How the tool estimates the call sensitivity: every knob, every assumption.
The calculator produces an assumption-dependent embedded-call teaching estimate. Under the hood it is a small Hull-White lattice teaching engine that decomposes a high-yield bond into bullet minus an American-style issuer call, approximated on a discrete exercise grid. This page walks through the model, the eight variables, the strike function, the decomposition math, the assumptions, and a worked example matching the base case.
The doctrine (start here)
An optional-redemption clause gives the issuer a contractual right to retire its debt. A make-whole formula sets the price of exercising that right before the first scheduled call date. It protects the lender relative to an otherwise cheaper early call, but it is not an absolute prohibition on redemption and does not guarantee that early exercise is uneconomic.
The economic analogy is a borrower-held call on its own debt. Identify the permitted exercise dates, redemption formula, accrued-interest entitlement, notice requirements and exceptions before estimating its value.
The calculator compares three bonds with identical coupons, maturity, discounting and rate scenarios:
- Bullet: no optional redemption at all.
- Post-NC only (noMW): no redemption before first call, then the specified p50 / p25 / par schedule. This is a model counterfactual, not a claim that deleting a legal make-whole clause creates hard non-call protection.
- Full: pre-first-call redemption at the modeled make-whole price, followed by the same step-down schedule.
All numbers remain teaching estimates. Equity claws, special redemptions, tenders, partial calls, tax calls and transaction-specific exceptions are outside this model.
The model
The rate process is one-factor Hull-White:
The time-dependent shift fits the selected risk-free discount factors on the lattice. This is an internal curve fit, not market calibration: the tool does not load an observed yield curve, calibrate rate volatility or infer credit dynamics from prices.
The analytic zero-coupon formula supplies the conditional risk-free discount factors used in the make-whole teaching approximation. The implementation uses stable exponential differences and the explicit zero-mean-reversion limit. Zero rate volatility is a supported deterministic benchmark.
The recombining trinomial tree has 48 time steps per year. Its conditional first two moments use the standard short-rate discretization, with inward branching at the mean-reversion boundary and an Arrow-Debreu forward fit to the selected initial curve. Transition probabilities are checked for nonnegativity and unit sum. Coupons and first-call/step-down dates fall exactly on the grid.
Exercise is allowed at each interior grid date, including the first-call date, but not at issue or maturity. This is a Bermudan-grid approximation to an American-style right, not continuous-time exact pricing. The grid and conditional analytic strike introduce discretization error. Base-case totals at 24, 48, 96 and 192 steps per year are 5.22294, 5.22166, 5.22078 and 5.22025 points respectively; this example is not a universal error bound.
The order matters: an exercise decision never cancels a coupon due on the same date. Between coupon dates the redemption payment includes accrued interest. The implementation is independently checked against direct discounted cash flows in the deterministic limit and an analytic semiannual bullet benchmark. The historical standalone reference used the previous coupon convention and is not a current validation oracle.
Numerical reference: QuantLib Hull-White curve fitting and callable-bond coupon/exercise ordering. The browser engine is independently authored; a shared convention is not a claim of identical engines.
The eight variables
Each slider maps to one input to the Hull-White engine. Ranges are the supported public controls; the pure engine validates its inputs rather than silently accepting arbitrary terms.
On smaller screens, scroll the table horizontally to see every column.
