It is important for prospective bond buyers to know how to determine the price of a bond because it will indicate the yield received should the bond be purchased. In this section, we will run through some bond price calculations for various types of bond instruments. Bonds can be priced at a premium , discount , or at par. If the bond's price is higher than its par value, it will sell at a premium because its interest rate is higher than current prevailing rates.

If the bond's price is lower than its par value, the bond will sell at a discount because its interest rate is lower than current prevailing interest rates. When you calculate the price of a bond, you are calculating the maximum price you would want to pay for the bond, given the bond's coupon rate in comparison to the average rate most investors are currently receiving in the bond market.

Required yield or required rate of return is the interest rate that a security needs to offer in order to encourage investors to purchase it. Usually the required yield on a bond is equal to or greater than the current prevailing interest rates.

Compare between online brokers to see which one offers the best Bond Screener in our Brokerage Review Center. Fundamentally, however, the price of a bond is the sum of the present values of all expected coupon payments plus the present value of the par value at maturity. Calculating bond price is simple: Remember that to calculate present value PV - which is based on the assumption that each payment is re-invested at some interest rate once it is received--we have to know the interest rate that would earn us a known future value.

For bond pricing, this interest rate is the required yield. If the concepts of present and future value are new to you or you are unfamiliar with the calculations, refer to Understanding the Time Value of Money. Here is the formula for calculating a bond's price, which uses the basic present value PV formula:. The succession of coupon payments to be received in the future is referred to as an ordinary annuity , which is a series of fixed payments at set intervals over a fixed period of time.

Coupons on a straight bond are paid at ordinary annuity. The first payment of an ordinary annuity occurs one interval from the time at which the debt security is acquired. The calculation assumes this time is the present.

You may have guessed that the bond pricing formula shown above may be tedious to calculate, as it requires adding the present value of each future coupon payment. Because these payments are paid at an ordinary annuity, however, we can use the shorter PV-of-ordinary-annuity formula that is mathematically equivalent to the summation of all the PVs of future cash flows.

This PV-of-ordinary-annuity formula replaces the need to add all the present values of the future coupon. The following diagram illustrates how present value is calculated for an ordinary annuity:. Each full moneybag on the top right represents the fixed coupon payments future value received in periods one, two and three. Notice how the present value decreases for those coupon payments that are further into the future the present value of the second coupon payment is worth less than the first coupon and the third coupon is worth the lowest amount today.

The farther into the future a payment is to be received, the less it is worth today - is the fundamental concept for which the PV-of-ordinary-annuity formula accounts.

It calculates the sum of the present values of all future cash flows, but unlike the bond-pricing formula we saw earlier, it doesn't require that we add the value of each coupon payment. For more on calculating the time value of annuities, see Anything but Ordinary: Calculating the Present and Future Value of Annuities and Understanding the Time Value of Money. By incorporating the annuity model into the bond pricing formula, which requires us to also include the present value of the par value received at maturity, we arrive at the following formula:.

In our example we'll assume that coupon payments are made semi-annually to bond holders and that the next coupon payment is expected in six months.

Here are the steps we have to take to calculate the price:. Determine the Number of Coupon Payments: Because two coupon payments will be made each year for ten years, we will have a total of 20 coupon payments. Determine the Value of Each Coupon Payment: Because the coupon payments are semi-annual, divide the coupon rate in half.

The coupon rate is the percentage off the bond's par value. Determine the Semi-Annual Yield: From the above calculation, we have determined that the bond is selling at a discount; the bond price is less than its par value because the required yield of the bond is greater than the coupon rate.

The bond must sell at a discount to attract investors, who could find higher interest elsewhere in the prevailing rates. In other words, because investors can make a larger return in the market, they need an extra incentive to invest in the bonds. Accounting for Different Payment Frequencies In the example above coupons were paid semi-annually, so we divided the interest rate and coupon payments in half to represent the two payments per year. You may be now wondering whether there is a formula that does not require steps two and three outlined above, which are required if the coupon payments occur more than once a year.

A simple modification of the above formula will allow you to adjust interest rates and coupon payments to calculate a bond price for any payment frequency:. Notice that the only modification to the original formula is the addition of "F", which represents the frequency of coupon payments, or the number of times a year the coupon is paid. Therefore, for bonds paying annual coupons, F would have a value of one. Should a bond pay quarterly payments, F would equal four, and if the bond paid semi-annual coupons, F would be two.

Pricing Zero-Coupon Bonds So what happens when there are no coupon payments? For the aptly-named zero-coupon bond, there is no coupon payment until maturity.

Because of this, the present value of annuity formula is unnecessary. You simply calculate the present value of the par value at maturity.

Here's a simple example:. Determine the Number of Periods: Unless otherwise indicated, the required yield of most zero-coupon bonds is based on a semi-annual coupon payment. Therefore, the number of periods for zero-coupon bonds will be doubled, so the zero coupon bond maturing in five years would have ten periods 5 x 2. You should note that zero-coupon bonds are always priced at a discount: Pricing Bonds between Payment Periods Up to this point we have assumed that we are purchasing bonds whose next coupon payment occurs one payment period away, according to the regular payment-frequency pattern.

