Spot Rates - Fixed Income

CFA Level I
#1

Absolutely clueless as to how to work this out.

Could someone please explain the intuition behind this?

Thanks so much!

Suppose that the six-month spot rate is equal to 7% and the two-year spot rate is 6%. The one-and a half-year forward rate starting six months from now has to:

A) be less than 6%. B) lie between 6% and 7%. C) be more than 6%.

#2

I suspect that answer C) should be “be more than 7 %”.

I was talking about this yesterday evening with a couple of candidates I tutor. It’s pretty intuitive; i.e., no calculations necessary (although you can do calculations if you want to). The average rate for 2 years (four 6-month periods) is 6%. The rate for one 6-month period is 7%, so what do the other three 6-month rates have to average to bring the total down to 6%? Answer: they have to be less than 6%: the correct answer is A).

As an approximation, you can solve this:

(1)7% + (3)r = (4)6%

7% + 3_r_ = 24%

3_r_ = 17%

r = 5.667%

#3

You’ve smashed it again!

Cheers mate.

#4

Thanks!

(By the way, did the no-calculation argument help enough, or did you have to (or want to) see the calculation?)

#5

I wouldn’t mind seeing the calculation.

In fact, I know you’ve written articles on other topics (I’ve been frequenting your site lately) – do you happen to have anything on calculation forward from spot and spot from forward rates?

Thanks!

#6

Yup. In Technicolor:

http://financialexamhelp123.com/calculating-forward-rates-from-spot-rates/

http://financialexamhelp123.com/calculating-spot-rates-from-forward-rates/

#7

Unfortunately I needed the calculation too.

Finance doesn’t always come as intuitively to me as others!

Could you perhaps try to re explain the intuition part without calculations in another way?

The part that gets me in the question is the

_ The one-and a half-year forward rate starting six months from now has to: _

I’m never quite sure which rates to use and where.

Thanks

#8

If we’re measuring time in half-years, then you can think of the 2-year spot rate of 6% as a series of four half-year rates:

3%, 3%, 3%, 3%.

However, to make it easier to see why this works, I’ll pretend that the 6% rate applies to a half-year, not to a whole year. (The only effect is that I won’t need to divide annual rates by 2 to get semiannual rates, nor multiply semiannual rates by 2 to get annual rates; my life is a lot easier when I don’t have to multiply or divide.) So, let’s think of the 2-year spot rate as four half-year rates:

6%, 6%, 6%, 6%

We’re also told that the 6-month spot rate is 7%, and we want to calculate (or, at least, estimate) the 18-month forward rate starting 6 months from now. If we call that forward rate F, then we have another sequence of half-year rates:

7%, F, F, F

These two sequences of discount rates have to give the same result (i.e., the PV has to be the same whether we discount for 4 periods at 6%, or we discount for 1 period at 7%, then for 3 periods at F). So the sequence

7%, F, F, F

has to average 6%, just as the sequence

6%, 6%, 6%, 6%

does.

What does that tell us about F? Well, when you average a bunch of numbers, if some of them are above the average, others have to be below the average. Here, 7% is above the average (6%), and the other three numbers are all the same, so they each have to be below the average; i.e., F < 6%.

#9

What’s the answer in the book? I got 5.748% as follows:

$1 invested at 7% for 6 months = $1.035

$1 invested at 6% for 2 years = $1 * (1.03)^4 = $1.087

The amount earned between 6 months and 2 years is = 1.087 / 1.035 = $1.051

So, six months from now, until the end of the 2 years, I made $0.051. The annualized rate for 1.5 years = (1.051)^(1/1.5) - 1 = 5.748%.

I hope this is right, lol. It’s been awhile.

#10

You can solve for F in the following equation to calculate the forward rate from 2 spot rates.

(1.07)^0.6 (1+f)^1.5 = (1.06)^2

#11

Thanks S2000,

Got there eventually!

Your second explanation is really good.

Thanks