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. In other words, each cash flow is discounted at the appropriate Treasury spot rate plus the Z-spread. The Z-spread is also known as a static spread.

## 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:



P

=

C

1

(

1

+

r

1

+

Z

2

)

2

n

+

C

2

(

1

+

r

2

+

Z

2

)

2

n

+

C

n

(

1

+

r

n

+

Z

2

)

2

n

where:

P

=

Current price of the bond plus any accrued interest

C

x

=

Bond coupon payment

r

x

=

Spot rate at each maturity

Z

=

n

=

Relevant time period

\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)2nCnwhere:P=Current price of the bond plus any accrued interestCx=Bond coupon paymentrx=Spot rate at each maturityZ=Z-spreadn=Relevant time period

For 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%. The formula would be set up as follows:



$104.90 =$

5

(

1

+

2.5

%

+

Z

2

)

2

×

1

+

$5 ( 1 + 2.7 % + Z 2 ) 2 × 2 +$

105

(

1

+

3

%

+

Z

2

)

2

×

3

\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

With the correct Z-spread, this simplifies to:



$104.90 =$

4.87

+

$4.72 +$

95.32

\begin{aligned} \$104.90 = \$4.87 + \$4.72 + \$95.32 \end{aligned}

$104.90=$4.87+$4.72+$95.32

This implies that the Z-spread equals 0.25% in this example.

### Key Takeaways

• The zero-volatility spread of a bond tells the investor the bond’s current value plus its cash flows at certain points on the Treasury curve where cash-flow is received.
• The spread is used by analysts and investors to discover discrepancies in a bond’s price.

A Z-spread calculation is different than a nominal spread calculation. 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 Zero-volatility spread (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 valuation of a security instead of a single-point metric, such as a bond’s maturity date.