qbank spot/forward rate question

CFA Level I Fixed Income
#1

OK so this should be easy but I’m getting it wrong. Idk if I’m braindead from studying or if its a mistake.

The one-year spot rate is 7.00%. One-year forward rates are 8.15% one year from today, 10.30% two years from today, and 12.00% three years from today.

The value of a 4-year, 11% annual pay, $1,000 per bond is closest to:

A) $1,060

B) $1,052

C) $984

I’m getting B) $1,052. Apparently the answer is A?

#2

My answer is C = 984,

PMT = 110,

discount this by 7% in Y1, 8.15% in Y2, and so on,

#3

You don’t discount future cash flows by individual forward rates, you discount them by spot rates.

#4

You need to convert all the forward rates to spot rates. After doing so, you get

1 year spot rate = 7.00%

2 year = 7.57%

3 year = 8.47%

4 year = 9.35%

Then you discount 110 for each spot rate; ie. 1.07^1, 1.0757^2 etc.

#5

Yup: the answer’s A.

PV = $110/1.07 + $110/(1.07×1.0815) + $110/(1.07×1.0815×1.103) + $1,110/(1.07×1.0815×1.103×1.12)

= $102.80 + $95.06 + $86.18 + $776.46

= $1,060.50

By the way, the quick way to do this is:

  • $1,110 ÷ 1.12 = $991.07
  • $991.07 + $110 = $1,101.07
  • $1,101.07 ÷ 1.103 = $998.25
  • $998.25 + $110 = $1,108.25
  • $1,108.25 ÷ 1.0815 = $1,024.74
  • $1,024.74 + $110 = $1,134.74
  • $1,134.74 ÷ 1.07 = $1,060.50
#6

hmmmmm OK so it’s not acceptable to take the geometric mean and use that as the ytm because that would only be the discount value for the yr 4 cash flow. I think I understand.

Thanks!

#7

Thanks S2000magician.

I like how you skip the step of the taking the geometric mean and then adding 1 and raising the denominator to a power by simply leaving the whole term alone.

#8

My pleasure.

I’m a mathematician, so I’m lazy by nature.

If I have a bunch of 1-year forward rates, I can just discount back one year, add the payment, discount back another year, add the payment, and so on. My poor brain can’t handle anything more complicated.

#9

Ya… Oh man! What a dud! Thnx for posting this Q.