Discount factors and rate conventions

Interest-rate curves provide the discount factors used to value future payments and the forward rates used to project floating coupons. QuantSupport exposes these queries through InterestRatesTermStructure<T>, so a pricer can use a calibrated curve, a flat curve or a composite funding curve through the same interface.

A curve’s rate conventions determine how discount factors translate into quoted rates. Understanding these conventions is useful both when constructing a market and when interpreting a forward-rate query.

Write \(P(0,t)\) for the discount factor from reference date zero to date \(t\). For an interval with year fraction \(\tau\),

\[ P(t_1,t_2)=\frac{P(0,t_2)}{P(0,t_1)},\qquad L(t_1,t_2)=\frac{P(0,t_1)/P(0,t_2)-1}{\tau}. \]

The second expression is a simple annualized forward rate. The equivalent continuously compounded forward is \(-\log(P(t_1,t_2))/\tau\). Both describe the same accumulation; their numbers need not agree. A curve’s interpolation clock and a coupon’s accrual basis must also be distinguished when translating between them.

Rate definitions

RateDefinition combines day counter, compounding and frequency. InterestRate uses that definition to translate a rate into compound and discount factors. For year fraction \(\tau\), principal conventions are

\[ C_{\rm simple}=1+r\tau,\qquad C_{\rm periodic}=(1+r/m)^{m\tau},\qquad C_{\rm continuous}=e^{r\tau}. \]

Mixed simple/periodic conventions additionally switch behavior according to their interval rule. Frequency matters for periodic compounding even when payment dates follow another schedule. The first-swap example illustrates why identical displayed percentages under different conventions are not equal financial prices.

Day counters include ACT/360, ACT/365, 30/360 variants, Actual/Actual and Business/252. The day counter measures the accrual fraction between dates, while the calendar determines any business-day adjustment to those dates.

ActualActual splits at calendar-year boundaries and includes every complete intervening year. Thirty360US adjusts February month-ends after its 31st-day adjustments, differing from common US/NASD rules. Validate these conventions against the required dates; their API references describe the implemented rules. The examples here use ACT/360 or ACT/365.

Curve representations

DiscountTermStructure<T> stores dated discount factors. Its first factor is one at the reference date, and date/value vectors must agree. Pricing values are interpolated on year fractions. With log-linear interpolation,

\[ \log P(t)=(1-w)\log P(t_i)+w\log P(t_{i+1}), \quad w=\frac{t-t_i}{t_{i+1}-t_i}, \]

so the instantaneous forward is constant between adjacent nodes. Linear and cubic-spline choices describe different interpolation models. Extrapolation is a separate permission: log-linear continuation extends the terminal forward, while linear/cubic choices use their implemented linear extrapolation. Disabled extrapolation makes out-of-domain queries errors.

FlatForwardTermStructure<T> represents the term structure with one rate and rate definition. An explicit ZeroRateTermStructure interpolates zero rates, and tenor-based zero curves resolve pillar dates from the reference date. Choose the representation and interpolator together, since they determine the implied shape between pillars.

A spread representation can express a continuously compounded adjustment:

\[ s(t)=-\frac{\log(P_{\rm target}(t)/P_{\rm base}(t))}{t}, \qquad P_s(t)=e^{-s(t)t}. \]

CompositeTermStructure multiplies component factors, \(P=P_bP_s\). Matching reference dates and day counters keep the component clocks consistent. This is useful for separating benchmark and funding inputs, and is supported through serializable curve configurations as well as direct construction.

Pricing nodes and risk pillars

A calibrated curve has discount-factor nodes used for interpolation and market quote pillars used for sensitivity reporting. Calibration connects the two: changing a deposit, swap or bond quote changes the solved discount factors and hence the value of a trade. On some directly constructed curves, the pricing nodes themselves are the independent risk inputs.

Pillars<DualFwd> exposes labels, values and tape-registration operations. Calibration supplies a Jacobian to connect pricing nodes to independent market inputs. A derivative labeled OIS_USD_SOFR_5Y is therefore a derivative with respect to that quoted contract, not necessarily a five-year zero rate or a discount-factor node. The next chapter derives that connection.