Section 1.2
The three price drivers
Essential question
Which factors let us calculate the price of a zero-coupon bond?
As we saw in the previous chapter, the price of a zero-coupon bond is a function of its maturity and of the discount rate.
That discount rate is in fact made up of two distinct parts: the risk-free rate and the issuer's risk premium. The price of a zero-coupon bond therefore depends on three factors.
r = risk-free rate + issuer's risk premium
Three factors, three sensitivities
The price of a zero-coupon bond reacts to a move in each of these three independent factors. A longer maturity, a rise in rates or a perceived deterioration of the issuer all push the price down.
1 · Maturity
Maturity ↑ → price ↓
The price falls as the maturity lengthens: every extra year moves the repayment one more compounding period away. Extend the maturity by one year, all else equal, and the price of the 100 / 5-year zero-coupon bond at 3.30% goes from 84.79% to 82.04%.
2 · The risk-free rate
Rate ↑ → price ↓
The price falls when the reference rate of the currency rises. If the risk-free rate rises by 100 basis points, the price of the same 100 / 5-year zero-coupon bond goes from 84.79% to 80.65%.
3 · The risk premium
Premium ↑ → price ↓
The price also falls when the risk premium increases. If the risk premium rises by 100 basis points, the price of the same 100 / 5-year zero-coupon bond goes from 84.79% to 80.65%.
Table of the price of a zero-coupon bond by maturity and applicable rate
| Maturity | 0% | 1% | 2% | 3% | 4% | 5% |
|---|---|---|---|---|---|---|
| 1 year | 100.00 | 99.00 | 98.02 | 97.04 | 96.08 | 95.12 |
| 2 years | 100.00 | 98.02 | 96.08 | 94.18 | 92.31 | 90.48 |
| 3 years | 100.00 | 97.04 | 94.18 | 91.39 | 88.69 | 86.07 |
| 5 years | 100.00 | 95.12 | 90.48 | 86.07 | 81.87 | 77.88 |
| 7 years | 100.00 | 93.24 | 86.94 | 81.06 | 75.58 | 70.47 |
| 10 years | 100.00 | 90.48 | 81.87 | 74.08 | 67.03 | 60.65 |
What about negative rates?
Nothing requires r to be positive. In a negative-rate environment, which the euro area and Switzerland went through from 2015 to 2022, the price of a zero-coupon bond can exceed 100% of the nominal: at −0.40% over five years it is worth 102.02. The investor then pays more than what will be recovered at maturity, and the budget available for options becomes negative.
Rate sensitivity and convexity
Saying that the price falls when rates rise is not enough: we need to be able to assess by how much, in order to measure the product's exposure to that factor and the associated risk. The unit of measure used in rates markets is the DV01 (dollar value of 1 basis point): the change in the price of the zero-coupon bond for a rise of one basis point, that is 0.01%, in the discount rate.
On our reference zero-coupon bond, 100 at 5 years at 3.30%, one extra basis point of rate takes the price from 84.7894% to 84.7470%. The DV01 is therefore about 0.042% of the nominal, roughly four cents per 100 of nominal. It is proportional to the residual maturity: a 10-year zero-coupon bond has a DV01 about twice as high as a 5-year one.
The DV01 is computed directly from the price and the residual maturity, and the price change for any rate move follows by multiplying it by the number of basis points of the move.
DV01 ≈ P × T × 0.0001
| Step | Calculation | Result |
|---|---|---|
| Starting price | 100 × e−0.033 × 5 | 84.79% |
| DV01 | 84.79 × 5 × 0.0001 | 0.0424% |
| Size of the move | Δr | 50 bp |
| Estimated change | − DV01 × 50 | − 2.12% |
| New estimated price | 84.79 − 2.12 | 82.67% |
| Exact recalculated price | 100 × e−0.038 × 5 | 82.70% |
The estimate announces 82.67% when the exact price is 82.70%: three hundredths of a percent of difference for a 50 basis point shock. For small rate moves, the sensitivity is enough to estimate the new price. The larger the shock, however, the further this approximation drifts from the real price. The reason is convexity.
Indeed, sensitivity is not stable. The DV01 is computed from the price, and the price changes as soon as rates move. On our 10-year bond, a rise of 100 basis points costs 7.19% of price at the outset; but once the price has fallen to 65.05, the next 100 basis points only cost 6.51%. Each slice of increase weighs a little less than the one before, and symmetrically each slice of decrease earns a little more.
The DV01 therefore describes the slope of the curve at the point where we stand. It is exact for an infinitesimal move, and it degrades as soon as the move widens, because it assumes a constant slope. Convexity is precisely the measure of that drift: it quantifies how fast the sensitivity itself decreases when rates rise. Graphically, the price / rate relationship is a curve bending upwards, and the DV01 is its tangent: the straight line always passes below the curve, in both directions.
The gap therefore always works in favour of the holder, and it grows with the maturity as well as with the size of the shock.
Price of a 10-year zero-coupon bond as a function of the rate
Nominal 100, maturity 10 years, rates from 0 to 10%. The exact price curve bends upwards; the tangent at the point 3.30% / 71.89 is the straight line traced by the DV01. The two coincide at the point of tangency and diverge on either side, the line always passing below. At 8%, the tangent announces 38.10 when the real price is 44.93, nearly 7% of difference: this is the limit of the linear approximation, and the reason why convexity has to be taken into account on large moves.
To keep in mind for what follows
A 10-year capital-guaranteed product contains a 10-year ZC, whose DV01 is twice that of a 5-year one. It is therefore very sensitive to rate moves, a fact that the words "capital-guaranteed" often make the investor forget when looking at a portfolio statement after a rise in rates. We come back to this in section 1.6.
Key message
Three parameters, maturity, the risk-free rate and the risk premium, all push the price down when they rise. On the secondary market the price converges towards 100 but reacts in the opposite direction to rates.
Question for reflection
You hold a 10-year capital-guaranteed product issued two years ago, when rates were at 1%. They are at 4% today. Your statement shows a value well below 100 even though the underlying has not moved. The mark to market loss is real and immediate. What becomes of it if you hold the product to maturity?
Question 1 / 5
What is the discount rate used to value a zero-coupon bond issued by a bank made of?
Question 2 / 5
The 5-year risk-free rate rises by 100 basis points, the issuer's risk premium staying unchanged. The price of the 100 / 5-year zero-coupon bond goes from 84.79% to 80.65%. What would a 100 basis point rise in the risk premium alone have given?
Question 3 / 5
What does the DV01 of a zero-coupon bond measure?
Question 4 / 5
Two zero-coupon bonds from the same issuer, one at 5 years, the other at 10 years. What can be said about their DV01?
Question 5 / 5
On the 100 / 5-year zero-coupon bond at 3.30%, the DV01 estimate announces 82.67% after a rise of 50 basis points, whereas the exact price is 82.70%. Where does that gap come from?