# Fixed income valuations- Calculation of forward rates from spot rates

Hello,

I am having difficuty in grasping the concept behind calculation of forward rates for different periods based on the spot rates.

Here is a sample question.

Years to maturity Spot rates

0.5 4%

1 4.4%

1.5 5%

2 5.4%

a)What is the 6 month forward rate one year from now?

b)What is the 1 year forward rate 1 year from now?

c) What is the value of a 2 year,4.5% coupon semiannual pay bond?

First, you have to consider that the spot rates from your example are on a semiannual bond basis. If you want to calculate 6 month forward rates you need to divide them by 2.

Also very helpful is to draw a timeline (!) with the given rates and the rates you are searching for.

a) 1FR0.5

(1+0,044/2)^2 * (1+1FR0.5) = (1+0.05/2)^3

Solving for 1FR0.5:

1FR0.5 = ((1.025)^3 / (1,022)^2) - 1 1FR0.5 = 0.031

Convert this to an annual yield by multiplying it by 2 gives you a 1FR0.5 of 6.21%.

b) 1FR1

(1+0,044/2)^2 * (1+1FR1)^2 = (1+0.054/2)^4

Solving for 1FR1:

(1+1FR1)^2 = ((1.027)^4 / (1,022)^2) 1FR1 = 0.032

Convert this to an annual yield by multiplying it by 2 gives you a 1FR1 of 6.41%.

c) Arbitrage-free bond pricing: calculate the present value of the bond using above spot rates (bear in mind that you need to divide them by 2).

PV = 2.25/1.02 + 2.25/1.022^2 + 2.25/1.025^3 + 102.25/1.027^4 = 98.3634

Regards,

Oscar

Oscar, I think you mistaked, the coupon is annual, so the pay would be 2.25 and the price would be 98.36. It makes sense being that the average spot rate (annualized) is higher than the bond coupon.

Regards

You are right, I corrected it above.

Thanks for the hint.

Oscar

Assuming that the rates are BEYs:

a) [(1 + 5%/2)³ / (1 + 4.4%/2)² − 1] × 2 = 6.2053%

b) [√((1 + 5.4%/2)^4 / (1 + 4.4%/2)²) − 1] × 2 = 6.4049%

c) Assuming a \$1,000 par, \$22.50/(1 + 4%/2) + \$22.50/(1 + 4.4%/2)² + \$22.50/(1 + 5%/2)³ + \$1,022.50/(1 + 5.4%/2)^4 = \$983.63.

Why are we dividing the BEYs by 2? The question says to assume that all spot rates are calculated on a semiannual basis (not sure if that means semiannual bond basis or what…)

Also, for b), why are you taking the square root? Is there another way to solve the problem without the square root? I’ve never seen it before in the Schweser materials.

Thanks.

Interest rates are always – _ Always! _ – quoted as annual rates. When they say that the 6-month spot rate is 4%, what they mean is that the 6-month spot rate is 4% _ annually _; thus, the effective rate for 6 months is 4% / 2 = 2%. If you receive cash flows every 6 months you need to convert the annual rate into a 6-month rate to discount the cash flow. If the rates are BEYs, the 6-month effective rate is half the annual rate.

You have a 2-year (4-period) spot rate of 5.4% (BEY) and a 1-year (2-period) spot rate of 4.4% (BEY). First, you convert each of those annual rates to 6-month rates by dividing them by 2. Then you compound the first for 4 periods (call the result F) and the second for two periods (call the result T). You then divide F by T to get the annual (effective) rate (properly, growth factor: 1 + the rate) starting one year from today and ending two years from today; that’s a 2-period rate. To get the effective semiannual rate you take the square root of that number; you’re _ uncompounding _ an annual number to get a semiannual number. Finally, you double that semiannual (effective) rate to get the annual (BEY) rate.

No.

I can’t be responsible for the shortcomings of other prep providers’ materials.

Perhaps you should look at my website; I have a lot of articles on Level I material, and it doesn’t have any of the mistakes that I’ve seen in other prep providers’ materials. You can get a sample of the articles here: http://www.financialexamhelp123.com/sample-articles/.

You’re quite welcome.

What if we are trying to find the effective rate for 3 months (i.e. effective quarterly rate)? Is it just 4%/4 = 1%?

Then, we would be able to determine the EAY of a quarter pay bond by: (1 + effective quarterly rate)^4 - 1? This EAY would be higher than that of an annual pay bond because of the compounding effect.

However, as per our discussion on the other thread (sorry for carrying two different discussions simultaneously), YTMs are typically based on BEY.

Is any part of my rationale incorrect?

Ok - I’m slowly grasping this question. So we are taking the square root to “uncompound” a 2 period (annual) number to get the semiannual number. Then, we multiply that by 2 to get the BEY.

No.

I can’t be responsible for the shortcomings of other prep providers’ materials.

Perhaps you should look at my website; I have a lot of articles on Level I material, and it doesn’t have any of the mistakes that I’ve seen in other prep providers’ materials. You can get a sample of the articles here: http://www.financialexamhelp123.com/sample-articles/.
[/quote]

I’ll definitely take a look at the website! Thanks!

What if we are trying to find the effective rate for 3 months (i.e. effective quarterly rate)? Is it just 4%/4 = 1%?

Then, we would be able to determine the EAY of a quarter pay bond by: (1 + effective quarterly rate)^4 - 1? This EAY would be higher than that of an annual pay bond because of the compounding effect.

However, as per our discussion on the other thread (sorry for carrying two different discussions simultaneously), YTMs are typically based on BEY.

Is any part of my rationale incorrect?

Ok - I’m slowly grasping this question. So we are taking the square root to “uncompound” a 2 period (annual) number to get the semiannual number. Then, we multiply that by 2 to get the BEY.

I’ll definitely take a look at the website! Thanks!

It’s not 4%/4.

First, the 4% rate is a 6-month rate. I have no idea what the 3-month rate is, but it appears that it will be less than 4%; the yield curve slopes upward.

In any case, they’ll likely quote it as a BEY. Suppose that it’s 3%. You’d divide that by 2 to get a semiannual effective rate of 1.5%, then uncompound that as:

(1 + QER)² = 1.015

1 + QER = √1.015 = 1.007472

QER = 0.007472 = 0.7472%

You are correct that to get the effective annual (3-month) rate, you will compound this rate for 4 quarters.

What’s the difference between a) and b)? I know we are taking the square root of a two-period EAY to get the effective semiannual rate in b).

Why doesn’t that occur in a)? Is it because (1 + 0.05/2)^3 / (1 + 0.0404/2)^2 giving us the effective semiannual rate already, while (1 +0.054/2)^4 / (1+0.0404/2)^2 is giving us the EAY that we need to square root in order to get the effective semiannual rate?

Exactly.