Sep 18

Forward rates and spot rates

by David Harper, CFA, FRM, CIPM

FRM |

forwardRate3

FRM Learning Outcome (2007)

  • LO 22.3: Calculate forward rates from a series of spot rates.

The forward rate is a prediction about future spot rates

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:

  • The six month spot rate is 4.3%
  • The one year spot rate is 4.55%
  • We want to solve for the six month forward rate, in six months

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.

 

No arbitrage means you are indifferent to holding for one year or rolling over at the forward

Compare two scenarios:

  1. You invest for one year, at the one year spot rate, or
  2. You invest for six months, at the six month spot rate. Then you "roll over" into the the six month forward rate (for another six months).

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:

forwardppt

 

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:

forward1

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):

forwardshow

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.

EditGrid Spreadsheet by bt/admin.

Comments

  1. James Cox 23 Nov 2008

    Thank you for making this post, I’m studying the CFA and I found your analysis particularlyu helpful…

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