Bond Pricing (present value) - Finance - How to calculate (formula) - Finance Dictionary





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Uploaded on Jan 12, 2012

Bond Pricing - Bonds have coupon payments and principal repayments that all occur in the future. Therefore the way to find the present value of a bond is to compare the dollars that are to be received in the future with dollars today. For example, if someone offered you $100 today and you could invest it in an account that earned 10% per year; how much would the account be worth one year from now? This is pretty simple. It would be worth $100 X 1.10% = $110. What we are essentially saying is that $110 one year in the future is equivalent to $100 today. What would $100 today be worth two years from now at a 10% interest rate? It would be worth $100 X 1.10 X 1.10 = $100 X 1.10^2 = $121. So $121 two years from today would be worth $100 today. These examples are just to help you understand the valuation of bonds conceptually. If someone told you they would give you $121 two years from today, how would you figure out what that $121 would be worth today? If the interest rate is 10% and you would receive this payment two years from today then the present value would be calculate as follows: $121/(1.21^2) = $100. What we just did was discount $121 by the interest rate which we rose to the power of 2 because it was to be received 2 years from today.

OK, you now have a conceptual understanding but this needs to be explained in more detail. Remember, with a bond you are receiving multiple coupon payments in the future and a repayment of the principle at the maturity date. To find the present value of a bond you would have to discount all the future coupon payments and the final repayment of principle by the Yield to Maturity (rate offered by the market for a similar bond) and add them up.

Let's assume we have a bond with an annual coupon payment of $50 and a par value of $1000. Let's also assume that there are 3 more years until maturity and the yield to maturity (rate of similar bonds currently offer in the market) is 7%. How would we calculate the present value? First let's list all the information that is necessary to make the present value calculation.

Par Value = $1000
YTM = 7%
coupon payment = $50
periods until maturity = 3

The present value of the bond would be calculated as follows :

(50/1.07) + (50/1.07^2) + (50/1.07^3) + (1000/1.07^3) = $947.51

If this bond made semiannual payments then we would multiply the number of years remaining to maturity by 2 so there would be 6 periods. We would also divide the YTM and the coupon payments by by 2 so the YTM would be 3.5 and the coupon payments would be $25.

For example imagine that there is a semiannual bond with a par value of $1000, a coupon rate of 5% and still has 3 years until it reaches maturity. Now imagine that the YTM is 7%. How would we calculate the present value of this bond?

We need to find the number of payment periods, the amount of the payments and the YTM for each period. If there are 3 years left to maturity and the payments are made semiannually then there are 6 payment periods remaining. If the bond pays a 5% annual coupon and the par value is $1000 then it pays .05 X $1000 = $50 in interest payments annually. If these payments are made semiannually then it makes two payments per year so the coupon payments are $50/2 = $25. Finally, we would divide the YTM by 2 to get the YTM for the period so it would be 7%/2 = 3.5%.
Par Value = $1000
YTM = 3.5%
coupon payments = $25
periods to maturity = 6

The present value of the bond would be calculated as follows :

(25/1.035) + (25/1.035^2) + (25/1.035^3) + (25/1.035^4) + (25/1.035^5) + (25/1.035^6) + (1000/1.035^6) = $946.71

As you can see the present value of the bond is less than the face value. This is because the bond is paying a lower coupon rate than that which is offered by the market for a similar bond. Why would an investor purchase a bond for $1000 that only pays 5% when they could buy a similar bond that would pay 7%. There is less demand for these bonds and therefore the prices fall until they are correctly priced.

Below is an excerpt of U.S. Treasury quotes for bonds and notes from WallstreetJournal.com. The bonds have maturities up to 30 years and the notes have maturities up to 10 years. Take a look at the highlighted issue. We can see that it has a coupon rate of 1.375%. Par value is $1000 therefore it pays interest of $13.75 per year in two semiannual installments so it would make two payments of $6.88 each year. The payments are made in January and July of each year. The bid and ask prices are quoted as percentages of par value. The par value is $1000. Therefore the bid price of the highlighted issue is 100.3047% of par value which is $1003.05. The last column labeled Asked Yield is the bonds yield to maturity based on the ask price.



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