# Spot and Foward Rates

Being challenged by this material under Fixed Income. Has anyone found any sites that do a good job of explaining the foward and spot rate conversion material?

Thanks

Elan has a lot of their Level I fixed income material on YouTube. I remember I really liked their explanation of this stuff.

It’s not particularly difficult, especially if you draw a timeline.

Suppose that you have a payment due in 3 years, and you want to know the present value of that payment.

You can discount it in one step from year 3 to today, using the 3-year spot rate: s3; divide by (1 + s3)³.

You could also discount it from year 3 to year 2 (using the 1-year forward rate starting 2 years from now: 1f2), then discount it from year 2 to today (using the 2-year spot rate: s2); divide by (1 + 1f2)(1 + s2)².

You could also discount it from year 3 to year 1 (using the 2-year forward rate starting 1 year from now: 2f1), then discount it from year 1 to today (using the 1-year spot rate: s1); divide by (1 + 2f1)²(1 + s1).

The key idea is this: no matter how you discount it, you have to get the same present value; if you don’t, there’s an arbitrage opportunity.

Thus,

(1 + s3)³ = (1 + 1f2)(1 + s2)²

If they give you s2 and s3, you can solve for 1f2.

Also,

(1 + s3)³ = (1 + 2f1)²(1 + s1)

If they give you s1 and s3, you can solve for 2f1.

If you draw a timeline and write the spot rates and forward rates for the different time periods, you’ll see how easy this is. Seriously, if you draw it out, it’s not all that hard.

thanks!

My pleasure.

What is the real difference between YTM, spot rate and forward? Given all three which is a more accurate discounting factor to use?

A YTM is a single discount rate that is used to discount all of the cash flows on a bond, discounting them all back to today. The yield curve that comprises YTMs for all maturities is called the par curve.

A spot rate is a discount rate for a single cash flow at a given maturity, discounting it back to today. Each cash flow for a bond is discounted at the spot rate appropriate for the maturity (time to receipt) of that cash flow. The yield curve that comprises spot rates for all maturities is called the spot curve; it is derived from the par curve by bootstrapping.

A forward rate is a discount rate for a single cash flow at a given maturity, discounting it back to an earlier time (not necessarily today). There are forward rates for any maturity back to any other time between today and that maturity. The yield curve that comprises single-period forward rates for all maturities is called the forward curve; it is derived from the spot curve.

They are all equally accurate; whether you use one or another depends on what you’re trying to do, as described above.

The spot rate is the YTM on a zero coupon bond.

The difference between forward rates and spot rates is that spot rates start at time = 0 whereas a forward rate can start at any time period.

So a ‘2 year spot rate of 5%’ is the same thing as a ‘2 year forward rate, 0 years from today’.

Elan has a really good video on it.