Zero-Volatility Spread (Z-Spread) Formula and Calculation (2024)

Suppose you're considering two investments: a super-safe government bond and a riskier corporate bond. The zero-volatility spread (z-spread) helps you understand how much money you'd need to make from the riskier bond over its lifespan to make it worthwhile.

In short, the z-spread puts a number to the adage that the greater the risk, the greater the potential reward. The z-spread determines the added yield (or interest) an investor can earn from a corporate bond compared with a risk-free government bond. It measures the extra incentive or reward for taking on the risk of investing in a particular bond.

Also known as the static spread, this concept is often made to sound quite tricky, but it's intuitive. Suppose you're fresh out of college and offered a safe job with a local government agency, which promises a decent income and job security. It's an important agency, and it has never had job cuts in its entire history. Meanwhile, you're also offered a job with a startup business that a friend is forming. It offers potentially far higher pay through bonuses and stock options, but there's also a risk the company will struggle or even go out of business. How much more would you need to be offered to take the startup job? To make a decision, you need to know more than the initial salary difference; you also need to know the different effects they would have over your lifetime since you're comparing two career paths. So, how much would it take, on average, for each paycheck to make it worth the risk?

Calculating that difference related to higher and lower-risk bonds is precisely what the z-spread does. Below, we show you how it's measured and why it's crucial to any bond investor.

What the Zero-Volatility Spread (Z-Spread) Tells You

The z-spread is the constant spread that needs to be added to a benchmark yield curve to make the price of a bond equal to the present value of its cash flows. Below is a sample spread:

To understand this concept, let's return to our job analogy. Suppose you took the stable government job, but your friend calls again asking you to move over to the startup. The z-spread is like calculating the extra compensation package (salary, bonuses, and benefits) the startup needs to offer each year, over your entire career, to make the risky move worthwhile.

In bond terms, the government job is the yield curve of risk-free Treasury securities. These government securities are considered risk-free since the U.S. government has always paid such debts. The startup job is analogous to the greater risk, but potentially higher reward, of the corporate or other non-government bond you're considering. The z-spread gives you a measure of how much more in yield the riskier bond must offer at every point along its life cycle to match the value of the safe Treasury bonds.

Here are key aspects of the z-spread:

1. It's a comprehensive measure: Unlike simpler spread calculations, the z-spread accounts for the entire yield curve, not just at a single point.
2. Quantifies your risk: It effectively measures the premium investors demand for taking on the additional risk of a non-Treasury security.
3. Present value comparison: The z-spread ensures that the present value of the bond's cash flow, when discounted at the Treasury yield plus the spread, equals the bond's market price.
4. Constant spread: The "zero-volatility" in the name of the spread refers to how the spread remains constant across all maturities of the yield curve.

As such, the z-spread is different from the nominal spread. A nominal spread calculation uses one point on the Treasury yield curve (not the spot-rate Treasury yield curve) to determine the spread at a single point that will equal the present value of the security's cash flows to its price.

The z-spread helps analysts discover if there is a discrepancy in a bond's price. Because the z-spread measures the spread that an investor will receive over the entirety of the Treasury yield curve, it gives analysts a more realistic value of a security instead of a single-point metric, such as a bond's maturity date.

Formula and Calculation for the Zero-Volatility Spread

To calculate a z-spread, an investor must take the Treasury spot rate at each relevant maturity, add the z-spread to this rate, and then use this combined rate as the discount rate to calculate the price of the bond. The formula to calculate a z-spread is as follows:$\begin{aligned} &\text{P} = \frac { C_1 }{ \left ( 1 + \frac { r_1 + Z }{ 2 } \right ) ^ {2n} } + \frac { C_2 }{ \left ( 1 + \frac { r_2 + Z }{ 2 } \right ) ^ {2n} } + \frac { C_n }{ \left ( 1 + \frac { r_n + Z }{ 2 } \right ) ^ {2n} } \\ &\textbf{where:} \\ &\text{P} = \text{Current price of the bond plus any accrued interest} \\ &C_x = \text{Bond coupon payment} \\ &r_x = \text{Spot rate at each maturity} \\ &Z = \text{Z-spread} \\ &n = \text{Relevant time period} \\ \end{aligned}$​P=(1+2r1​+Z​)2nC1​​+(1+2r2​+Z​)2nC2​​+(1+2rn​+Z​)2nCn​​where:P=CurrentpriceofthebondplusanyaccruedinterestCx​=Bondcouponpaymentrx​=SpotrateateachmaturityZ=Z-spreadn=Relevantperiod​

Example

Assume a bond is currently priced at $104.90. It has three future cash flows: a $5 payment next year, a $5 payment two years from now, and a final total payment of $105 in three years. The Treasury spot rate at the one-, two-, and three-year marks are 2.5%, 2.7%, and 3%.

We need to find the z-spread that equates the bond price to the present value of these cash flows. Those cash flows are determined by discounting the two $5 cash flows and the final $105 payment to find their present value. This is done by dividing each cash flow by one plus the sum of the Treasury spot rate for that period (2.5%, 2.7%, and 3.0% in our example) and the z-spread, raised to the power of two times the number of the period:

$\begin{aligned} \$104.90 = &\ \frac { \$5 }{ \left ( 1 + \frac { 2.5\% + Z }{ 2 } \right ) ^ { 2 \times 1 } } + \frac { \$5 }{ \left ( 1 + \frac { 2.7\% + Z }{ 2 } \right ) ^ { 2 \times 2 } } \\ &+ \frac { \$105 }{ \left ( 1 + \frac { 3\% + Z }{ 2 } \right ) ^ {2 \times 3 } } \end{aligned}$$104.90=​(1+22.5%+Z​)2×1$5​+(1+22.7%+Z​)2×2$5​+(1+23%+Z​)2×3$105​​

We can then calculate the net present value of the three cash flows:

4.8696 + 4.7176 + 95.3157 = 104.9029

Given that our bond price is $104.90 and the calculated present value of the cash flows is about $104.9029, we adjust the z-spread slightly to match the bond price exactly. In this case, the z-spread is found to be the following:

$140.9029 - 140.9000 = 0.0029, or 0.29%.

The Bottom Line

The z-spread is an important tool for assessing the relative value of bonds. It represents the constant spread that, when added to the Treasury spot rate at each cash flow point, equates the bond's price to the present value of its cash flows.

This spread offers a more detailed valuation compared with single-point metrics like the nominal spread, as it accounts for the entire yield curve. By understanding and utilizing the z-spread, investors, and analysts can more accurately identify price discrepancies and make more informed decisions about bond investments.