Forward rates and spot rates
by David Harper, CFA, FRM, CIPM
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:
- You invest for one year, at the one year spot rate, or
- 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:
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.
Comments
Thank you for making this post, I’m studying the CFA and I found your analysis particularlyu helpful…
I’m sorry but can I ask what is causing the spot rate curve and forward rate curve to slope down at first and then curve up (and then slope downward again)?
Hello. I’m just wondering why we are dividing the 4.55% by 2 then raising the sum to the power of 2 . Are we assuming semiannual compounding? I was under the impression that we are earning the one year rate of 4.55% for the year, then setting this as equivalent to investing in the 6 month rate and rolling over to the 6 month forward rate.
hmm what a calculation that i can never understand. these interest rates, market rates, sale predictions etc all are out of my mind. i belong to a wedding registry company that also handles the bridal lingerie and provides you with excellent wedding planner. you may contact me in case of need.
Beth, I think you are right. Otherwise the effective annual rate would be different. Therefore, I think the formula should have been
(1+ one year rate)^2 = (1 + six month spot rate)(1+six month forward rate)
I used this formula and got the same result for the forward rate, i.e 2.4%
Opps! my bad. I got it all wrong. I mean
(1 + one year rate) = ( 1 + six month spot rate/2)(1+ six month forward rate/2)
Which gives 2.35%.
Kindly help me answer this question. Thank you.
On October 1st, 2009 Fruity Fresh, a food processing company in Ghana made a sale of fruit juices to a firm in the USA. Total sale of the juices amounted to GBP 500,000. Payment of the sale will be effected on 1st January, 2010. The contract requires that the USA firm makes payment in USD. Fruity Fresh has however, incurred some costs in GBP and is considering two alternatives to deal with exchange rate fluctuation that has become rampant in recent times as a result of the credit crunch and its resultant recession in the USA.
The first alternative is to enter the forward market and sell USD equivalent of GBP 500,000 for GBP at the 90-day forward rate quoted on the 1st October, 2009. Under this arrangement the company will receive a definite amount in pound sterling in January as determined by the forward rate on 1st October.
Alternative two is to borrow now from a bank the USD amount that the firm requires, such that the principal plus the interest will equal what the company will be receiving on January 1st 2010. Interest rate on the deal is 6%. By borrowing from the bank Farm Direct will receive USD and with that, it can immediately purchase GBP at the October 1st spot market. It can then invest the GBP RECEIVED AT AN INTEREST RATE of 2% in the UK. When Fruity Fresh receives the proceeds of its export in January USD, it can then use the funds to liquidate the GBP loan incurred in October (assuming there are tax liabilities in both countries and no significant changes in exchange rates on 1st January, 2010).
The third alternative is to make no attempt to cover the exchange risk involved in waiting the three months for the receipt of the of the USD. Under this alternative the Fruity Fresh will be paid the GBP 500,000 equivalent in USD at whatever spot rate prevails on January 1st, 2010.
Spot rate for GBP/USD on the 1st of October 2009 =1.5986
90-day forward rate for GBP/USD ON 1ST October 2009 = 1.5720
Expected rate for GBP/USD on 1st January, 2010 =1.5700
QUEATIONS
HOW much USD will Fruity Fresh receive on October, 2009.
What is the net proceed received under each of the alternatives?
Which alternative will you recommend to Fruity Fresh? Give reasons.
What is the yield for the forward deal? Is the forward rate a discount or a premium.
Kindly help me answer this question. Thank you.
On October 1st, 2009 Fruity Fresh, a food processing company in Ghana made a sale of fruit juices to a firm in the USA. Total sale of the juices amounted to GBP 500,000. Payment of the sale will be effected on 1st January, 2010. The contract requires that the USA firm makes payment in USD. Fruity Fresh has however, incurred some costs in GBP and is considering two alternatives to deal with exchange rate fluctuation that has become rampant in recent times as a result of the credit crunch and its resultant recession in the USA.
The first alternative is to enter the forward market and sell USD equivalent of GBP 500,000 for GBP at the 90-day forward rate quoted on the 1st October, 2009. Under this arrangement the company will receive a definite amount in pound sterling in January as determined by the forward rate on 1st October.
Alternative two is to borrow now from a bank the USD amount that the firm requires, such that the principal plus the interest will equal what the company will be receiving on January 1st 2010. Interest rate on the deal is 6%. By borrowing from the bank Farm Dire
Leave a Comment