PROJECT STRATA

Methodology

How Strata calculates every metric — and where more than one method exists, which we use and why. Click an entry to expand.

Fixed Income

Fixed Income · rate sensitivity

Duration

How much a bond's price moves when rates move. Four measures, each handling a case the one below it can't.

01Macaulayfoundation

The weighted-average time to the bond's cashflows — when, on average, you're paid back.

D_{\mathrm{Mac}} = \dfrac{\sum_t t\,\mathrm{PV}(\mathrm{CF}_t)}{P}

It's a time in years, not a price sensitivity — you can't apply it directly to a yield move.

▲ why a step upMacaulay is a time in years; dividing by

1+y/k

turns it into a price sensitivity — the % move per 1% change in yield.

02Modifiedworking

The % price change for a 1% parallel yield move (Strata computes it via a 1bp finite difference).

D_{\mathrm{mod}} = \dfrac{D_{\mathrm{Mac}}}{1 + y/k}

Assumes the cashflows are fixed — wrong for callables, putables and MBS, whose cashflows move with rates.

▲ why a step upModified assumes fixed cashflows; Effective re-prices under

\pm\Delta y

, so callables and MBS whose cashflows change with rates are handled honestly. (For a vanilla bond,

D_{\mathrm{eff}} = D_{\mathrm{mod}}

03Effectiveadvanced◢ default

Re-price the bond at yield

-\Delta y

and

+\Delta y

and take the symmetric slope — because it re-prices, an embedded option's effect on the cashflows is captured directly.

D_{\mathrm{eff}} = \dfrac{P_{-} - P_{+}}{2\,P\,\Delta y}

Needs a full revaluation at each shock — overkill for a plain vanilla bond where it just equals modified duration.

▲ why a step upEffective bumps the whole curve in parallel; Key-rate bumps each tenor on its own — because the 2y and 30y rarely move together — and the per-tenor buckets sum to the effective duration.

04Key-ratecurve-aware◢ default

Sensitivity to a 1bp shift at each tenor (2/5/10/30y) separately, spread to neighbours by a triangular kernel; the buckets sum back to effective duration.

\mathrm{KRD}_i = -\dfrac{1}{P}\dfrac{\partial P}{\partial z_i}, \qquad \sum_i \mathrm{KRD}_i \approx D_{\mathrm{eff}}

Needs a full curve and per-tenor revaluation — overkill for a single vanilla bond.

▸ Variables

D_{\mathrm{Mac}}

Macaulay duration — the cashflow-weighted average time to repayment (years)

D_{\mathrm{mod}}

Modified duration — % price change per 1% yield move

D_{\mathrm{eff}}

Effective duration — sensitivity from re-pricing under

\pm\Delta y

\mathrm{KRD}_i

Key-rate duration at tenor

T_i

(sums to effective duration)

P,\ P_{-},\ P_{+}

Current dirty price, and the price after a yield cut / rise of

\Delta y

y,\ k,\ \Delta y

Yield, coupon periods per year, and the size of the yield shift

\mathrm{CF}_t,\ t

Cashflow at time

t

, and the time to it

△ The insight

Each rung doesn't replace the one below — it removes an assumption: Macaulay→Modified drops the time-units, Modified→Effective drops fixed-cashflows, Effective→Key-rate drops the parallel-shift. You climb only as far as the bond demands.

What Strata computes

Strata quotes Effective Duration for any bond — it's honest whether or not there's optionality — and Key-Rate Duration for curve risk. Modified duration is computed via a 1bp finite difference; Macaulay and modified are the rungs you climb to understand the number, but a vanilla approximation isn't what you'd quote on a callable.

Assumptions

•Modified duration uses a symmetric 1bp finite-difference bump; effective duration re-prices under a curve shock.
•Key-rate durations are verified against QuantLib parallel-bump effective duration in CI ( $\sum \mathrm{KRD}_i \approx D_{\mathrm{eff}}$ ).

Limitations

—Duration is first-order — for large yield moves pair it with convexity.
—Macaulay/Modified assume fixed cashflows; only Effective and Key-rate capture embedded-option effects.

vs Industry Tools

Bloomberg DUR / KRD: Same measures; Bloomberg's default key-tenor set usually adds short-end keys (0.5Y, 1Y).

Excel MDURATION() / DURATION(): Return the analytic parallel-shift duration only; Key-rate decomposes that into per-tenor buckets.

↗Convexity ↗DV01 ↗CS01 ↗Credit Spread

Go deeper

CFA — Fixed-Income Risk & Return

Fixed Income · rate sensitivity

Convexity

The curvature of the price-yield relationship — the second-order correction duration alone misses on large yield moves.

