Continuously Compounded Zero Coupon Bond Rates

Zero-coupon bonds mature like normal bonds but sell at a discount rather than offering interest payments.

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Bonds are financial instruments that allow investors to earn money by lending money to a government or corporation for a set period of time. Typically, bond issuers (governments and corporations) reward bondholders (investors) with interest payments called "coupons" over the course of a bond's term before returning the principal amount, or "face value," of the bond to the holder once the bond matures.

When it comes to interest-paying bonds, an investor's profit—assuming they hold the bond for the entirety of its term and do not resell it—is the total value of the coupon payments they receive over the course of a bond's term. So what, then, is a zero-coupon bond, and how do investors make money with them?

What Are Zero-Coupon Bonds and How Do They Work?

Like regular bonds, zero-coupon bonds are financial securities that mature over time, and their face (par) value is paid to their holder at the end of their term. Unlike coupon-paying bonds, however, zero-coupon bonds do not provide periodic coupon payments—hence the name.

Since they don't offer interest payments, these bonds are sold at a discount to incentivize investors to purchase them. For this reason, they are sometimes referred to as "deep discount" bonds. The difference between a zero-coupon bond's discounted purchase price and its face value is its investor's profit.

What Is Imputed Interest?

This difference between a zero-coupon bond's purchase price and its face value (the investor's profit if they hold to term) is often referred to as imputed interest. Impute means "to assign or ascribe to," so imputed interest essentially means "implied interest ascribed to a bond despite a lack of actual interest payments."

So, where a normal bond's yield is the total value of all of its interest payments, a zero-coupon bond's yield is its imputed interest, or the amount by which its face value was discounted when it was sold.

How Do Zero-Coupon Bonds Differ From Regular Bonds?

Unlike traditional bonds, zero-coupon bonds are not fixed-income debt instruments. While normal bondholders receive interest payments on a regular basis (usually twice a year or annually), zero-coupon bondholders never receive interest payments. Instead, they receive a single lump sum of cash upon the bond's maturity (or sale, if they sell it before it matures).

Additionally, zero-coupon bonds are more volatile than traditional bonds. This means that their value tends to goes up more than the value of typical bonds when interest rates drop (and vice versa)

How Are Zero-Coupon Bonds Priced?

The longer the time until a bond matures, the riskier it is, so generally, bonds with longer terms sell at steeper discounts (i.e., have higher yields). Typically, the following formula is used to calculate the sale price of a zero-coupon bond based on its face value and maturity date.

Zero-Coupon Bond Price Formula

Sale Price = FV / (1 + IR) ^N

Where:

• FV is the face value of the bond.
• IR is the imputed interest rate (expressed as a decimal).
• N is the number of years until the bond matures.

Zero-Coupon Bond Pricing Example

If an investor wanted to make 5% imputed interest on a zero-coupon bond with a face value of $15,000 that matures in four years, how much would they be willing to pay?

Sale Price = FV / (1 + IR) ^N
Sale Price = $15,000 / (1 + 0.05) ⁴
Sale Price = $15,000 / (1.05) ⁴
Sale Price = $15,000 / (1.05) ⁴
Sale Price = $15,000 / (1.05) ⁴
Sale Price = $12,295.08

So, in this scenario, an investor would pay $12,295.08 today in exchange for $15,000 in four years.

How Is the Income From a Zero-Coupon Bond Taxed?

Despite the fact that zero-coupon bonds don't actually pay interest, bondholders are still subject to any applicable taxes on their imputed interest each year. This means that they must pay taxes on income they will not receive until their bond matures.

For instance, if a zero-coupon bond was sold at a $100 discount and matures in four years, its holder would have to pay the applicable bond interest tax rate on $25 worth of the bond's total $100 yield each year.

Depending on the issuer of a bond and where it is purchased, its imputed interest payments may or may not be taxable at federal, state, and local levels. Sometimes, these taxes can be deferred if a bond is purchased through a tax-deferred retirement account like an IRA or 401K. Since many individuals have lower income once retired, they may fall into a lower income tax bracket, so deferring taxes with a retirement account can help minimize the effect of taxes on bond yield.

Where Can You Buy Zero-Coupon Bonds?

Zero-coupon bonds, like other bonds, can be purchased directly from the governments and corporations that issue them, through banks or brokers, or on secondary markets.

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Source: https://www.thestreet.com/dictionary/z/zero-coupon-bond

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