Section 1.1
Definition and mechanics
Essential question
How does a zero-coupon bond work, and what is it for inside a structured product?
The vast majority of structured products contain a zero-coupon bond: a bond that pays no interest during its life and repays 100% of the nominal at maturity.
Definition
A zero-coupon bond is a debt security that pays no coupon during its life and gives the right to repayment of the nominal on a single date, its maturity date. Its value follows from one single future cash flow, discounted at the issuer's rate for the maturity concerned.
A zero-coupon bond can come in two shapes. Either it repays 100 at maturity and is therefore bought below 100 (at a discount), in a world of positive rates. Or it is bought at 100 and repays 100 plus the capitalised interest. Both forms describe the same mechanics; throughout the programme you will see that in most cases we think in terms of the "discount" version when building structured products.
How is its price calculated?
The value of a zero-coupon bond today is its expected value at maturity, discounted. The discount rate is the issuer's rate for the maturity concerned.
Suppose we want to price a zero-coupon bond that repays 100 in 5 years, with rates at 3.3% p.a.:
P = N × e−r×T
The whole difficulty of pricing a zero-coupon bond lies in the choice of the interest rate r. Sections 1.2 and 1.3 are devoted to it.
The head calculation
It is useful to be able to estimate the price of a zero-coupon bond quickly. For that we use an approximation: the discount is roughly r × T. Our 5-year ZC at 3.3%? Discount ≈ 16.5% (3.3% × 5), so price ≈ 83.5. That is off by 1.3%, but it gives you a fast estimate.
P ≈ N × (1 − r × T)
| Rate r | Maturity T | r × T | Head calculation | Exact formula | Gap |
|---|---|---|---|---|---|
| 1.0% | 2 years | 2.0% | 98.00 | 98.02 | −0.02 |
| 2.0% | 3 years | 6.0% | 94.00 | 94.18 | −0.18 |
| 3.0% | 5 years | 15.0% | 85.00 | 86.07 | −1.07 |
| 3.3% | 5 years | 16.5% | 83.50 | 84.79 | −1.29 |
| 5.0% | 7 years | 35.0% | 65.00 | 70.47 | −5.47 |
| 4.0% | 10 years | 40.0% | 60.00 | 67.03 | −7.03 |
| 5.0% | 15 years | 75.0% | 25.00 | 47.24 | −22.24 |
The higher the rates and the longer the maturity, the wider the gap between the estimate and reality.
A useful clarification
There is no such thing as one rate but a yield curve: the 5-year rate is not the 1-year rate. Every maturity has its own, and the curve can be flat, steep or inverted. The single r we use here is the rate of the product's maturity: for a single cash flow, it is the only one that matters.
How does a zero-coupon bond behave during its life?
A zero-coupon bond is liquid: the investor can exit before maturity, so the security has a price at every moment of its life. That price is computed on the residual maturity and on today's rates.
Assuming rates do not move, the valuation of the zero-coupon bond will head towards its redemption price in a straight line.
In practice the value of a zero-coupon bond never rises in a straight line, and its price will change not only with the passing of time but also with rate moves. For instance, if two years after the launch of our zero-coupon bond rates are no longer 3.3% but 4.5%, the price will not be 90.57%, it will be 87.37%.
A single cash flow, discounted: the "discount" version
Example: nominal 100, maturity 5 years, issuer rate 3.3% p.a. The solid dark line is the theoretical path, the one the security would follow if rates stayed at 3.3%: it reaches 100 without incident, passing through 90.57 after two years. The red line is the observed price: it moves continuously with rates, is worth 87.37 at two years after their rise to 4.5%, and moves back above the theoretical path when rates come down again. The two necessarily meet at 100 at maturity. You will handle these parameters yourself in the simulator of section 1.4.
What is a zero-coupon bond for inside a structured product?
This building block does not always serve the same final purpose. In a capital-guaranteed product it is what secures a given level of capital at maturity. In a product that sells optionality, a Reverse Convertible for instance, it acts as collateral for the commitment taken by the investor. See part 5.
In the coming chapters we will also come back in detail, for each of the products we study, on the role and the impact of the zero-coupon bond in the way the product works.
At this stage, let us simply keep in mind that this zero-coupon bond, or more commonly the interest-rate component, is an integral part of structured products.
Every product we will see in the rest of the programme rests on this building block. We will find it in each of them, with its two functions, guarantee or collateral, and with its sensitivities to market parameters: maturity, the level of rates and the issuer's funding.
Key message
A zero-coupon bond has one single cash flow: the nominal at maturity. Its price today is that nominal discounted; the gap between the two is the investor's return.
Question for reflection
Two zero-coupon bonds from the same issuer, one at 2 years, the other at 10 years. Which one is bought at the lower price, and why? Run both head calculations at 3%, then check with the formula: on which one does the approximation mislead you the most?
Question 1 / 5
A 5-year zero-coupon bond with a nominal of 100 is issued by a bank. What sets it apart from a conventional bond of the same maturity?
Question 2 / 5
The issuer's 5-year rate is 3.30%, under continuous compounding. What is the price of the zero-coupon bond with a nominal of 100?
Question 3 / 5
You want to estimate that price in your head, without a calculator. What does the approximation give, and in which direction is it wrong?
Question 4 / 5
Two years after issue, the issuer's rates have gone from 3.30% to 4.50%. The zero-coupon bond is quoted at 87.37. How should the gap with the theoretical path of 90.57 be read?
Question 5 / 5
For what purpose do we find this building block in the structured products studied in the rest of the programme?