Bond Calculator

What a bond is, and what you are buying

A bond is a loan, with the roles reversed from what you might be used to. Instead of borrowing, you are the lender: you hand money to a government or a company, and in return they promise to pay you a fixed amount of interest at regular intervals, and to give you back the full face value on a set date in the future. The regular interest payments are called coupons, and the final repayment date is the maturity.

So when you buy a bond, what you are really buying is a stream of future payments: a run of coupons stretching out over the years, and a lump sum at the end. That stream is fixed and known in advance. This calculator takes the details of that stream, the face value, the coupon, the maturity, and the yield the market requires, and works out what the whole thing is worth today.

Why a bond's price is a present value

Here is the central idea. Money arriving in the future is worth less than money in your hand now, because money now can be invested to grow. So to value a bond, you cannot simply add up all its future coupons and its face value; you have to translate each future payment into what it is worth today, then total those up. That translation is called discounting, and the total is the present value.

The rate you discount at is the yield: the return investors currently demand for lending to this issuer over this period. Each coupon gets discounted back to today by that yield, with payments further in the future discounted more heavily, and the face value at maturity gets discounted back too. Add the present value of all the coupons to the present value of the face value, and you have the bond's price. In short: price = present value of the coupons + present value of the face value. That is exactly the sum this calculator performs, coupon by coupon.

A worked example

Take a bond with a face value of 1,000, a 5% coupon paid semi-annually, ten years to maturity, and a market yield of 6%.

Paid twice a year, the 5% coupon delivers 25 every six months, across 20 periods, and each period is discounted at half the annual yield, so 3% per period. Discounting all 20 coupon payments back to today totals about 371.94, and discounting the 1,000 face value back over those 20 periods gives about 553.68. Add them and the bond's price is roughly 925.61. Notice it came out below the 1,000 face value. That is no accident: the bond pays a 5% coupon while the market wants 6%, so buyers will only take it at a discount, one just deep enough to lift their overall return up to the 6% they could get elsewhere. Had the market yield instead been exactly 5%, matching the coupon, the price would have landed precisely at 1,000.

The most important rule: price and yield move in opposite directions

If you take away one thing about bonds, make it this: a bond's price and its yield move in opposite directions. When the market's required yield rises, the bond's price falls; when the required yield drops, the price climbs. This is the single most important principle in all of bond investing, and the example above shows it in miniature.

And the reason is not some quirk of markets; it is just the arithmetic of discounting. A bond's future payments are fixed. The only thing that changes is the rate you discount them at, and discounting any fixed future amount at a higher rate makes it worth less today. Picture it plainly: you own a bond paying a 5% coupon, and tomorrow new bonds start paying 7%. Nobody will pay full price for your 5% bond when they can get 7% fresh, so its price has to slide until its total return matches the going 7%. This is why rising interest rates push existing bond prices down, and it is the heart of what people mean by interest-rate risk. The yield that makes this whole relationship tick is the yield to maturity, which our bond yield to maturity calculator works out from a bond's price.

Premium, par, and discount

The comparison between a bond's coupon rate and the market's yield decides, in one stroke, whether it trades above, at, or below its face value, and the three cases have names.

When the coupon rate equals the yield, the bond is worth exactly its face value, and it trades at par. When the coupon rate is higher than the yield, the bond pays more than the market demands, so it is worth extra and trades at a premium, above face value. When the coupon rate is lower than the yield, as in our example, the bond pays too little to be worth full price, so it trades at a discount, below face value. This is not a separate set of rules to memorise; it falls straight out of the present-value calculation. A generous coupon relative to the market lifts the present value above face; a stingy one drops it below. Par, premium, and discount are simply the three ways the coupon can compare to the yield.

Coupon frequency

Bonds do not all pay their coupons on the same schedule, so the calculator lets you choose: annually, semi-annually, quarterly, or monthly. In much of the bond market, semi-annual payments are the standard, which is why a bond's annual coupon is usually split into two.

Frequency matters to the calculation because it changes both the size and the timing of each payment. A 6% annual coupon paid semi-annually becomes two payments of 3% of face value, each discounted at half the annual yield over twice as many periods. More frequent payments mean smaller, more numerous cash flows, and because you receive some of your money a little sooner, frequency has a modest effect on the price. The calculator handles this for you: tell it how often the coupon pays, and it slices the coupon and the discounting into the right periods automatically.

What the price does and does not include

The figure this tool gives you is the bond's fair value: what its stream of future cash flows is worth today, discounted at the yield you enter. That is the clean, fundamental price, and it is exactly the number you want for understanding how a bond is valued and how its price responds to yields.

It is worth knowing that a real-world transaction can carry a little more on top. When a bond is bought between coupon dates, the buyer typically also pays the seller the interest that has built up since the last coupon, known as accrued interest, so the cash that changes hands can differ slightly from the clean price here. Prices in the market also reflect things a present-value formula cannot see, chiefly the issuer's credit quality, the risk that it might not pay, which is why a shaky issuer's bonds yield more and cost less. None of that changes the method; it simply means the calculator gives you the sound theoretical price, the anchor from which those real-world adjustments are measured. For the income side of the picture, our bond current yield calculator shows what a bond pays relative to its price.

Questions people ask

How is a bond's price calculated?

A bond's price is the present value of its future cash flows: each coupon payment and the face value at maturity, discounted back to today at the market's required yield. Summing the discounted coupons and the discounted face value gives the price.

Why would a bond be worth less than its face value?

Because its coupon rate is lower than the yield the market currently requires. Buyers will only accept the smaller coupon at a discount to face value, one that raises their overall return to the market rate. Such a bond is said to trade at a discount.

What happens to a bond's price when interest rates rise?

It falls. A bond's payments are fixed, so when required yields rise, those fixed payments are discounted more heavily and are worth less today. This inverse relationship between price and yield is the core principle of bond investing.

What do par, premium, and discount mean?

They describe a bond's price relative to its face value. At par, the price equals face value, which happens when the coupon rate equals the yield. At a premium, the price is above face value, when the coupon exceeds the yield. At a discount, it is below, when the coupon is less than the yield.

References

The valuation of a bond as the present value of its coupons and face value, and the inverse relationship between a bond's price and its yield, follow FINRA and the standard fixed-income text by Fabozzi below.

1. FINRA. Understanding Bond Yield and Return. finra.org
2. Fabozzi, F. J. Bond Markets, Analysis, and Strategies (bond pricing and the price-yield relationship). Pearson.