# Value Bond using spot rates

I got confused by seeing people using different methodologies to price semiannual bonds using spot rate;

in the first case the 6 months payment is being calculated as:

(coupon/2)/ (1+ annual spot rate)^0.5

the 2nd case:

(coupon/2)/ (1+(annual spot rate/2))^1

1st case:

Let’s consider a hypothetical bond with a par value of 100, that pays 6% coupon semi-annually issued on January 15th, 2015 and set to mature on January 15th, 2016. The bond will pay a coupon on July 15th, 2015 and January 15th, 2016. The par amount of 100 will also be paid on the January 15th, 2016.

To make things simpler, lets assume that we know the spot rates of the treasury as of January 15th, 2015. The annualized spot rates are 0.5% for 6 months and 0.7% for 1 year point. Lets calculate the fair value of this bond.

He is pricing the bond using the following formula:

3/pow(1+0.005, 0.5) + (100 + 3)/(1+0.007)
105.27653992490681

but the other example:

Let’s take another example. Suppose we have a bond that matures in 2 years, that has a coupon rate of 6%, and pays coupon semi-annually. The spot rates are 3.9% for 6 months, 4% for 1 year, 4.15% for 1.5 years, and 4.3% for 2 years.

The cash flows from this bond are \$30, \$30, \$30, and \$1030.

The value of the bond will be calculated as follows:

Bond value = \$30/(1+3.9%/2)^1+\$30/(1+4%/2)^2+\$30/(1+4.15%/2)^3+\$1030/(1+4.3%/2)^4

Bond value = \$1032.45

Do you have a question?

Case #1 would work if interest rates were stated as effective annual (1 compounding per year), while case #2 would work if interest rates were stated as nominal semi-annual (2 compoundings per year, i.e. good old bond equivalent).

Yes, why for the same problem/calculation, different formulas have been used. Especially for the semi-annual coupon payment,

#1
3/ (1+ 0.005)^0.5

#2
30/ (1+ 0.039/2)^1

Spot rate divided by 2 raised on power 1(number of periods)

Which one is correct?

Aren’t those the same, annualized spot rates aka zero coupon yields, one compounding per year?

In the first case the spot rates are quoted as effective annual rates, whereas in the second case they’re quoted as nominal annual rates (in this case, BEY).