Question on Bond Valuation?

Question

I don't get the difference between the effective annual interest rate and nominal rate, how does it make the answer for this question any different??
the question is: you are considering a 10 year 1,000 par value bond. its coupon rate is 9 percent, and interest is paid semi-annually. If you require an "effective" annual interest rate (not a nominal rate) of 8.16 percent, then how much should you be willing to pay for the bond?
I used a financial calculator to do the following:
since its semi-annual i calculated payment as (PMT): .09/2 x 1000 = 45
Number of years (N) = 10 x 2 = 20
Interest (I/Y) = 8.16/2 = 4.08
Future Value (FV) = 1000
and i computed PV which equals = 1,056.68
PV is max price investor is willing to pay for the bond, did i get it right, and how is effective vs. nominal interest rate make my answer any different? rather, what does it even mean?

Answer 1

The effective rate basically converts the nominal rate to its rate with annual compounding. Coz as you know rates can be stated as compounded monthly, quarterly, semi-annually or annually. Converting the rate makes everything comparable.

Example:
10% compunded annually = 10%
10% compunded semi-annually = 10.25% (naturally the higher number of compounding the higher the effective rate)

Formula for computing effective rate is (1+i/n)^n - 1

where i = nominal rate and n = no. of times compounded in a yr.

In your example, you can convert the 8.16% effective rate to a nominal rate by using the formula above and get 8%.

0.0816 = (1+ i/2)^2 -1
(0.0816-1)^1/2 = 1+i/2
1.04 - 1 = i/2
0.04 (2) = i
0.08 = i = 8%

Now using your financial calculator you compute for the PRICE by inputting:
PMT = 9 (coupon rate)
i = 8 (nominal rate)
then enter a 10 year period and compute for the Price (keys depend on your calculator. for hp its f PRICE)

You will get 106.795. You notice I no longer asked you to input the par value of 1000 because the hp financial calculator assumes a PAR of 100. That is why to get to the right Price you have to multiply by 10 to get the correct price of $1,067.95.

The bond calculation in the HP calculator also assumes a semi-annual compounding.

Answer 2

I look at your calculation again. I believe you have answered it correctly. The gentleman above found the price assuming 8%, not 8.16!

My xplanation is this. The Nominal rate is the stated rate on the bond, which is 9%

Now, because we are buying this bond at a premium price..
$1056.88 when we will only GET $1000 an maturity, it means that the EFFECTIVE rate of interest is LESS
than the 9% per year. SO while we will physically get the 9% interst in actual fact every 6 months, we will lose 56.88 as well.
Therefore the effective rate is less than 9%, and in this case we found that the price equates to an effective rate of 8.16%

Answer 3

Megan got it right.

Danny is completely wrong. If there are N periods per year, then the nominal rate is N times the one period rate. This ignores complounding.

The effective rate takes compounding into consideration. The effective rate is: [(1+nominal / N)^N - 1]

Give Megan the points.