# What is the Convexity of a Zero-Coupon Bond?

Convexity is a measure of the curvature in the relationship between bond prices and bond yields.

It demonstrates how the duration of a bond changes as the interest rate changes.

This concept is particularly important in the context of zero-coupon bonds, which are bonds that do not pay interest during the life of the bond.

Instead, investors buy zero-coupon bonds at a discount from their face value, which is the amount a bond will be worth when it “matures” or comes due.

Rather, payment comes in the form of price appreciation rather than coupon payments.

## Key Takeaways – Convexity of a Zero-Coupon Bond

- Convexity is a measure of the curvature in the relationship between bond prices and bond yields, and it demonstrates how the duration of a bond changes as the interest rate changes.
- Zero-coupon bonds are unique financial instruments issued at a discount to face value and do not make periodic interest payments. Their convexity is higher than that of coupon bonds with the same maturity.
- Convexity adjustment is important for zero-coupon bonds due to their high convexity. It is a correction applied to the bond price to account for convexity and can significantly affect the bond’s price, especially when interest rates are volatile.
- The convexity of a zero-coupon bond is the time to maturity squared.
- Understanding convexity is highly useful for managing risk and making informed investment decisions.

## Understanding Zero-Coupon Bonds

Zero-coupon bonds are unique financial instruments.

They are issued at a discount to face value and redeemed at face value at maturity.

The difference between the purchase price and the face value represents the investor’s return.

The lack of periodic interest payments distinguishes zero-coupon bonds from conventional bonds.

## Convexity and Its Importance

Convexity is a measure of the sensitivity of the duration of a bond to changes in interest rates.

In other words, it measures the rate of change of the bond’s duration.

Convexity is an important concept in bond pricing because it allows bond traders to assess the risk associated with changes in interest rates.

## Convexity of Zero-Coupon Bonds

The convexity of a zero-coupon bond is particularly interesting.

Because zero-coupon bonds do not make periodic interest payments, all of their duration and convexity comes from the single payment at maturity.

This makes the convexity of a zero-coupon bond higher than that of a coupon bond with the same maturity.

## Calculating Convexity

The formula for calculating the convexity of a bond is complex and involves several variables, including the bond’s price, yield, and the time to each cash flow.

For zero-coupon bonds, the formula simplifies because there is only one cash flow at maturity.

The convexity is the second derivative of the price of the bond with respect to interest rates, whereas duration is the first derivative.

## Convexity Adjustment

Convexity adjustment is a correction applied to the price of a bond to account for the convexity of the bond.

This adjustment is particularly important for zero-coupon bonds due to their high convexity.

The convexity adjustment can significantly affect the price of a zero-coupon bond, especially when interest rates are volatile.

## Zero-Coupon Bond Example

Consider a zero-coupon bond with a face value of $1000, with an annual yield of 10%.

The bond matures in five years.

### Price

This price represents the amount an investor would need to pay to purchase the bond today.

With a zero-coupon bond due to the lack of coupon payments, it is calculated as:

P = F / (1+r)^n

The price of the zero-coupon bond with a face value of $1000, an annual yield of 10%, and 5 years to maturity is approximately $620.92.

### Duration

The duration of a zero-coupon bond is simply its time to maturity, which is 5 years in this case.

Duration is a measure of the sensitivity of the price of a bond to changes in interest rates.

### Convexity

The convexity measure indicates how the duration of the bond changes as the interest rate changes.

It demonstrates how the duration of a bond changes as the interest rate changes.

Specifically, convexity is the rate that the duration changes along the price-yield curve, and is therefore a measure of the bond’s sensitivity to interest rate changes.

For a zero-coupon bond, the formula for convexity is simply the time to maturity squared. This is because a zero-coupon bond’s cash flows consist of a single payment at maturity, so its duration and convexity calculations are straightforward.

The reason it’s the time to maturity squared is due to the mathematical derivation of the convexity formula.

In general, the formula for convexity is:

**Convexity **= 1 / P∗(1+y)^2 * ∑ t∗(t+1)∗Ct∗(1+y)^-t

Where:

- $P$ is the bond price,
- $y$ is the yield or interest rate,
- $n$ is the number of periods,
- $C_{t}$ is the cash flow in period $t$

For a zero-coupon bond, there is only one cash flow at the end of the period (at maturity), so the formula simplifies to *$n∗(n+1)$*, and since *$n+1$* is approximately equal to *$n$* for large *$n$*, it simplifies further to *n^$_{2}$*, which is the time to maturity squared.

The higher the convexity, the more sensitive the bond price is to changes in interest rates.

This can be beneficial in a falling interest rate environment (as bond prices would increase more than they would for a bond with lower convexity), but can also be a risk in a rising interest rate environment (as bond prices would decrease more).

### Summary

So, to summarize:

- The price of the zero-coupon bond is approximately $620.92.
- The duration of this bond is 5 years.
- The convexity of the bond is 25.

## Conclusion

Understanding the convexity of a zero-coupon bond is important for investors/traders.

It provides insight into how the price of the bond will change as interest rates fluctuate.

This knowledge can help investors manage risk and make more informed investment decisions.