| Slider | Symbol | Range | What it does |
|---|---|---|---|
| Tenor | T | 3-10 yrs | Time to legal maturity. Sets the horizon of the tree. |
| Non-call period | n | 1-5 yrs | Years during which the strike is the HW make-whole PV, not par. |
| Coupon | C | 3-12% | Annual fixed coupon rate, paid in equal semiannual installments. |
| Risk-free rate | rf | 1-8% | Anchor of the flat initial curve. HW is fitted to PM(0,T)=exp(−rfT). |
| Credit spread | s | 0-1500 bps | Initial continuously compounded discount spread. The selected tightening reduces it over time; no explicit default or recovery is modeled. |
| Rate volatility | σr | 0-250 bps | HW short-rate vol in absolute bps. Sets the space step Δx = σr√(3Δt). |
| Mean-reversion | a | 0-1.00 | HW pull-back speed. Zero gives the Ho-Lee limit; higher values make rate shocks fade faster. |
| Spread tightening | Δs | 0-400 bps | Deterministic overlay, capped at initial spread. Applied tightening increases linearly to first call, then stays constant; ending spread is never negative. |
The five pre-set structures (5NC2, 6NC3, 7NC3, 8NC3, 10NC5) are shortcuts for the (T, n) pair. The other six sliders keep whatever value they had.
The strike function
This example protects coupon payments through first call, plus the first-call redemption price, not merely principal at final maturity. Per 100 par, p50 = 100 + 50% × annual coupon amount, p25 = 100 + 25% × annual coupon amount. At 7%, these are 103.5 and 101.75. They are total clean redemption prices, not premiums of 50 or 25 points.
A publicly filed example of redemption terms sets a 1% minimum premium, protects payments to the first-call date, excludes already-accrued interest from those protected payments and pays accrued interest separately. That is a documented example, not a universal template.
The accrued portion is removed from the next future coupon before discounting and paid at redemption. Any coupon due exactly on the redemption date is accounted for separately outside the call cap. The 101 floor applies to the clean redemption amount.
At and after first call the clean strike is p50 for one year, p25 for the next year, then par; maturity itself repays par plus its final coupon. A short tenor can mature before all step-down stages have occurred.
Reference-yield limitation: the engine discounts each protected payment using its conditional Hull-White zero-coupon factor with a continuously compounded 50-bp add-on. Actual Treasury-reference provisions may specify a single reference yield, interpolation, a minimum reference maturity, semiannual compounding and a particular day count. This curve-based approximation is not an exact calculation of an indenture make-whole amount. A transaction-specific calculation must follow its wording.
The Schedule tab shows clean prices at an unchanged flat reference curve. It is neither the expected future strike nor a simulated path; the valuation engine still uses state-dependent conditional curves at lattice nodes.
The decomposition
Once the tree returns the three prices, the tool splits the total optionality into two intuitive slices:
The Value Split is an additive, sequential counterfactual decomposition: first add post-NC rights, then add pre-first-call rights. These are not two independently priced options, nor probabilities or discounted cash flows attributed to actual exercise dates. Changing the order of adding rights can change their allocated increments because the exercise opportunities interact. The displayed increments sum exactly before rounding; percentage shares are not applicable when total option value is zero.
A worked example (the base preset)
Load the tool with the default 7NC3 preset:
- Tenor: 7 years
- Non-call: 3 years
- Coupon: 7.0%
- Risk-free rate: 3.5%
- Credit spread: 300 bps
- Rate volatility σr: 100 bps
- Mean-reversion a: 0.10
- Tightening scenario: 200 bps
At 48 steps per year, the semiannual-coupon engine produces the following reconciliation (per 100 par):
Read that carefully: even with a 200 bps tightening scenario, the make-whole window remains effectively out-of-the-money in the base case. That’s not a bug. That’s the point of the shield. The borrower’s economically relevant option is the post-NC step-down call.
The Sensitivity tab varies one input at a time, keeping the other selected inputs fixed. Applied tightening is capped at initial spread; raising the requested tightening past that point no longer changes the scenario.
Solve for par targets the bullet reference, including the selected tightening. In this base case it solves an initial spread of approximately 487.659 bps, rounded to the control’s 0.1-bp precision. The displayed residual shows the remaining rounding error. This does not solve the callable bond to par, and some coupons/rates admit no nonnegative solution.
