Yield Curve Arithmetic

Curves are deflators, not rates

A yield curve is fundamentally a price deflator — it tells you what a future dollar is worth today. When you combine two curves (e.g. a base curve and a spread curve), you're compounding two deflators: a dollar deflated by the base curve and then by the spread curve. Mathematically, discount factors multiply:

\[D_{\text{combined}}(t) = D_{\text{base}}(t) \cdot D_{\text{spread}}(t)\]

Multiplying discount factors is equivalent to adding rates in continuous (log) space — exp(a) × exp(b) = exp(a + b) exactly. + on curves means "compound the deflators."

Demonstration

using FinanceModels, FinanceCore

base   = Yield.Constant(0.05)   # 5% annual effective
spread = Yield.Constant(0.02)   # 2% annual effective (e.g. credit spread)
combined = base + spread

# The combined curve multiplies discount factors:
t = 10.0
df_base   = discount(base, t)      # 1.05^-10 = 0.61391
df_spread = discount(spread, t)    # 1.02^-10 = 0.82035
df_combined = discount(combined, t) # 0.50362

df_base * df_spread  # 0.50362 — matches combined

# The combined rate is NOT simply 7%:
discount(Yield.Constant(0.07), t)  # 0.50835 — slightly different

# The difference: compounding 5% and 2% as deflators gives
# (1.05 × 1.02) - 1 = 7.1%, not 7%:
1.05 * 1.02  # 1.071

Why the difference?

Adding nominal rates ignores the cross-term. If you invest $1 at the base rate and the spread applies on top:

\[(1 + r_{\text{base}})(1 + r_{\text{spread}}) = 1 + r_{\text{base}} + r_{\text{spread}} + r_{\text{base}} \, r_{\text{spread}}\]

The cross-term (0.05 × 0.02 = 0.001) is what gets dropped when you simply add 0.05 + 0.02 = 0.07. Over a single year this is 10 bps; over 10 years it compounds.

In continuous space, the cross-term doesn't exist — exp(a) × exp(b) = exp(a + b) exactly — so adding continuous zero rates is the correct way to combine deflators. curve_a + curve_b creates a CompositeYield that, at each time t:

1. Extracts continuous zero rates from each curve: $z_i = -\log D_i(t) / t$
2. Adds the zero rates: $z = z_1 + z_2$
3. Returns the combined DF: $D(t) = e^{-z \, t}$

Subtraction works analogously — curve_a - curve_b divides discount factors, giving the implied spread curve.

Working with spread curves

Use curve subtraction to find the spread between two curves:

base   = Yield.Constant(0.05)
target = Yield.Constant(0.07)

# The spread that, when added to base, reproduces target's discount factors:
spread = target - base
zero(spread,1)  # Continuous(0.01887...) — not exactly 2% in nominal terms

# Verify round-trip:
discount(base + spread, 10.0) ≈ discount(target, 10.0)  # true

# You can also add the spread Rate directly:
discount(base + spread.rate, 10.0) ≈ discount(target, 10.0)  # true

This pattern — using curve subtraction and adding the result back — ensures you're always working in the correct space, regardless of which rate convention (Periodic, Continuous, etc.) the inputs use.

Curves versus rates

The key is that combining Rates is not the same thing as combining curves (AbstractYieldCurves):

a = Yield.Constant(Periodic(0.05,1) + Periodic(0.02,1))
b = Yield.Constant(Periodic(0.05,1)) + Yield.Constant(Periodic(0.02,1))
a != b # true

discount(a, 10)  # 0.50835 — rate addition (drops cross-term)
discount(b, 10)  # 0.50362 — deflator compounding (correct)

This is the point made above - combining discount curves results in a different discount path than a curve of combined rates. See the Migration Guide for details on how this changed in v5.

Scaling curves: * and /

curve * α scales every continuous zero rate by a scalar factor α. This is the "power-of-DF" operation:

\[D_\alpha(t) = e^{-\alpha \, z(t) \, t} = \bigl[D(t)\bigr]^\alpha\]

This creates a ScaledYield.

A common use case is after-tax yields. If the tax rate is 21%, the after-tax curve is curve * 0.79:

pretax = Yield.Constant(Continuous(0.05))
aftertax = pretax * 0.79

discount(aftertax, 10) ≈ exp(-0.05 * 0.79 * 10)  # true

Division is the inverse — curve / α scales by 1/α, useful for grossing up to a pre-tax equivalent:

aftertax = Yield.Constant(Continuous(0.0395))
pretax = aftertax / 0.79

discount(pretax, 10) ≈ exp(-0.0395 / 0.79 * 10)  # true

Multiplication is only defined between a curve and a scalar. Multiplying two curves together is not a meaningful operation and will raise a MethodError.

Convenience: curves with scalars and rates

You can add or subtract a scalar or a Rate directly to/from a curve. The scalar is wrapped in a Yield.Constant:

m = fit(Spline.Linear(), ZCBYield.([0.04, 0.05, 0.06], [1, 5, 10]), Fit.Bootstrap())

# These are equivalent:
m + 0.01
m + Yield.Constant(0.01)

# With an explicit rate type:
m + Continuous(0.01)
m + Yield.Constant(Continuous(0.01))

Operation summary

Performance

CompositeYield and ScaledYield evaluate both underlying curves on every call to discount. If you are using a composite curve in a hot loop, consider pre-computing the combined zero rates at the tenors you need and fitting a single spline curve to the result.