Pricing Zero-Coupon Bonds

These results also demonstrate that there is an inverse relationship between yields and bond prices:. For a bond that makes semi-annual coupon payments, the following adjustments must be made to the pricing formula:. As an alternative to this pricing formula, a bond may be priced by treating the coupons as an annuity; the price is therefore equal to the present value of an annuity the coupons plus the present value of a sum the face value.

This method of valuing bonds will use the formula: The bond in the previous example can be priced using this alternate bond valuation formula as follows:. A zero-coupon bond does not make any coupon payments; instead, it is sold to investors at a discount from face value. The difference between the price paid for the bond and the face value, known as a capital gain , is the return to the investor. The pricing formula for a zero coupon bond is:. What is the market price of this bond? In order to be consistent with coupon-bearing bonds, where coupons are typically made on a semi-annual basis, the yield will be divided by 2, and the number of periods will be multiplied by There are different types of yield measures that may be used to represent the approximate return to a bond.

These include:. This equals the rate of return earned by a bond holder known as the holding period return if:. It is impossible to solve for the yield to maturity algebraically; instead, this must be done using a financial calculator or Microsoft Excel.

Advanced Bond Concepts: Yield and Bond Price

The variables in brackets fv, type and guess are optional values; the value of type is set to zero if it is not specified. Guess can be used to provide an initial estimate of the rate, which could potentially speed up the calculation time. Note that either pv or fv must be negative, and the other must be positive. The negative value is considered to be a cash outflow, and the positive value is considered to be a cash inflow. Also note that entering semi-annual periods and coupon payments will produce a semi-annual yield; in order to convert this into an annual yield on a bond-equivalent basis , the semi-annual yield is doubled.

For a bond that is callable, the yield to call may be used as a measure of return instead of the yield to maturity. The process is similar to computing yield to maturity, except that the maturity date of the bond is replaced with the next call date.

How to Convert Bond Price to Yield

This is because yield to call is based on the assumption that the bond will be called on the next call date. The face value is replaced with the call price since this is the amount that the investor will receive if the bond is called.

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  • Bond Basics: Yield, Price And Other Confusion.

What is the yield to call? In this case, the bond will mature in eight years, but it can be called in three years. The yield to call is computed as follows:. The current yield is simpler measure of the rate of return to a bond than the yield to maturity. The current yield is computed as:. This measure has the benefit of simplicity. It suffers from the drawback that it does not account for the time value of money. Excel contains a set of specialized bond functions that can be used to account for several complications that arise in bond pricing, such as day-count conventions.

Money market instruments e. Note that the settlement date and maturity date are represented as numerical values in Excel. The date January 1, is represented as 1; all later dates represent the number of days that have passed since January 1, As an example, suppose that a bond is sold on June 15, with a maturity date of June 15, In order to fully understand why that is the value of the bond, you need to understand a little more about how the time value of money is used in bond pricing, which is discussed later in this article.

If interest rates were to fall in value, the bond's price would rise because its coupon payment is more attractive.

1) What are Bonds?

For example, if interest rates fell to 7. The further rates fall, the higher the bond's price will rise, and the same is true in reverse when interest rates rise. In either scenario, the coupon rate no longer has any meaning for a new investor. However, if the annual coupon payment is divided by the bond's price, the investor can calculate the current yield and get a rough estimate of the bond's true yield.

More complex calculations are needed to see the full picture of a bond's yield. A bond's yield to maturity YTM is equal to the interest rate that makes the present value of all a bond's future cash flows equal to its current price. These cash flows include all the coupon payments and its maturity value. Solving for YTM is a trial and error process that can be done on a financial calculator, but the formula is as follows:.

However, if the coupon payments were made every six months, the semi-annual YTM would be 5. In the case of a semi-annual coupon payment, the effective annual yield EAY would be calculated as follows:. If an investor knows that the semi-annual YTM was 5.

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  • How to Convert Bond Price to Yield | Finance - Zacks;

There are a few factors that can make finding a bond's yield more complicated. For instance, in the previous examples, it was assumed that the bond had exactly five years left to maturity when it was sold, which would rarely be the case. When calculating a bond's yield, the fractional periods can be dealt with simply; the accrued interest is more difficult. For example, imagine a bond has four years and eight months left to maturity. It represents the return you would expect if you held a bond for a year.

You can calculate current yield using the following formula:. This formula gives accurate results as long as you pay par value for the bond. The modified current yield formula takes into account the discount or premium at which you buy the bond, and is calculated as:. The adjusted current yield of 6.

Keep in mind: If you buy a bond between coupon payments, you should use the dirty price in place of the market price in the equation. Zero-coupon bonds have only one coupon payment, so we have to use a different calculation. We can calculate its current yield by plugging these values into the following formula:. Because calculating YTM is complicated and time consuming, investors both private and professional typically rely on financial calculators to do the math.

Still, some investors prefer to tackle the math on their own. Before revealing the equation, here are three things to keep in mind:. Remember, the cash flow is the amount you receive for each coupon payment. Instead of picking random numbers, we remember that when a bond is priced at par, the interest rate is equal to the coupon rate.