Let's explore the relationship between par, spot, and forward rates.
In the examples below, we will assume annual coupon payment frequency to make things easier. But same principles apply to semiannual coupons, we would just have to do everything in terms of semiannual periods rather than years.
The par rates are yields to maturity on bonds that are priced at par value. Being yields to maturity, they apply the same rate to all the cash flows of the bond, represented here in the graphic.
In the example to the right, if the par rates are 1,2, and 3 percent, for years 1 to 3 respectively, that means that a 1% coupon 1 year bond is priced at par, because the appropriate discount rate is 1%,
a 2% coupon 2 year bond is priced at par, because the appropriate discount rate is 2%,
and a 3% coupon 3 year bond is priced at par for the same reason.
As you recall from principles of finance, when the discount rate is equal to the coupon rate, the bond is priced at par.
Based on the information contained in the par curve, we can calculate the spot rates. Spot rates are rates that apply only to specific maturity cash flows. So 1 year spot only applies to cash flows we expect to receive in 1 year, 2 year spot rate applies only to cash flows we expect to receive in 2 years, and 3 year spot rate applies only to cash flows we expect to receive in 3 years. This is represented in the graphic here, where each rate only applies to a particular cash flow.
So to find the price of a bond using spot rates, each expected cash flow is discounted by its respective spot rate. Based on this information, we can compute what the spot rates have to be - by using coupon bonds priced at par, and making sure they are still priced at par if we use spot rates, we can find the correct spot rates.
In the example to the right, the 1 year spot rate is easy to determine since it is equal to the 1 year par rate. Since all the cash flows for a 1 year bond happen at time 1, the par discount rate is the same as the 1 year spot rate.
To find year two spot rate, the setup is that a 2 year par bond should be priced at par using spot rates. The price of it is the present value of all expected cash flows. The first coupon that we expect to receive in 1 year we discount using the 1 year spot rate, which we already know is 1%. The second cash flow which is repayment of principal plus the 2% coupon, we discount using the unknown 2 year spot rate. Since we know everything except for the 2 year spot rate, we can rearrange the equation and solve for the 2 year rate that makes this true. The elementary algebra is left as an exercise to the student. :)
Then once we know the 1 and 2 year spot rates, we do the same thing to find the year 3 spot rate, by setting up the price of 3 year par bond to be equal to 100 and using the spot rates that we know so far. Again, we solve for the only unknown.
Finally, the 1 year forward rates are rates that apply to only particular 1 year time periods. The 1 year rate that starts right now at time 0 is named the 0y1y rate, and we already know that is 1%.
To find the 1y1y forward rate, that is the 1 year rate that starts at the end of year 1 and beginning of year 2, we find using the no-arbitrage argument. The idea is that someone who invests right now for 2 years and earns the 2 year annual spot rate, should make exactly the same amount of money as someone who invests right now for one year, and earns the 1 year spot rate, and also locks in the 1y1y forward rate for the second year in the forward market, to reinvest the proceeds at the end of the year at that forward rate.
Since both are risk-free investments, they should generate the same total return.
In the equilibrium condition, on the right we have the future value of a dollar invested right now at 2 year spot rate, and on the left we have the future value of a dollar invested right now at the 1 year spot rate, and reinvested at the locked in forward rate for another year.
Again, we solve for the unknown 1y1y rate using elementary algebra, which is also left as an exercise to the student.
For the 2y1y rate, a similar equilibrium condition applies. We can choose to either invest for 3 years right now at the 3 year spot rate, or invest for 2 years right now at the 2 year spot rate, and also lock in the 2y1y rate in the forward market for reinvesting after 2 years are out. We can solve for the unknown 2y1y forward rate as before.
If we did everything right, then the spot curve and the forward curve should both produce the same bond prices. Let's quickly double check using a 3 year zero coupon bond.
Using the spot rates, we just discount the face value for 3 years using the 3 year spot rate, and find the price of 91.4 or so.
Using the forward rates, we discount the face value for 3 years using each of the forward rates for 1 year, and find exactly the same price, so things are looking good.
You can also double check by finding the price of the 3 year par bond using the spot and forward rates, and if you did everything right, both methods should produce a price of exactly 100 and price the bond at par.
I hope this was helpful - if you have questions, the discussion boards are waiting for you!