# yield measures (fixed income)

A question from CFA Textbook 5, P541

Can anyone please explain? I don’t really understand it….

The par curve is the YTM for a coupon-paying bond: if the bond were issued at par, it would have to pay that coupon rate to be priced at par.

The spot curve is the discount rate for a single payment at a specified maturity.

The forward curve is the single-period discount rate for a single payment.

A simple example will illustrate why C is the correct answer.

Suppose we have annual-pay bonds (to make things easier), that the 1-year par rate is 3%, the 2-year par rate is 4%, and the 3-year par rate is 4.5%.

Calculating the 1-year spot rate is easy: it’s the 1-year par rate, 3%. (A 1-year bond has only one payment: principle plus coupon in one year.)

Before calculating the 2-year spot rate, let’s think about it a bit. We have a 2-year bond paying a coupon rate of 4% (to be priced at par); we can get the price (\$1,000) if we discount all of the payments at 4% (the par rate). We will also get the price if we discount the first payment (coupon) at the 1-year spot rate and the second payment (coupon + principle) at the 2-year spot rate. We’d discount the first payment at 3% (less than the par rate), so we’ll have to discount the second payment at (slightly) more than the par rate; thus, the 2-year spot rate will be slightly more than the par rate.

Now, let’s calculate it:

\$1,000 = (\$40 / 1.03) + (\$1,040 / (1 + S2)²).

If we solve for S2, we get S2 = 4.0202%.

For the 3-year spot rate, we know that we can discount all payments at 4.5% to get the price, or discount the first at 3%, the second at 4.0202%, and the third at the 3-year spot rate; thus, the 3-year spot rate has to be higher than 4.5% You can think of the 3-year par rate as (some sort of) a weighted-average of the 1-year, 2-year, and 3-year spot rates; the 1-year and 2-year rates are less than 4.5%, so the 3-year has to be more than 4.5%.

The calculation is:

\$1,000 = (\$45 / 1.03) + (\$45 / (1.040202)² + (\$1,045 / (1 + S3)³).

If we solve for S3, we get S3 = 4.5384%.

Thus, you can see that when the par curve slopes upward, the spot curve lies above it.

You can do a similar analysis for the forward curve.

A cash flow two years from now can be discounted two ways: you can discount it for two years at the 2-year spot rate, or you can discount it for one year at the 1-year forward rate starting 1 year from now (to get the PV of that payment 1 year from now), then discount it for one year at the 1-year spot rate. No matter which way you do it, the present value (today) has to be the same. (Why?)

Discounting for two years at the 2-year spot rate (S2) means dividing by (1.040202)². Discounting one year at S1 and 1 year at 1f1 means dividing by [(1.03)(1 + 1f1)]; those have to be equal, so:

(1.040202)² = (1.03)(1 + 1f1)

Before solving we note that if we discount for 2 years at 4.0202% each year, then if we discount one year at 3%, we’ll need to discount the next year at about 5% to get the same result.

Solving for 1f1 gives 1f1 = 5.0505%.

Discounting for three years at the 3-year spot rate of 4.5384% will be the same as discounting for the two years at the 2-year spot rate (4.0202%), then one year at 1f2; thus:

(1.045384)³ = (1.040202)²(1 + 1f2)

Discounting for three years at 4.5% vs. two years at 4% and one year at 1f2: 1f2 should be about 5.5%.

Solving for 1f2 gives 1f2 = 5.5825%

So, when the spot curve slopes upward, the forward curve is above it.

wow…!! Thank you so much!

My pleasure . . . but you didn’t answer my question:

Hey S2000magician, sorry that I just saw your question! Didn’t realize that I should answer it…