Assumptions & what they cost you
Gaussian rates and discrete exercise
The short rate may become negative. Very high volatility and low mean reversion can generate economically extreme scenarios. Inward boundary branching keeps lattice probabilities valid; it is a numerical approximation, not an economic rate floor. A finite exercise grid can miss between-date opportunities. Grid refinement is checked, but no universal pricing tolerance or calibrated parameter set is claimed.
No explicit default, recovery or market-access model
A deterministic discount spread is not a hazard-rate model. The issuer never defaults in the state tree, and access to refinancing is assumed whenever exercising is beneficial. There is no universal survival-probability multiplier that repairs this: default, recovery, rate/credit dependence and exercise policy interact, so the direction and size of a transaction-specific valuation difference are not established here.
Frictionless exercise
The issuer exercises optimally on the grid. Notice periods, funding fees, tax effects, partial calls and operational constraints are excluded. Restricting the same exercise opportunities or adding costs cannot increase the issuer option under an otherwise identical model; the size of that effect still requires explicit modeling.
Semiannual coupons and accrued interest
Coupons are paid every half-year on an issue-date time axis. Linear accrual separates clean redemption prices from dirty payments; payment-date coupons are never lost to the exercise cap. Exact settlement dates, business-day adjustment, stub coupons, record-date entitlement, ex-coupon periods and document-specific day-count conventions are not represented.
Deterministic credit-spread scenario
The exact spread integral is used between grid dates, avoiding a time-step-dependent left-endpoint bias. It is a selected deterministic discounting scenario, not a stochastic credit process or a probability forecast. A fixed-cash-flow bullet therefore has an analytic price independent of rate volatility and mean reversion once the initial curve is fitted.
Selected make-whole terms, not universal market drafting
The 50-bp reference add-on, 101 clean floor and p50 / p25 / par schedule are example assumptions. Different terms can materially change results. The model does not calculate make-whole treatment after acceleration, bankruptcy, a mandatory redemption or an equity claw.
What is outside this tool: TLB soft call
A loan soft-call provision can require a fee on specified repricing transactions during a protection window. It is not a European binary option: a qualifying event can occur at different times within the window, the fee base may cover only affected principal, and carve-outs can block the fee.
For example, filed credit-agreement terms address qualifying prepayments and pricing amendments within six months and link the fee to the affected loan amount. The actual definition, exceptions, repricing purpose and lender-replacement treatment must be read before modeling an event.
The fee cash flow is not the whole value of the borrower’s refinancing option. This rate-lattice calculator does not estimate either a soft-call fee value or a repricing-event probability.
Where this sits in the LevFin Book
The tool and this page are a hands-on implementation of one argument from Part VII: Contingent Rights and Teaching Sensitivities in the LevFin Book. Selected covenant rights are economically option-like. Each selected term must be mapped to its holder, legal states, exercise conditions, blockers and interactions before an assumption-dependent teaching estimate is applied. The rest of Part VII extends that discipline to soft calls, incremental debt, builder baskets, restricted-payment carve-outs and mandatory prepayment events.
If this tool clicks for you, the book is the rest of the shelf.
Common questions
A Hull-White lattice compares nested issuer exercise rights with semiannual coupons and accrued interest. Its discrete grid approximates American-style exercise. Fitting the selected flat initial curve is not market calibration.
Tenor, non-call period, coupon, risk-free rate, credit spread, rate volatility, mean reversion, and an optional spread-tightening scenario. The methodology page walks through each knob and what it costs you.
The optional-redemption clause may give the issuer a right to retire the debt before first call. The make-whole formula sets the contractual redemption price and protects the lender relative to a cheaper call; it does not prohibit exercise.
A soft-call fee depends on qualifying repricing events, their timing, affected principal and exceptions. This rate-lattice teaching model does not estimate that event process or its fees.
Want the whole framework in one place? Part VII of the LevFin Book takes the tool’s doctrine and applies it across the option-like economics in a real credit agreement, ending in a single worked deal where selected terms are mapped, tested, and converted into bps with clear methodology labels.