# Understanding the Zero-Volatility Spread (Z-Spread): Formula & Calculation

![Zero-Volatility Spread (Z-Spread): The constant spread that will make the price of a security equal to the present value of its cash flows.](https://www.investopedia.com/thmb/aOKErYoPECi10-4QnSLE1MlFvhk=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/zspread.asp-final-7c8853cd68c54e53b2a1c5ea65b52506.png)

*— Investopedia / Julie Bang*

> **Definition**
>
> The zero-volatility spread (z-spread) is the constant spread that makes the price of a security equal to the present value of its cash flows when added to the yield at each point on the spot rate Treasury curve where cash flow is received.

Consider two investments: a safe government bond and a riskier corporate bond. The z-spread shows how much more you need to earn from the riskier bond to make it worth the risk.

Simply put, the z-spread quantifies the idea that with greater risk comes the potential for greater 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.

### Key Takeaways

*   The zero-volatility spread (Z-spread) is a measure that quantifies the additional yield a bond must provide over a risk-free Treasury bond to align its price with the present value of its cash flows.  
    
*   Unlike the nominal spread, the Z-spread accounts for the entire yield curve, providing a comprehensive evaluation of the risk-return tradeoff for bonds.  
    
*   A higher Z-spread indicates a greater credit risk, as investors demand more compensation to account for the perceived risk of holding a particular bond.  
    
*   By analyzing the Z-spread, investors can identify price discrepancies and assess the relative value of bonds, optimizing their portfolios for better risk-return balance.  
    
*   In the context of mortgage-backed securities (MBS), the Z-spread is valuable because it evaluates the bond's yield over the whole yield curve, addressing the complex and varied cash flows associated with these securities.

Also known as the [static spread](https://www.investopedia.com/terms/s/static_spread.asp), 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.

## Insights from the Zero-Volatility Spread (Z-Spread)

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](https://www.investopedia.com/articles/investing/073113/introduction-treasury-securities.asp) 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.

Unlike the nominal spread, which uses a single point on the Treasury yield curve, the z-spread considers the full curve. The nominal spread calculates the spread at one point to match a bond's cash flow value with its price.

Analysts use the z-spread to spot price [discrepancies](https://www.investopedia.com/ask/answers/052115/what-difference-between-optionadjusted-spread-and-zspread-reference-mortgagebacked-securities-mbs.asp) in bonds. It provides a realistic value by measuring the spread over the whole Treasury yield curve, unlike single-point metrics.

## Understanding the Zero-Volatility Spread Formula and Calculation

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: $\begin{aligned} &\text{P} = \frac { C_1 }{ \left ( 1 + \frac { r_1 + Z }{ 2 } \right ) ^ {2n} } + \frac { C_2 }{ \left ( 1 + \frac { r_2 + Z }{ 2 } \right ) ^ {2n} } + \frac { C_n }{ \left ( 1 + \frac { r_n + Z }{ 2 } \right ) ^ {2n} } \\ &\textbf{where:} \\ &\text{P} = \text{Current price of the bond plus any accrued interest} \\ &C_x = \text{Bond coupon payment} \\ &r_x = \text{Spot rate at each maturity} \\ &Z = \text{Z-spread} \\ &n = \text{Relevant time period} \\ \end{aligned}$

#### 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:

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

We can then calculate the [net present value](https://www.investopedia.com/terms/n/npv.asp) 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%.  

## Why Is the Z-Spread Important for Mortgage-Backed Securities?

The z-spread is particularly useful for analyzing mortgage-backed securities ([MBS](https://www.investopedia.com/terms/m/mbs.asp)) because it considers the spread across the entire yield curve, providing a clearer picture of the bond's value. Given the complexity and varied cash flows of an MBS, the z-spread helps investors assess the true yield and risk by discounting each cash flow at the correct rate.

## Can a Z-Spread Be Negative?

Yes, the z-spread on a bond can be negative if it's trading at a [premium](https://www.investopedia.com/terms/p/premiumbond.asp) compared with [risk-free](https://www.investopedia.com/terms/r/riskfreeasset.asp) Treasury securities. This would indicate that investors are willing to accept a lower yield because of factors such as superior credit quality, higher liquidity, or other attractive bond features.

## How Does the Z-Spread Help in Credit Risk Assessment?

The z-spread provides insights into the credit risk of a bond by comparing its yield to the risk-free Treasury yield curve. A higher z-spread indicates a higher perceived credit risk, as investors demand additional compensation for taking on that risk. Conversely, a lower z-spread suggests lower [credit risk](https://www.investopedia.com/terms/c/creditrisk.asp).

## How Can the Z-Spread Help Bond Investors?

In bond portfolio management, the z-spread helps assess the relative value of different bonds. By comparing the z-spreads of various securities, investors can identify relatively undervalued or overvalued bonds, optimize the portfolio's risk-return profile, and make more informed decisions about buying or selling bonds to achieve their investment objectives.

## How Does the Z-Spread Differ From the Option-Adjusted Spread (OAS)?

The Z-spread measures the spread over the Treasury yield curve without considering any [embedded options](https://www.investopedia.com/terms/e/embeddedoption.asp) in the bond, such as call or put options (e.g., a convertible bond, a callable bond, or a putable bond). If present, the option-adjusted spread ([OAS](https://www.investopedia.com/terms/o/optionadjustedspread.asp)) adjusts for these options, providing a more accurate reflection of the bond's true spread by accounting for the value of these options. Essentially, the OAS is the z-spread adjusted for option risk.

## 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](https://www.investopedia.com/terms/n/nominal_yield_spread.asp), 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.