I’m a bit confused regarding the method of these two questions. Why do we in the first one just divide by spot rate and in the second we take spot rate x forward rate?
MODULE QUIZ 57.1
If spot rates are 3.2% for one year, 3.4% for two years, and 3.5% for three years,
the price of a $100,000 face value, 3-year, annual-pay bond with a coupon rate of
4% is closest to:
A. $101,420.
B. $101,790.
C. $108,230.
Given the following spot and forward rates:
Current 1-year spot rate is 5.5%.
1-year forward rate one year from today is 7.63%.
1-year forward rate two years from today is 12.18%.
1-year forward rate three years from today is 15.5%.
The value of a 4-year, 10% annual-pay, $1,000 par value bond is closest to:
A. $870.
B. $996.
C. $1,009.
In #1, you have a series of cash flows that are basically spot. Therefore, all you have to do is discount year t’s cash flows by the t-year spot rate. Simple and easy!
In #2, you can either invest at the t year spot rate (which is NOT provided for t>1) or you can invest at the 1 year spot and reinvest at the forward rate for the next 3 years. These are just 2 different paths to get you to the same end result.
Let’s say you wanna invest $1,000 for 2 years. I have a 1 year spot rate handy, but I can’t give you a 2 year spot quote. However, to get you to the SAME END RESULT, I will take whatever accumulated balance you have at the end of year 1 and REINVEST at the forward rate.
End of year 1 AV = 1,000 x 1.055 = $1,055
End of year 2 AV = 1,055 x 1.0763 = $1,135.50
So if I need to pay a coupon at time 2, you would discount it using the SAME PATH you used to roll up the AV at time 2.
I don’t really understand why but I guess I have to just memorize that when it comes to the case when it is both spots and future rates I just have to forget about N and just multiply them all together. When it is spot rates I just use them for the specific year and raise it to n. It feels like this whole test is a lot about just memorizing things.
You must raise the Discount Rate for a certain number of years because the spot rates given to you as annual rates.
If you invest $100 at a 2-year rate of 5%, that means that you will earn 5% after one year, not after two.
Same concept when discounting, when you are discounting by the 2-year rate, you are discounting back one year at a time, thus you must raise the discount rate to account for both years.
I thought you were talking as a generality. I see what you’re asking now.
The reason you don’t raise in the second example is because you’re not given two-year, three-year, or four-year rates. You are given a whole bunch of one-year rates.
Today, the one-year rate is 5.50%. Next year, the one-year rate will be 7.63%. In two years, the one-year rate will be 12.18%. And in three years, the one-year rate will be 15.50%.
In the first example, you are given n-year rates, which are always quoted in annual terms, thus you must compound them to make the discount match n. In the second example, you are given one-year rates. You don’t need to raise these because they are already annual rates.
The math works out, if you want to try convert the next year’s one-year rate into today’s two-year rate, two years from now’s one-year rate into today’s three-year rate, and three years from now’s one-year rate into today’s four-year rate. These should all line up to enforce arbitrage.
If you do it that way, then you will need to raise by n.