Outline a one-factor interest rate model and give examples of factors that are commonly used in it.
The idea
A one-factor interest rate model assumes that a single random variable drives every rate and bond price, so once you know how that factor moves you know how each security reprices. The factor is usually a chosen yield or rate: the short-term rate, the yield on a benchmark par bond such as the 10-year, or one spot rate. The model then either shifts all yields by the same amount, a parallel shift, or moves each yield by a fixed multiple of the factor, called a yield beta. Its appeal is that one number summarizes the rate scenario, so hedging is simple. Its weakness is that changes in slope and curvature are ignored, so a position hedged against the factor can still lose when the curve twists. A factor is a risk driver such as a yield, never a contractual term like a coupon or maturity.
Formulas
Change in value under a one-factor model with yield betas
ΔV = −DV01 × β × Δf
- DV01
- the position's price change per basis point of its own yield
- β
- the yield beta, the bond's yield change per unit factor change
- Δf
- the change in the factor, in basis points
With a parallel-shift model β is one. Sum across positions, with short positions entering negatively; slope changes are not captured.
A practice question
In fixed-income risk management, what defines a one-factor interest rate model?
- Each bond is priced from its own independent yield, so the model needs one factor per security.Why not: Confuses one factor in total with one factor per bond; a one-factor model has a single common driver.
- A single random variable drives changes in all rates and prices, so one move in that factor fixes every security's price change. - correct
- Interest rates are assumed constant, so no random factor is needed to value a bond.Why not: Describes a deterministic rate assumption; a one-factor model is stochastic, with one source of randomness.
- A bond is valued using only its coupon rate as the input that changes over time.Why not: Treats a contractual term as the risk driver; a coupon is fixed and does not move with the market.
A one-factor model reduces the whole rate environment to one random variable, such as a chosen yield. All prices are then functions of that single factor, which is why one DV01 or one duration summarizes the exposure.
A harder one
A portfolio is long a 2-year bond position with DV01 of USD 9,000 and short a 10-year bond position with DV01 of USD 9,000. Under a one-factor model with parallel shifts the portfolio is hedged. The curve then flattens, with 2-year yields up 40bp and 10-year yields up 10bp. Which describes the model's prediction and the actual outcome?
- The model predicts no gain or loss, but the portfolio loses USD 270,000 because the slope change is not captured by the factor. - correct
- The model predicts no gain or loss, and the portfolio also makes no gain or loss, since the DV01s offset.Why not: Assumes DV01 offsets work for any curve move; they offset only for the parallel shift the model assumes.
- The model predicts no gain or loss, but the portfolio gains USD 270,000 because the short position dominates.Why not: Sign error: the long 2-year position loses 9,000 x 40, more than the short 10-year gains 9,000 x 10.
- The model predicts no gain or loss, but the portfolio loses USD 450,000 because both legs lose.Why not: Adds the two moves as losses; the short 10-year position gains when yields rise, so it offsets part of the loss.
Long 2-year loses 9,000 x 40 = 360,000. Short 10-year gains 9,000 x 10 = 90,000. Net is a loss of 270,000. The parallel-shift model sees zero because it has no slope factor.
The hardest tier
A trader hedges a long 30-year bond position with a short 2-year position of equal DV01, USD 8,000 each, under a one-factor parallel-shift model. Over the next month the 2-year yield rises 30bp while the 30-year yield is unchanged. The profit or loss on the hedged position is closest to:
- A loss of USD 240,000Why not: Sign error: the short 2-year position gains when its yield rises, and the long 30-year position has no price change.
- A gain of USD 240,000 - correct
- Zero, because the hedge removes interest rate riskWhy not: Assumes a DV01 hedge works whatever the curve does; it protects only against the parallel moves assumed by the one-factor model.
- A gain of USD 2,400Why not: Treats 30bp as 0.30 and multiplies by DV01, instead of using 30 basis points.
The short 2-year leg gains 8,000 x 30 = USD 240,000. The long 30-year leg is unchanged. A slope change is a second factor that the model does not hedge.
In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.
Start free →More in Applying Duration, Convexity, and DV01
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