C = \dfrac{1}{P}\dfrac{\partial^2 P}{\partial y^2} \approx \dfrac{P_{+} + P_{-} - 2P}{P\,(\Delta y)^2}

▸ Variables

C

Convexity, in years-squared

P

Current dirty price

P_{+},\ P_{-}

Price after a yield rise / cut of

\Delta y

(Strata uses a 1bp bump)

\Delta y

Size of the symmetric yield shock (default 1bp)

What Strata computes

Strata computes convexity by central finite difference at the same 1bp bump it uses for duration: $C \approx (P_{+} + P_{-} - 2P)/(P\,(\Delta y)^2)$ . For a vanilla bond convexity is positive, so the duration estimate always under-states the gain on a rate cut and over-states the loss on a rise — convexity adds the missing curvature back.

Assumptions

•Second derivative approximated by central difference at the same 1bp shock as duration.
•Cashflows are fixed — for callables/MBS, effective convexity (re-pricing) is needed and can turn negative.
•Positive for standard option-free bonds (the price-yield curve bows toward the origin).

Limitations

—It's a second-order term — the price approximation $\Delta P/P \approx -D_{\mathrm{mod}}\,\Delta y + \tfrac12 C\,(\Delta y)^2$ still truncates the Taylor series at order two.
—Fixed-cashflow convexity misses embedded optionality; a callable bond can show negative effective convexity near the call strike, which this formula won't see.
—The finite-difference value is sensitive to the bump size — too small and floating-point noise dominates, too large and the curvature is itself averaged away.

vs Industry Tools

Bloomberg CNVX: Matches for most standard fixed-rate bonds; effective convexity diverges once optionality is present.

↗Duration ↗DV01

Go deeper

CFA — Fixed-Income Risk & Return

Fixed Income · rate sensitivity

DV01

The change in a position's value for a 1bp move in yield — duration translated from a percentage into hard currency.

\mathrm{DV01} = D_{\mathrm{mod}} \times \mathrm{MV} \times 0.0001

▸ Variables

\mathrm{DV01}

Currency value of a 1bp yield move (in the notional currency)

D_{\mathrm{mod}}

Modified duration of the bond

\mathrm{MV}

Market value of the position,

\mathrm{MV} = (P/100)\times\text{notional}

What Strata computes

Strata reports DV01 as a positive magnitude — the size of the P&L for a 1bp move — and applies the direction by convention at use: rising yields are a loss for a long position, a gain for a short. It is derived from the same modified duration Strata computes via a 1bp finite difference, so the number is consistent with the duration shown alongside it.

Assumptions

•Linear (first-order) approximation through duration — convexity is ignored, so it over/under-states for large moves.
•Direction is a sign convention applied to the magnitude: a long position loses as yields rise.
•Market value uses the current dirty price scaled to the position's notional.

Limitations

—First-order only: for moves beyond a handful of basis points, pair it with convexity.
—It's a parallel-yield sensitivity — for curve risk, the per-tenor analogue is key-rate DV01 (the dollar version of key-rate duration).
—DV01 scales with market value, so two bonds with equal duration but different prices/notionals carry different DV01 — compare on a like-for-like notional.

vs Industry Tools

Bloomberg DV01: Same concept; Bloomberg may use a slightly different scaling (per million notional vs per position) — check the unit.

↗Duration ↗CS01

Go deeper

CFA — Fixed-Income Risk & Return

Fixed Income · relative value

CS01

The change in a position's value for a 1bp move in credit spread, with the benchmark curve held still — DV01's credit-side twin.

\mathrm{CS01} = P(s + 1\text{bp}) - P(s)

▸ Variables

\mathrm{CS01}

P&L impact per 1bp spread move (in the notional currency)

P(s)

Price at the current credit spread

s

P(s + 1\text{bp})

Price after widening the spread by 1bp

s

Credit spread over the benchmark curve, in basis points

What Strata computes

Strata computes CS01 only in Curve+Spread pricing mode — the only mode where a credit spread is a separate, bumpable input — as a one-sided difference: re-price with the spread widened 1bp, holding the UST benchmark fixed. This isolates credit-specific risk from rate risk, so a desk can hedge the two legs independently.

Assumptions

•One-sided difference (spread widened by 1bp only), not a symmetric central bump.
•The benchmark (UST) curve is held constant — only the spread moves.
•Available in Curve+Spread mode; the All-in YTM and Pure Curve modes have no separable spread to bump.

Limitations

—Captures credit-spread risk only — it says nothing about benchmark-rate risk, which is DV01's job.
—First-order: a parallel 1bp spread bump, so it misses spread-curve steepening and any spread convexity.
—Undefined outside Curve+Spread mode, where the spread isn't an independent input.

vs Industry Tools

Bloomberg CS01 / spread DV01: Same concept (P&L per 1bp of spread); conventions on one-sided vs symmetric bumps can produce small differences.

