Bond Pricing: Z-Spread

This is a series of short posts examining the bond markets and some of the key pricing, risk and quoting concepts (with EXAMPLES!) –
        The Yield Curve
        Duration/Convexity and DV01
        Spread Quoting
        Z-Spread
        Roll PnL

An alternative to the spread-to-benchmark for a credit-risky bond that tries to capture the full shape of the yield curve is the Z-Spread.

Since govvies are ‘risk-free’, we should be able to recover the market price of a govvie by discounting each of its cashflows using the discount factor coming from the point on the yield curve that the payment occurs*. If we perform the same calculation for a higher-yielding corporate bond, but offset the yield curve that we use for discounting by a fixed amount \Delta at each cashflow payment such that the sum of the PVs of the cashflows matches the bond’s traded price, then this \Delta is the bond’s Z-Spread (in this case, interpolation IS used to find discount factors at each time).

A yield curve implied from govvies, and another offset by a constant Z-Spread of 1.1%. On the right axis, the corresponding discount curves

In the graph above, we show the effect of this parallel shift of the yield curve on the discount factors that we use to discount the payments at each time coming from the risky coupon. A larger spread between the two yield curves leads to a more rapidly falling discount curve and lower prices for credit bonds.

Putting this together, we can find the Z-Spread by fixing \Delta to solve the equality

(1)   \begin{align*} C(t) &= \sum^{N}_{i=1} CF(t_i) \cdot \frac{1}{\Bigl(1 + ( r_{gy}(t_i) + \Delta ) \Bigr)^{t_i}} \end{align*}

where C(t) is the market price of the bond, CF(t_i) are the cashflows at each time t_i, and r_{gy}(t_i) is the govvie yield at time t_i interpolated from the yield curve (compare this to the equality we had to solve to calculate YtM, discussed in an earlier post).




What is the advantage of the Z-Spread over the plain spread to benchmark? Firstly, as we interpolate discount factors to each coupon payment date, we come up with a measure that behaves continuously as maturity and coupon payment dates increase, where spreads jump as benchmarks bonds change along the curve. Secondly, the Z-Spread gives us some measure of how much pickup we are getting over the risk-free payment we would receive for our bond, so Z-Spread gives us a good quantity for comparing similar maturity bonds from different issuers or bonds at different points on the curve from a single issuer (‘Relative Value’ ideas involving going long one bond and short another are often quoted in terms of the Z-Spread pickup that the Asset Manager will achieve). Practically, as seen below, spread and Z-Spread are typically fairly similar when maturity date and payment date of the benchmark and the corp bond are close to overlapping.

*Note the subtle implication here – the yield curve shows the return for a zero coupon bond, coupon-bearing bonds will display a slightly different yield than the yield curve implies for their maturity, as discussed in an earlier post. This effect is very, very small except for long-dated bonds with large coupons, as shown in the example.

Example: assuming the yield curve for USD has the values shown in the table in the image below, calculate the Z-Spread for a BB-rated corp bond paying a 6% coupon with 4 years left to maturity and trading at $104 (paying biannual coupons, assume it recently paid a coupon and this is reflected in the pricing).

Data to go with the Z-Spread example

Answer: Since the bond pays bi-annual coupons, we’ll need to discount a $3 payment at each 6m interval out to 4 years. The table is missing a 3.5y point so let’s interpolate linearly between 3y and 4y, giving 3.1% ytm for that point. Discount factors can be approximated as (1+r_y+\Delta)^t where t is time in years, and the sum-product of these with the relevant cashflow payments should be the traded price of $104, which hopefully leads you to a calibrated Z-Spread of \Delta=1.804%

Bonus: Consider the govvies used to build the yield curve. Imagine that they all paid 5% coupons (biannual payments). Discount each payment according to the yield curve to determine what their traded price should be today, and then use the YtM formula from the Yield Curve post to determine their yield to maturity. Observe that is is close to, but not the same as, the yield curve’s yield (eg. for the bond maturing in 3y, the YtM should be 2.96% – so note that coupon bearing bonds should have a YtM BELOW that of a ZCB for an upward-sloping yield curve, reflecting that these bonds return a portion of your investment earlier than maturity as coupons, which the market is pricing ass less risky).

