The Par Curve and Bootstrapping

The spot curve is where you are, and the forward curve is where things are going ... sort of. There's also a __par curve__, which is made up of yields on coupon-paying bonds trading at par. Again, these are sovereign bonds so that there can be a default risk-free assumption.

Think about the structure of coupon-paying bonds. How might you look at these bonds and see only zero-coupon bonds?

Right!

No, that's not a good idea. Those cash flows are important pieces of information. Instead, just imagine each cash flow as a little zero-coupon bond.

So if you did that, you might look at annual coupon bonds on the par curve with these yields to maturity. | Maturity | 1 Year | 2 Years | 3 Years | 4 Years | |---|---|---|---|---| | YTM | 3.00% | 4.00% | 5.00% | 6.00% | From this information, you can derive the spot curve through a process called __bootstrapping__. The idea is that there is a single spot rate for each maturity, which will allow these yields to exist arbitrage-free.

To make sense of this, think about a one-year zero-coupon bond that yields 3%, and you can invest any amount of money you would like. There is also a savings account with identical risk where you can choose to place your money, earning 3% interest, as long as you left it there for a full year. How would you compare the yields of these two offers?

Yes! They are. After all, 3% is 3%. So look again at the par curve information given. | Maturity | 1 Year | 2 Years | 3 Years | 4 Years | |---|---|---|---|---| | YTM | 3.00% | 4.00% | 5.00% | 6.00% | If you purchase a two-year bond, you'll get a 4% yield. So it's really not different than imagining that you invest a single currency unit, and that investment gets you 0.04 in Year 1, and then 1.04 in Year 2. Just interest and principal.

No. By purchasing a zero-coupon bond, you earn a 3% yield. That's not worse than the 3% savings account.

No. By purchasing a zero-coupon bond, you earn a 3% yield. That's no better than the 3% savings account.

If that's clear, then bootstrapping from this point will be pretty straightforward. Just set up that exact investment of a currency unit, discounting at the spot rates. The one-year spot rate $$z_1$$ you know, since it is the 3.00% yield on the one-year zero-coupon bond. But $$z_2$$ has to be found with $$\displaystyle 1 = \frac{0.04}{1 + 0.03} + \frac{1 + 0.04}{[1 + z_2]^2} $$. Solve for $$z_2$$, and you have done some bootstrapping. What do you get?

Not quite. Perhaps the numerator of the final term got ignored here. Be careful with your algebra steps.

Correct! A little algebra is required to end up with $$z_2$$, but it can be solved for explicitly if you'd like. $$\displaystyle z_2 = \left[ \frac{1 + 0.04}{1 - \frac{0.04}{1 + 0.03}} \right]^{1/2} - 1 \approx 0.0402 = 4.02 \% $$

No. This could be the result of stopping with a squared term and subtracting 1.

From this point, just continue. The Year 3 spot rate can be found by bootstrapping with the rates $$z_1$$ and $$z_2$$. $$\displaystyle 1 = \frac{0.05}{1 + 0.03} + \frac{0.05}{[1 + 0.0402]^2} + \frac{1 + 0.05}{[1 + z_3]^3} $$ This leads to $$\displaystyle z_3 \approx 5.07 \% $$. And then you can get $$z_4$$ in the same way. $$\displaystyle 1 = \frac{0.06}{1 + 0.03} + \frac{0.06}{[1 + 0.0402]^2} + \frac{0.06}{[1 + 0.0507]^3} + \frac{1 + 0.06}{[1 + z_4]^4} $$ $$ z_4 \approx 6.16 \% $$ What can you say about the spot rate curve so far?

No. There is information here. Keep in mind that bootstrapping derives spot rates from the par curve.

Not quite. That would only be the case if the spot rates found so far were pretty much the same.

Exactly! Bootstrapping to this point has shown spot rates which are increasing with longer maturities. That's typical. Most of the time, a spot curve is upward sloping. Rarely it will be downward sloping, and sometimes it will even be humped, where the intermediate maturities are offering the highest yields. With only a few rates so far, you can't make that determination yet. But upward sloping is a pretty good guess in most cases.

To summarize: [[summary]]

Ignore the coupons

Imagine each cash flow as a little zero-coupon bond

They are the same

The zero-coupon bond offers a lower yield

The zero-coupon bond offers a higher yield

2.00%

4.02%

8.20%

Nothing

It's fairly flat

It is upward-sloping

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