Spot Rates and Forward Rates

If the time value of money (TVM) concepts are clear to you, this will be a nice extension. In fact, start back there at TVM. It's always fun. If you calculated the present value of one currency unit to be received in three years, assuming a discount rate of 5%, you would come up with a present value of $$\displaystyle PV = \frac{1}{(1 + 0.05)^3} = 0.863838 $$. Ah, the memories.

Now just rename a couple of things, and you can use this same idea with spot rates and forward rates. The spot rate is denoted $$z_N$$, and that's a yield to maturity to time $$N$$. Once you use that rate to discount a currency unit back to the present, you end up with a __discount factor__, denoted $$DF_N$$. What would $$DF_N$$ represent to you?

No. The future here is just 1.

Not quite. The discount rate is $$z_N$$, not $$DF_N$$.

That's right. $$\displaystyle DF_N = \frac{1}{(1 + z_N)^N} $$ It's really nothing new, just a notational change here. In fact, you may have even seen discount factor tables that combine various discount rates and times for general TVM use. This is really the same thing.

From this point, you can obtain the spot rate given for each time period. This is called the __discount function__, and it can be plotted to make the __spot yield curve__. So these two things are essentially the same. The spot yield curve shows risk-free yields. What sort of bond do you think would work best?

No. This sort of bond wouldn't be risk-free.

No. A bond trading at par would have cash flows that could make the calculation a little tougher.

Exactly! More specifically, an option-free and default-risk-free zero-coupon bond.

Once you have that, you're ready to take a step forward. Or many steps. A __forward rate__ can be determined today for some time period in the future. For example, perhaps you're interested in a one-year loan that starts two years from today. That would be a forward rate of $$\displaystyle f_{2,1}$$ that's priced at $$\displaystyle F_{2,1}$$. The general notation for a forward rate is $$\displaystyle f_{A,B-A}$$. For the forward price, it is $$\displaystyle F_{A,B-A}$$. Looking at the example, what do you think $$A$$ means?

Yes! A forward rate is one that starts at $$A$$ and then goes for $$B-A$$, finally ending from today at time $$B$$. If you have the spot rate curve, there's really only one rate that will work for this forward rate without allowing for arbitrage. So with the spot rate curve, you can build the forward rate curve.

No. That's actually $$B$$.

No, the length of the loan is $$B-A$$.

Finishing the example will certainly help at this point. So maybe the two-year spot rate is 4%, and the three-year spot rate is 5%. That means you can invest for two years at 4% per year, or three years at 5% per year. The forward rate is $$\displaystyle f_{2,1}$$. What do you think has to be true about the forward rate?

No. It couldn't be in this range. Think of what rate you would need to grow the two-year investment into the three-year investment with one year of growth.

Absolutely! It has to be, since the lower 4% return has to "catch up" with the 5% return through this forward rate. To step away from zero-coupon bonds for a second, you could invest one currency unit at each rate, and you'd have $$\displaystyle FV_{2yr} = 1(1 + 0.04)^2 = 1.0816 $$ $$\displaystyle FV_{3yr} = 1(1 + 0.05)^3 = 1.1576 $$. The rate that connects these two would be 7.0289%. $$\displaystyle \frac{1.1576}{1.0816} - 1 = 0.070289 $$

No. That's too low. Think of what rate you would need to grow the two-year investment into the three-year investment with one year of growth.

And here's how you can see that in action with spot rates and zero-coupon bond prices, using the __forward pricing model__. $$\displaystyle DF_B = DF_A \times F_{A,B-A} $$ Using the example, this states that the price of a three-year zero-coupon bond equals the price of a two-year zero-coupon bond times the forward price. Sure, it has to since everything needs to stay arbitrage-free. This means $$\displaystyle \frac{1}{(1 + 0.05)^3} = \frac{1}{(1 + 0.04)^2} \times F_{2,1} $$. So the forward price is found after some simplification as $$\displaystyle F_{2,1} = \frac{0.863838}{0.924556} = 0.934327 $$.

A zero-coupon bond priced at 0.934327 suggests a yield of $$\displaystyle \frac{1}{0.934327} - 1 = 0.070289 $$. So there's the forward rate again. It all fits together. How might you summarize this relationship?

No. In this example, you'll notice that one spot rate was divided by another to get the result desired.

You got it. That's exactly what just happened here, and that's how the forward rate model works. The logic is there as well. The discount factors in a ratio leaves the "gap" to be filled by the perfect forward contract price. Again, it's more time value of money, but even a relatively simple example does test your understanding.

No. The forward rate isn't found directly here; the forward price is the initial output.

To summarize: [[summary]]

The future value

The discount rate

The present value

A zero-coupon bond

A bond trading at par

A high-yield corporate bond

It's when the loan ends

It's when the loan starts

It's the length of the loan

It must be less than 4%

It must be greater than 5%

It must be between 4% and 5%

A discount factor is divided by another to create a forward rate

A discount factor is divided by another to create a forward price

A discount factor is multiplied by another to create a forward price

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