↗DV01 ↗Credit Spread

Go deeper

CFA — Fixed-Income: Yield & Spread Measures

Fixed Income · relative value

Credit Spread

How much extra yield a bond pays over a risk-free benchmark. Four measures, each handling a case the one below it can't.

01G-Spreadfoundation

The bond's yield over the government curve at its maturity — a single point.

\mathrm{G} = y^{cc} - r^{\text{govt}}(T)

Reads one point on the curve; a steep or kinked curve between now and

T

is invisible to it.

└I-Spreadvariant

Same single-point read, but over the swap curve instead of governments — the funding benchmark for many desks.

\mathrm{I} = y^{cc} - r^{\text{swap}}(T)

Use when the swap curve is your funding/relative-value benchmark rather than governments.

Uses Strata's estimated swap curve (SOFR + post-GFC spreads), not a live par-swap curve — indicative, not trade-grade.

▲ why a step upG/I read the spread at a single maturity; Z spreads a constant over every curve point, so it reflects the shape G/I miss. (Z = G when the curve is flat.)

02Z-Spreadworking◢ default

One constant spread added across the whole zero curve that re-prices the bond — so the curve's shape is in the number.

P_{\text{mkt}} = \sum_{t} \dfrac{CF_t}{\bigl(1 + z(t) + Z\bigr)^{t}}

Still assumes deterministic cashflows — it can't see embedded-option value in callables or putables.

└Asset-Swap Spreadvariant

The spread over the floating index in a par asset-swap package — coupon-vs-swap plus a price-to-par adjustment.

\mathrm{ASW} \approx (C - r^{\text{swap}}) + \dfrac{100 - P^{\text{clean}}}{A}

Use when the trade is structured as an asset swap (bond + matching IRS to par).

A par-construction approximation:

\approx

Z near par, but it diverges for deep-discount or premium bonds.

▲ why a step upZ assumes fixed cashflows; OAS reprices the bond in a short-rate lattice (

V_0

= backward-induction price), removing option value. (OAS = Z when there's no optionality.)

03Option-Adjusted Spreadadvanced◢ default

The spread over the curve inside a short-rate tree, so embedded-option value is priced out — callable and bullet bonds finally compare on equal terms.

P_{\text{mkt}} = V_0\bigl(z(t) + \mathrm{OAS};\ \sigma\bigr)

Model-dependent — BDT vs Hull-White give slightly different numbers — and it needs a rate-volatility

\sigma

assumption.

▸ Variables

y^{cc}

Bond yield-to-maturity, converted to continuous compounding:

y^{cc} = f\,\ln(1 + y^{\text{per}}/f)

y^{\text{per}}

Quoted (periodic) YTM at frequency

f

f

Coupon periods per year

r^{\text{govt}}(T)

Government zero rate interpolated at the bond's maturity

T

(UST, or ECB AAA for EUR)

r^{\text{swap}}(T)

Swap rate interpolated at

T

(Strata's estimated USD SOFR curve)

z(t)

Government zero rate at each cashflow time

t

(the whole curve, not one point)

Z

The constant spread added to every curve point that re-prices the bond

CF_t

Cashflow at time

t

P_{\text{mkt}}

Observed market (dirty) price

\sigma

Assumed short-rate volatility (the OAS tree input)

C

Bond coupon rate;

A

= the swap annuity (PV of 1bp over the schedule)

What Strata computes

Strata computes all four and shows them together. For an option-free bond you'd quote the Z-Spread; for a callable/putable bond the OAS (the engine is BDT for Student/API tiers, Hull-White 1F for Professional). G- and I-Spread are the single-point reads you climb through to understand them.

Assumptions

•All yields are converted to continuous compounding before differencing, so spreads are comparable across coupon frequencies.
•Z-Spread and OAS solve for the spread that re-prices the bond against Strata's zero curve (bisection).
•OAS prices embedded options in a calibrated short-rate tree at the chosen volatility.

Limitations

—G/I-Spread for non-USD/EUR bonds fall back to the UST benchmark — a cross-currency read; treat as indicative.
—I-Spread and ASW rest on Strata's estimated USD swap curve, not a live traded par-swap curve.
—OAS depends on the model + the volatility input — quote the engine alongside the number.

vs Industry Tools

Bloomberg YAS / OAS1: Same ladder (G/I/Z/OAS); Strata's I-Spread/ASW use an estimated swap curve, so expect small differences there.

↗CS01 ↗Bond Pricing ↗YTM Solver

Go deeper

CFA — Fixed-Income: Yield & Spread Measures

Fixed Income · valuation

Bond Pricing

Three ways to discount a bond's cashflows to a price. They differ by which inputs you have — a single yield, a full curve, or a curve plus a credit spread.

