## In this final look at spread, we define discount factors and the zero discount factor, before bringing all three articles together to define exactly what spread is

Last week we discussed BBSW and Swaps (click here for "What is spread? - Part 2"), and whilst explaining these items, the concept of a discount factor was brought up, as well as the concept of a forward rate (see the Appendix – calculating swap in last week’s article). This week’s article will extend the concept which will allow us to finally define what a spread is exactly. Again, there is a bit of maths involved, however don’t worry if you can’t follow the maths, hopefully the concepts will be clear enough for you to understand and you can leave the maths to us.

**What is a discount factor?**

A discount factor (DF) is a multiplier associated with a cashflow, which converts the value of that cashflow into its present value equivalent that is it converts the future cashflow into what it is worth right now. For example, if I invest in a bill of exchange (bank bill) with exactly one year to maturity paying 6%, where I receive $100 face value at maturity, my discount factor would be as follows:

We discount by dividing by 1 plus the coupon.

Therefore, for every $100 face value invested, I would be willing to pay $94.34, or:

Discount factors are useful because they allow us to price a security’s cashflows independently.

Many people are familiar with the concept of a yield, or an internal rate of return, which is often used to price a fixed rate bond or express its return. The yield essentially solves the sum of a series of cash flows to be received by the investor from the bond; thus, it weights each cash flow by where it occurs in time. It reflects the time value of money that is where the value of a $6 coupon in year one, is worth more to the investor than a $6 coupon in year two.

Although the yield method provides a way to compare investments, it can make it difficult to ascertain exactly how an investment’s returns are distributed over time. An alternative method we could employ would be to value each cash flow independently using discount factors, and then add these present values to arrive at a suitable price.

This might seem a little confusing, so let me give you an example:

Suppose that we have a fixed rate **bond** that pays semi-annually, with a fixed coupon of 6%, with exactly 1 year to maturity, which trades at par. Let us also assume the same issuer has a **bill of exchange**, which ranks the same in the issuer’s capital structure as the bond, which matures in exactly six months (i.e. when the next bond coupon is due) and is currently being priced at a yield of 5%.

The **bill of exchange** is priced as follows:

**DF= 1/(1+0.05/2)=0.9756** or for every $100 face value invested, one would be willing to pay $97.56 now.

On a yield basis, the **bond** would be priced as follows:

These two securities, despite being ranked the same in the capital structure, and thus theoretically displaying the same level of risk would produce two different answers as to what the discount factor should be at the first coupon date ( 6 months from today).

According the yield method, the present value of the $3 coupon due in six months is **$2.9126** ( $3 ÷ 1.03), however according to the bill price, the appropriate DF for that period is 0.9756, which would mean that when valued as a standalone cashflow, the $3 coupon should equal $3 × 0.9756 = **$2.9268**.

So which valuation method is correct? Since the two instruments rank pari-passu (rank the same in the capital structure), the bill of exchange maturity cashflow risk should be the same as the bond’s mid-year coupon risk. As an independent cashflow with a market price, the six month coupon on the bond should be valued using the discount factor derived from the bill of exchange price.

As we have market data to provide the appropriate discount factor for the first six months, we can effectively ‘hard code’ this into the calculation of the appropriate discount factor for the second six month coupon payment. So if the present value of the mid-year coupon is **$2.9268** based off the bill of exchange, then the present value of the final coupon and the bond’s principal must equal $97.0732 (i.e. $100 - **$2.9268**). Or the discount factor attributable to the final cash flow is $97.0732 ÷ $103 = 0.9425.

We have just stumbled onto two important concepts: zero coupon discount factors and bootstrapping.

Discount factors, as described above, are known as ‘zero-coupon discount factors’ as they are used to individually price cashflows (as opposed to discount factors generated from a yield). We look at the zero coupon discount factor, and the zero curve in more detail below.

Bootstrapping is the process whereby we use market information ( current security prices) to generate discount factors (and forward rates).

**Zero coupon discount factors and the zero curve**

The simple example above demonstrates how we can ascertain zero-coupon discount factors. Using the discount factors we can calculate zero coupon rates, from which we can then build a zero curve by calculating data points.

So how do we calculate a zero coupon rate? The relationship between discount factors and zero rates is as expressed below:

Using the example above, the half year zero-rate is calculated by:

The 1 year zero-rate is calculated by:

**Spread?**

The zero-coupon swap curve is derived using the published BBSW and Swap rate data combined with the bootstrapping method above described. If we used the discount factors from this curve to price market securities, in most situations, we would find that the swap curve does not solve for the security’s current market price. Why is this so? Previously we noted that the zero coupon swap curve reflects both the rate at which the major banks are indifferent between receiving or paying a fixed or a floating rate, and that it **also** imbeds a risk premium for the major bank as a swap counterparty (as there is a risk they may not honour their swap agreement position in the future).

Given this, it is unlikely that any individual security will price at exactly the same risk level; hence, a margin needs to be added to each zero rate – it is this margin that is the spread we often talk about.

Let’s use the above example to calculate a spread:

- We know that we have a bond with exactly 1 year to maturity that pays 6% semi-annually, and that is currently priced in the market at par ( $100)
- Let’s assume the zero-coupon 6 month BBSW-Swap zero rate is 3%
- Let’s assume the zero-coupon 1 year BBSW-Swap zero rate is 5%

The zero-coupon 6 month BBSW-Swap discount factor is calculated as follows:

The zero-coupon 1 year BBSW-Swap discount factor is calculated as follows:

If we price the bond using the BBSW-Swap zero discount factors we should pay the following:

**CF(1)×DF(1)+CF(2)×DF(2)=$3×0.9852+$103×0.9518=$2.96+$98.04=$100.99**

So why is the market paying a lower amount of $100 for this security? Well, compared to the BBSW-Swap curve, the security is perceived to be riskier; hence, there is a margin being added to the zero rates. We need a numeric solver to calculate this margin, but as it turns out the margin that solves for price given the BBSW-Swap zero rates supplied is 1.03%. Alternatively stated, the market demands an extra 1.03% over BBSW-Swap for the perceived risk of the issuer.

Let’s confirm the margin.

The zero-coupon 6 month discount factor is calculated as follows:

The zero-coupon 1 year discount factor is calculated as follows:

If we price the bond using the BBSW-Swap zero rates with the margin applied we should pay the following:

**CF(1)×DF(1)+CF(2)×DF(2)=$3×0.9802+$103×0.9423=$2.94+$97.06=$100.00**

**Conclusion**

So this second calculation proves the 1.03% spread demanded by the market to reflect the risk of this issuer. Often we would quote this issuer to clients as returning BBSW + 103bps (where one basis point – bps – is equal to 0.01%). As we noted above, the spread reflects the risk of the issuer. The higher the risk the issuer represents, the higher the spread the market will demand, and the higher the return to the investor will be. This is the basic risk and reward axiom.

This concludes our look at “What is spread?” Whilst there is a lot of maths involved (but don’t be afraid of the formulas in this paper, as I noted, you can leave the maths to us), hopefully you now have a greater understanding of this central tenant of fixed income.

If you understand the basic concept of a spread, you are able to analyse and compare fixed and floating rate investments – but we will look at that at a future date.