# Zero coupon bond valuation formula

This week we will introduce you to the basic language and mechanics of bond pricing. Though introductory, we attempt to cover a fair amount of content and with real world applications. Enroll for Free. From the lesson. About This Specialization Taught By. A sinking fund reduces the possibility of default ; default occurs when a bond issuer is unable to make promised payments in a timely manner.

Since a sinking fund reduces credit risk to bond holders, these bonds can be offered with a lower yield than an otherwise identical bond with no sinking fund. Bonds are issued by borrowers to raise funds for long-term investments; the main issuers of bonds in the U.

Treasury securities are issued by the U. These are free of default risk , which is the risk that the investor will not receive all promised payments. They are not taxed by state and local governments, but are taxed at the federal level. Another key difference between these securities is that Treasury bills are sold at a discount from their face value and redeemed at face value; Treasury notes and bonds are sold and redeemed at face value and pay semi-annual coupons to investors.

Corporations can raise funds by issuing debt in the form of corporate bonds. These bonds offer a higher promised coupon rate than Treasuries, but expose investors to default risk. The riskiest corporations offer the highest coupon rates to investors as compensation for default risk.

A municipal bond is issued by a state or local government; as a result, they carry little or no default risk. Occasionally, municipalities do default on their debts; in , the city of Detroit filed for bankruptcy as a result of being unable to pay its debts. Municipal bonds offer an extremely favorable tax treatment to investors. They are not taxed by federal, state or local governments as long as the bond holder lives in the municipality in which the bonds were issued. As a result, municipal bonds can be issued with very low yields.

Foreign bonds are issued by foreign governments and corporations and are denominated in dollars. If they are denominated in a foreign currency, they are known as eurobonds. Dollar-denominated bonds issued in the U. This formula shows that the price of a bond is the present value of its promised cash flows. The bond makes annual coupon payments. 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 Zero Coupon Bond: Pricing and Charactertistics.
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The pricing formula for a zero coupon bond is:. What is the market price of this bond? Of course, these data-construction methods introduce measurement error. Huston McCulloch at Ohio. Bliss updates this data set until the end of Both data sets consist of monthly observations over the whole postwar period. Estimations of affine models with zero-coupon bonds include Balduzzi et al. Read full chapter. Zero-coupon bonds , sometimes known as strips, have only one cash flow, the redemption payment on maturity. Hence the name: strips pay no coupon during their life. In virtually all cases zero-coupon bonds make one payment on redemption, and this payment will be par Therefore a zero-coupon bond is sold at a discount to par and trades at a discount during its life.

## Bond Valuation

For a bond with only one cash flow it is obviously not necessary to use 4. Equation 4. Because no interest is actually paid by a zero-coupon bond, the interest periods are known as quasi-interest periods. A quasi-interest period is an assumed interest period, where the assumption is that the bond pays interest. It is important to remember this because zero-coupon bonds in markets that use a semi-annual convention will have n equal to double the number of years to maturity. For annual coupon bond markets n will be equal to the number of years to redemption. We can rearrange 4.

What is its yield to maturity?

### What it is:

The yield of the bond is 5. Because this is a sterling bond it is assumed to have two quasi-coupon periods each year.

The calculation is 2. To obtain the bond-equivalent yield to maturity we double this figure, giving us a yield of 5. Consider the following zero-coupon bond stripped from a bond that pays semi-annual coupons: Maturity date: 1 December Settlement date: 1 August Yield: 5.

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- Bond valuation.
- An Introduction to Bonds, Bond Valuation & Bond Pricing.

The quasi-coupon dates are 1 June and 1 December each year. The price is then given by: The examples of zero-coupon bond yields illustrated in Example 4. We expect the yields to differ because two of them were annual yields while the other two were semi-annual yields. In the markets the convention is to convert one to the other to make them equivalent, in order to enable us to compare yields.

What happens if we convert the yield in 4. Our first thought must be that the yields should be equal because both bonds are trading at the same price. Zero-coupon bonds linked to the inflation do not pay coupons.

## Zero-Coupon Bond Definition & Example

Therefore, the unique adjustment is made to the principal. These types of bonds offer no reinvestment risk due to the absence of coupon payments and have the longest duration than other inflation-linked bonds. The value is given by Equation 6. The inflation adjustment on the redemption value can be derived as follows Equation 6.

Table 6. Zero Coupon Bonds or Discounted Debt Instruments are the financial contracts with the simplest cash flow pattern: They involve two separate cash flows depicted in Figure 3. Figure 3.

## 1.2 Zero-Coupon Government Bonds

Cash flow pattern of a zero coupon or discount debt instrument. Cash flow payments are assumed to coincide with contract commencement not conclusion! The redemption payment to the investor usually consists of the principal amount plus some interest in return for lending the initial amount for the given period. In practice, there is no distinction made between the interest and principal part of the redemption payment, when they are due on the same date.

There is no element of optionality, discretion, or dependence on underlying cash flows or accumulated asset values embedded in this type of contract, and in the absence of credit risk the investor holding the instrument to maturity would have complete certainty as to the cash flow he or she will receive in return for his or her initial investment. For the individual coupon period 3, Figure 3. The cash flow is fixed at the beginning and accrues monotonously over the entire period. The cash flow amount to be paid accrues from the beginning of the coupon period and is paid out at the end of it.