhere, like on Wikipedia, spot rates are said to apply only to very short term investments, as in those due to mature a few days from now. Assuming the term has an unambiguous meaning in finance, is it that, theoretically, spot rates can be defined for arbitrary periods into the future but in practice that never happens? Or is it something else?

Thanks

• There are plenty of different rates that apply for borrowing money for different time horizons: spot rates ($R_{t,T}$) cover periods $[t,T]$ whereas short rates ($r_t$) cover periods $[t,t+dt]$. Instantaneous forward rates ($f_{t,T}$) cover periods $[T,T+dt]$ and forward rates ($F_{t,T,S}$) cover periods $[T,S]$. There are indeed no limits how big the time horizon can be (in theory). – Kevin May 22 at 16:43
• Thanks, that's what I thought it meant. The spot in spot interest rate refers to the t in the interval [t, T], not the T. A lot of these textbooks shy away from using mathematical notation to define terms. Doing so would make it a lot easier for someone like me to pick this stuff up. My background is in math. – Harry May 23 at 19:46
• You nailed it. Including some more maths often makes things easier than hand waving'' explanations! Check out Brigo and Mercurio's (2006) book on interest rate modelling. It's quite good! – Kevin May 23 at 19:52
• Couldn't agree more. For simple concepts, precise definitions are infinitely more helpful than even intuitive ones. Thanks for all the help and the suggestion. Appreciate it. – Harry May 23 at 23:32

The confusion arises because “spot” is variousLy used in FX, equities and commodities to refer to the immediate/very-short-term price (before any forward adjustments for interest rates, dividends, contract rolls etc).

The same is sort-of the same with interest rates. If the spot 2 year rate is say 10% for a zero-coupon loan, then you would receive 1.21x back in two years time. Or a spot 10 year rate close to 7% would double your capital.

This is consistent because these rates might not be 10%, but be 9% or 11% tomorrow. So it’s an immediate rate to cover a future period. In a sense, it’s today’s average for a future period. Or you could think of it, if you had a variable rate mortgage, as the rate you’d get locking into a fixed rate.

This is obviously very different from saying something like “(short-term) rates will be 10% in 2 years time, or 7% in a decade’s time”. With a granular yield curve, one can infer what these forward rates should be (to prevent arbitrage); and these forward rates can be and are traded (on swap) today.

The most famous of these for economists is the “5y5y inflation breakeven” that the Fed used to talk about, Ie the take the spot 5y breakeven and the spot 10y breakeven, and work out what the breakeven in 5 years time for the next 5 years is (ie for years 5-9 from now).

But these are just derivatives of spot interest rates - what is the rate today covering the next T years. There being obviously multiple spots covering different values of T across the yield curve.

In fact “the yield curve is spot” is probably the simplest and shortest explanation possible :-) Given this, one can then slice-and-dice to deduce “expected” (that is, arb-free) forward interest rates for any period one might be Interested in.

• Okay, this cleared it up for me. Just to be sure, a spot interest rate at t = 0, can extend for any length of time into the future. It means a sum of cash invested today at a spot rate I for 2 years will grow to (1 + I)^2 in 2 years. Like you said, it's an immediate rate to cover a future period. So the "spot" in spot interest rate refers to the time at which the rate was determined, not the time at which an investment will mature. Thanks! – Harry May 23 at 19:43
• correct - there are spot 1m, 2m, 3m... 1y, 2y, 3y, 4y, 5y ... spot 50y rates. Strictly speaking, this rate is not a compound (1+r)^t figure but an IRR figure (ie an income of r% for t years and then capital back); but the two are equivalent in IRR terms. . – demully May 24 at 21:40

there are plenty of different situations where we consider the contract signing date, or the settlement date, depending on the product, see https://www.investopedia.com/terms/s/spot_rate.asp

The rule of thumb is: spot rate means now, at the present time, or at time t=0 by default when considering a fixed income pricing formula for instance

the rate at any time t between 0 and T is not spot rate, but forward rate, and should not be confused