Bond Pricing: Spread Quoting

This is a series of short posts examining the bond markets and some of the key pricing, risk and quoting concepts (with EXAMPLES!) –
        The Yield Curve
        Duration/Convexity and DV01
        Spread Quoting
        Z-Spread
        Roll PnL

Corporate bonds give a higher yield than govvies, reflecting their increased credit risk – the chance that the company defaults before the investor receives – their payment on the bond. The difference between these two is called the ‘Credit Spread’ – and typically the lower the credit rating of the company, the higher the yield investors will demand to purchase the bonds, so this spread will be larger. Because of this, corporate bonds are sensitive to two risk factors – interest rates in the bond’s currency, and the credit risk of the issuer.

If govvie yields (and hence interest rates) in a given currency increase, existing fixed rate bonds will be less valuable and the price will fall in order to increase yields to reflect the new govvie yield. Similarly, if the credit rating of the issuer declines, bonds it has issued see price declines as the spread to govvie bonds increases to reflect the new, increased credit riskiness of the bonds.

Investors seeking exposure to interest rates will typically buy govvies – and some will go long corporate bonds to gain an additional yield pickup over the risk-free govvies. On the other hand, specialist credit investors often seek exposure only to the credit riskiness of corporate bonds, and for these investors the sensitivity to interest rate risk is unwanted. To hedge this risk out, they will often trade bond ‘switches’, going long a corporate bond and simultaneously short the govvie benchmark bond with the closest maturity to the corporate bond in question, to neutralise the interest rate risk.

An example corp curve showing the spread over govvies, which typically increases with maturity, and is steeper for lower-rated bonds

Because switches are traded so often, for most highly-rated investment grade bond, dealers will quote them not by price or by yield, but instead by the spread between a bond and its govvie benchmark. The investor typically transacts both legs of the switch simultaneously with the dealer. A practical advantage of this way of quoting is that even if there are interest rate movements between the time of quoting and the time the trade is transacted, the spread between the two bonds is highly insensitive to them and rarely needs requoting (interest rates typically move much more frequently than credit spreads).

An example corp curve is shown above with the corresponding govvie curve in the diagram. Note that no interpolation is performed – a govvie with a nearby maturity is chosen as a benchmark, which can lead to small maturity mismatch.




Once a deal is agreed, the investor and the dealer must still agree on the Spread-to-Price conversion. This is done in two steps – the spread between the corporate bond and the govvie bond, plus the yield of the govvie, gives the equivalent YtM r_y for the corporate bond. This is then converted into a price by discounting all of the bonds payments using r_y as a risky discount factor.

Example: an AAA-rated corporate bond with three remaining annual coupon payments of 4%, which also returns notional of $100 at maturity (assume the most recent coupon was paid recently and the price already reflects this), is quoted bid-offer at 65-60 bps spread to the govvie benchmark which is quoted at 2.2% YtM, expires in 2.75 years, and pays biannual coupons at 1.8%. What is the approximate cash price to buy the corporate bond?

The only relevant quantities here are the corp spread to govvie, the govvie YtM, and the payments remaining on the corp. Since the govvie has a YtM of 2.2%, and the corp ask-quote is 60 bps (we’re buying the bond*, so the ask price is the relevant spread), we pay a price that implies a YtM of 2.8%

    \[\text{Price} = \frac{\$4}{(1 + 0.028)} + \frac{\$4}{(1 + 0.028)^2} + \frac{\$104}{(1 + 0.028)^3}\]

Performing the calculation this way round doesn’t require a solver – plugging in the r_y value from the quotes, we get a price of $103.41.

Note that the govvie here has a YtM higher than its coupon payments so is trading below par – despite its credit riskiness, the corp bond is more valuable due to its higher coupon payments. In yield terms the relative price relationship is much clearer.

*An interesting point to note from this example is that we typically think of dealer bid prices being lower than offer prices. This is of course true with spread-priced bonds, but as a lower price corresponds to a larger yield/spread, the larger spread is actually the bid price (ie. the YtM/spread that the trader would receive if she buys the bond from you)