So far, if we were to price a bond that pays semi-annual coupons and we purchased the bond today, our calculations would assume that we would receive the next coupon payment in exactly six months.

Of course, because you won't always be buying a bond on its coupon payment date, it's important you know how to calculate price if, say, a semi-annual bond is paying its next coupon in three months, one month, or 21 days. Determining Day Count To price a bond between payment periods, we must use the appropriate day-count convention. Day count is a way of measuring the appropriate interest rate for a specific period of time. This method counts the exact number of days until the next payment.

Bond Formulas

For example, if you purchased a semi-annual Treasury bond on March 1, , and its next coupon payment is in four months July 1, , the next coupon payment would be in days:. To determine the day count, we must also know the number of days in the six-month period of the regular payment cycle.

In other words, 60 days of the payment period - have already passed.

If the bondholder sold the bond today, he or she must be compensated for the interest accrued on the bond over these 60 days. This count convention assumes that a year consists of days and each month consists of 30 days.

As an example, assume the above Treasury bond was actually a semi-annual corporate bond. In this case, the next coupon payment would be in days. Determining Interest Accrued Accrued interest is the fraction of the coupon payment that the bond seller earns for holding the bond for a period of time between bond payments. The bond price's inclusion of any interest accrued since the last payment period determines whether the bond's price is "dirty" or "clean. In newspapers, the bond prices quoted are often clean prices.

However, because many of the bonds traded in the secondary market are often traded in between coupon payment dates, the bond seller must be compensated for the portion of the coupon payment he or she earns for holding the bond since the last payment. The amount of the coupon payment that the buyer should receive is the coupon payment minus accrued interest.

The following example will make this concept more clear. The next coupon payment after March 1, , is expected on June 30, What is the interest accrued on the bond? Determine the Semi-Annual Coupon Payment: Because the coupon payments are semi-annual, divide the coupon rate in half, which gives a rate of 3.

Determine the Number of Days Remaining in the Coupon Period: The seller, therefore, has accumulated 60 days worth of interest Calculate the Accrued Interest: Accrued interest is the fraction of the coupon payment that the original holder in this case Francesca has earned.

It is calculated by the following formula:. Now you know how to calculate the price of a bond, regardless of when its next coupon will be paid. Bond price quotes are typically the clean prices, but buyers of bonds pay the dirty, or full price. As a result, both buyers and sellers should understand the amount for which a bond should be sold or purchased.

In addition, the tools you learned in this section will better enable you to learn the relationship between coupon rate, required yield and price as well as the reasons for which bond prices change in the market.

Dictionary Term Of The Day. A measure of what it costs an investment company to operate a mutual fund. Latest Videos PeerStreet Offers New Way to Bet on Housing New to Buying Bitcoin?

This Mistake Could Cost You Guides Stock Basics Economics Basics Options Basics Exam Prep Series 7 Exam CFA Level 1 Series 65 Exam.

Sophisticated content for financial advisors around investment strategies, industry trends, and advisor education. Bond Pricing By Amy Fontinelle Share. Introduction Surviving Financially — for a Few Months or Longer Figure Out Which Debt to Pay Off First Will You Need a Retirement Drawdown Strategy? How to Get Health Insurance After Losing Your Job How Much Should You Pay in Life Insurance?

Glossary of Bond Terms

Job Hunting for People Over 50 Resumes and Interviews The Bottom Line DCF Analysis: Introduction Advanced Bond Concepts: Here is the formula for calculating a bond's price, which uses the basic present value PV formula: The following diagram illustrates how present value is calculated for an ordinary annuity: By incorporating the annuity model into the bond pricing formula, which requires us to also include the present value of the par value received at maturity, we arrive at the following formula: Let's go through a basic example to find the price of a plain vanilla bond.

Here are the steps we have to take to calculate the price: Plug the Amounts Into the Formula: A simple modification of the above formula will allow you to adjust interest rates and coupon payments to calculate a bond price for any payment frequency: Here's a simple example: To determine the value of a bond today - for a fixed principal par value to be repaid in the future at any predetermined time - we can use an Excel spreadsheet.

All bonds come with a coupon interest rate, which is the fixed annual interest a bond pays. Investors base investing decisions and strategies on yield to maturity more so than coupon rates. A guide to help to understand the simple math behind fixed-coupon corporate bonds. Use coupons strategically to score big savings on everyday purchases. Investment clubs have been around for several decades and are simply groups of people who get together to invest. Overdraft and credit cards will allow you to spend more money than you have, but whichever option you choose, be sure to Quitting your job does not trigger a taxable event for your k funds unless you elect to cash out your account and take Understand the relationship between a worker's earned income and benefits he or she receives from Social Security in retirement, Content Library Articles Terms Videos Guides Slideshows FAQs Calculators Chart Advisor Stock Analysis Stock Simulator FXtrader Exam Prep Quizzer Net Worth Calculator.

Work With Investopedia About Us Advertise With Us Write For Us Contact Us Careers. Get Free Newsletters Newsletters. All Rights Reserved Terms Of Use Privacy Policy.