Question about Bionic Turtle's 2009 FRM Program
07 Jan 2009
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Say we want to know what the market expects for the six month interest rate, but six months into the future. That's the six month forward rate six months from today. Then notation for this is 0.5 f 0.5 or 0.5 f m where 'm' refers to how many years forward. So, 0.5 f 2 refers to the six month forward rate starting in two years.
Assume the following:
We get the answer in the same way we bootstrapped the theoretical spot rate curve (term structure of interest rates): by assuming a no-arbitrage condition.
Compare two scenarios:
At the start of the year, you should be indifferent because both should produce the same value at the end of the year (not really, there are two subtle differences: first, your rollover gives you a choice at the end of six months, it is a more liquid alternatives. Second, you have reinvestment risk on your rollover; e.g., rates could go down in the next six months. But we'll ignore these differences).
The six month forward rate six months from today must make you indifferent to the choice:
Just remember we tend to deal in bond-equivalent yields; i.e., a 4.55% one year spot rate corresponds to 2.275% every six months, a 4.3% six month spot rate corresponds to 2.15% over six months. The solution in one variable:
Under these assumptions, one-half of the the six month forward rate solves to 2.4%. We double to translate into its bond-equivalent yield of 4.8%. Therefore, the spot rate curve implies a six month forward rate six months from today (the notation is 0.5 f 0.5) of 4.8%. That is the rate that would make is indifferent between investing at the one year spot or rolling over at six months. Here is the solution and the generic form for any six month forward, given the spot rate at (m) and (m+1):
This EditGrid spreadsheet (which can be downloaded into MS Excel or other formats. Select File > Save As...) also performs the same calculation. Except the inputs are the zero coupon bond prices; i.e., the six month bond price is $97.90 and the one year bond price is $95.60.
07 Jan 2009
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Comments
Thank you for making this post, I’m studying the CFA and I found your analysis particularlyu helpful…
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