A question from CFA Textbook 5, P541
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The correct answer is C.
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…
Briefly speaking, we can answer your question from two perspectives.
- The relationship between forwards rates and spot rates: (1+St)Tsquare=(1+f0)(1+f2)…(1+f(T-1) )
- Arbitrage rule – Law of One Price: if two securities or portfolios that have identical cash flows in the future, regardless of future events, should have the same price. In this way, if the PVs based on two methods are different, we can construct cash follow portfolios to arbitrage. As more and more people arbitrage, the forward rates and spot rates will soon reach the equation relationship as mentioned in #1.
Feel free to correct me if I am wrong. Thanks a lot for your instructions.
Can you please explain the equation in detail
Yup.