US Treasury Bond

 My answer is different from our guests.  We may have made different assumption, I don't know.  

The US government issued a Bond on June 1, 2010 to mature June 1, 2030. The coupon on this bond was 4% paid semi-annually on June 1 and on December 1. I wish to buy this bond on Monday, Nov. 2, 2015 at a yield of 3%. What is the Price that I should pay, as well as the accrued interest on this bond? Thank you.

Assumptions  

1)  The 4% and 3% are 6 monthly rates (not annual rates)

2)  Face value = Redeption value = $100

so

F=C=$100

r=4% =0.04

I = Fr = 100*0.04 = $4    That is, a $4 is paid every 6 months

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i= 3% = 0.03

Because you want less interest than the coupon rate this means that you are prepared to pay a premium - this means that you are prepared to pay more than $100 for the bond.

Now, you are buying the bond in  the middle of a coupon cycle so you need to first work out how much you would pay for it at the beginning of that cycle.

It is easiest for you to draw up a rough time line to display what is happening.

Now the beginning of the cycle BEFORE is 1st June 2015

The next period will begin on 1st Dec 15  (but we do not need that just yet)

From the 1st june15 till 1st June 30 is 15 years which is 30 rate perids

n=30

First off you need to work out the price that will yeild 3% back to that date.

\(P_1=C+\left[(I-Ci)\times \frac{(1-(1+i)^{-n}}{i}\right]\\ P_1=100+\left[(4-100*0.03)\times\frac{1-(1.03)^{-30}}{0.03}\right]\\ P_1=100+\left[1\times 19.60044135\right]\\ P_1=119.60044135\\ P_1=\$119.60\)

So, if you bought the bonds on 1st june 15 you would have paid $119.60

But you didn't.  You bought them later so you will get some ' free' interest period.

So you will be prepared to pay a little more.

So you need to work out how many free interest days that there are.

From the 1st of June to the 2nd Nov = 30+31+31+30+31+1 = 154 days

And how long it will be till the next cycle begins

And from 1st June to 1st December = 30+31+31+30+31+30=183 days

So the free portion is   154/183 of a cycle.

\(S=P_1(1+i)^{154/183}\\ S=119.60\times(1.03)^{154/183}\\ S=$122.61 \)

So you should pay a maximum of $122.61 per $100 Coupon.

Calculating the price of bond is a ralatively involved process. First, you have to figure out the the price of a stream of coupons. Second you have to find out the price of the par value of the bond at maturity, which is generally taken as $100. Then you add the two together to get the price of the bond.

But, what makes it even more complicated is the fact that the bond has to be priced on a specific date, i.e., the date of purchase, which is sometimes called "settlement date", which in this case is Monday, Nov. 2, 2015. So a special calculation has to be made for this odd period, from the last coupon date, June 1, 2015 to Nov. 2, 2015.

For this purpose, two different formulae are used to obtain the price of the bond: For the price of the coupons, this formula is used: PV=C[((1 + i)^n - 1)/((1 + i)^n.i)]. C=$4.00 par value of the coupon.

For the price of the par value of the bond at maturity, which is generally taken to be $100.00, this common formula is used: PV=100(1 + i)^-n. Where i=3/2=1.5, and n=number of semi-annual periods left in the life of the bond from Nov. 2, 2015 to June 1, 2030.

Putting all this together, it will be seen that the price of the bond on the purchase date of Nov. 2, 2015 will come to =$111.737 per $100. And the accrued interest=$1.683 per $100. This is for the period of June 1, 2015 to Nov. 2, 2015.

So, the total price of the bond is=$111.737 + $1.683=$113.420 per $100.00.

Melody:I congratulate you on trying to calculate the Bond price. Excellent effort. I give you 9 out 10 stars!!. The only mistake you made was to take the coupon rate of 4% and the desired yield of 3% as annual instead of semi-annual rates. The Treasury Bonds both in US and in Canada must be, by law, compounded semi-annaually and they are traded as such. I know this for a fact, because that is what I did as an Investment Banker for some 25 years. If you had just taken $4/2=$2 as paid every six months, and used 3%/2=1.5% for your calculation, you would have been spot on!. If you did that for 15 years, or 30 periods, you would have gotten $112.008. By law, the Bond prices and yields are carried to an accuracy of 3 decimal places. BRAVO!! Now, try to have your Pal, CPhill, try his hand and do the same thing in his spare time and see how close he gets!, but without taking a peek at your work!. I also have the advantage of having specialized software on my computer that does these calculations instantaneously. After all, I have to look after my own investments!!!!!!.

Nowhere  did you specify that this was an annual rate.

However in my answer I did specify that I had to make assumptions because the question had missing information.

My first assumption clearly stated that I had taken it to be the bond rate,which is 6 monthly.  I stated that it was not an annual rate.

My second assuption was what the face value and redemption value were.

Hence, if the interest rate is the only error.  It was your error as the question asker.  Not mine as the answerer.

Hence my answer is, by your own calculations, correct.