# qbank spot/forward rate question

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?

My answer is C = 984,

PMT = 110,

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

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

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.

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

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!

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.

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.

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