How to Calculate Bond Prices
Because bonds may be purchased in different principal amounts, a bonds price is quoted as a percentage of par, (“Par value” or “face value”) is the value of the bond printed on the bond certificate. It is the value used to calculate interest payments and the value of the principal paid to the bond holder at maturity, or the current price per $100 of principal. Let's see how we would quote the price of a bond with an invoice price of $963,701.
The price quoted for this bond in the market would be 96.370. From this point on in the hub prices will be quoted as a percentage of par, as shown below.
The fractional portion of the price of many bonds, such as U.S. Treasuries, is quoted in thirty-seconds of a percent, as shown below. The price we just calculated would therefore be quoted like this. Notice that a hyphen is used to separate the fractional part when it’s in thirty-seconds.
A U.S. Treasury bond has a calculated price of 102.1875. We would quoted the price in the market as:
x = .1875 x = 6
102.1875 = 102 6/32 = 102-06
If we are talking about a standard fixed-rate bond, we can treat the coupon stream as an annuity - that is, a series of evenly-spaced equal payments - and use this formula. The first part of the formula gives the present value of the coupon payments; the second gives the present value of the final principal payment.
We will use the formula to calculate the price of this bond. By the way, since price is always quoted per $100 of principal, we can simplify our calculations by using $100 as the principal amount. The calculation will then give us the price directly.
Bond: $1,000,000 U.S. Treasury note
Matures in 4 years
5% semi-annual coupon
Yield to maturity: 6.5%
CPN = 2.50, PRN = 100, n = 8, i = .065/2 = .0325
Price = 2.50( 1 - (1 + .0325)-8 ) + 100(1 + .0325)-8 = 94.790
Let's try another one.
Bond: Matures in 4 years
4% annual coupon
Yield to maturity: 5%
CPN = 4, PRN = 100, n = 4, i = .05
Price = 4( 1 - (1 + .05)-4 ) + 100(1 + .05)-4 = 96.454
Now let's look at a trade. A trader purchases a $1,000,000 bond at 98-12. Later that day, she sells the bond at 98-28. How do we calculate how much profit she has made?
Sold at 98-28 = 98.875
Purchased at 98-12 = 98.375
Profit per $100 = .500Total profit = (1,000,000/100) x .5 = $5,000
Finally we musn't forget the relationship between bonds and interest rates and bond yields and price.
$1,000,000 U.S. Treasury bond with 7.25% coupon
Matures in exactly 15 years
Purchase price: 91.484 @ 8.25% yield to maturity
If the yield for bonds of this type increases to 8.40%, the value of this bond will decrease. The realtionship between bond price and bond yield is inverse.
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