Approach	How it builds the distribution	Use when
All-in YTM◢ default	Discount every cashflow at one yield, compounding $m$ times a year — the quote-screen number when all you have is a single YTM.	you have (or are solving for) one yield-to-maturity and want the quote-screen price.
Pure Curve	Discount each cashflow at its own continuously-compounded zero rate off the curve — the term structure's shape is now in the price.	you have a full zero curve and want the curve-consistent, credit-free benchmark value.
Curve+Spread	Add a credit spread to the interpolated benchmark yield, then discount — separating rate risk (the curve) from credit risk (the spread).	you have a benchmark curve and a credit spread, and want to bump rates and credit independently (the mode CS01 needs).

Dirty price

P = \text{PV of all cashflows}

Every mode outputs the dirty (full) price per 100 par — the present value of every coupon and principal cashflow. The clean (quoted) price is this minus accrued interest.

▸ Variables

P

Dirty price (per 100 par) — the output of every mode

CF_i,\ t_i

Cashflow

i

and its time in years (from the coupon schedule)

y

Yield to maturity (annual, decimal) for the all-in mode

m

Coupon frequency per year (1, 2, 4 or 12)

r(t_i)

Interpolated zero rate at time

t_i

(continuous, decimal) for the curve mode

y_{\mathrm{UST}}

Interpolated UST benchmark yield at the bond's maturity

s

Credit spread over the benchmark, in basis points

△ The insight

All three discount the same cashflows — they differ only in the discount rate they build: one number, a whole curve, or a curve plus a spread. Curve+Spread is the only mode that splits rate risk from credit risk, which is why DV01 and CS01 live there.

What Strata computes

Strata defaults to All-in YTM because a single yield is the input you most often have from a quote screen, and it's the price the YTM solver inverts. It switches to Curve+Spread the moment you supply a benchmark curve and a credit spread (so it can show DV01/CS01 separately), and to Pure Curve when you want the credit-free, curve-consistent benchmark value.

Assumptions

•All-in YTM and Curve+Spread use periodic compounding at the coupon frequency; Pure Curve uses continuous compounding.
•ACT/365 Fixed day count for time fractions; no business-day adjustments; cashflows received on scheduled dates.
•The curve modes use log-linear interpolation in the zero rate between curve points (falling back to plain linear when a bounding rate is non-positive), with flat extrapolation beyond the curve.

Limitations

—All-in YTM collapses the term structure to one number — wrong for a steep or kinked curve.
—Pure Curve omits credit entirely; Curve+Spread restores it but only as a parallel spread shift.
—None of the three prices embedded optionality — for callables/putables that's the OAS/short-rate-tree machinery in the credit-spread ladder.

vs Industry Tools

Excel PRICE(): All-in YTM analogue; defaults to 30/360 — set basis=3 for ACT/365 to match Strata.

Bloomberg DPRICE / CDSW: DPRICE matches the all-in mode with the same day count + compounding; CDSW mirrors the curve mode (check the compounding convention).

↗YTM Solver ↗Day-Count Conventions ↗Accrued Interest ↗Credit Spread ↗CS01

Go deeper

CFA — Fixed-Income: Introduction to Valuation

Fixed Income · valuation

YTM Solver

The inverse of pricing: find the single yield that discounts a bond's cashflows back to its observed price.

\text{Find } y \ \text{ such that } \ P(y) = P_{\mathrm{target}}

▸ Variables

y

Yield to maturity being solved for

P(y)

Dirty price as a function of yield (the All-in YTM pricing map)

P_{\mathrm{target}}

Target dirty price,

= \text{clean} + \text{accrued}

What Strata computes

Strata solves YTM by bisection with bracket expansion, not Newton-Raphson: it brackets the yield in $[-5\%,\ +50\%]$ , doubles the upper bound up to 8 times if the root sits higher, then halves the interval to a $10^{-10}$ price tolerance (max 80 iterations). Bisection is chosen for robustness — it can't diverge or overshoot the way a derivative-based method can on a badly-conditioned bond.

Assumptions

•Initial bracket of $-5\%$ to $+50\%$ yield, with the upper bound doubling up to 8 times if needed.
•Convergence at a $10^{-10}$ price-error tolerance, capped at 80 iterations.
•Price is monotone in yield (a single root) — true for option-free fixed-rate bonds.

Limitations

—Bisection is robust but slower than Newton-Raphson — it converges linearly, not quadratically.
—Assumes a single monotone root; for callable/putable bonds with multiple yield measures, you solve YTC/YTP/YTW separately and take the worst.
—Returns a single all-in yield — it collapses any term-structure shape, exactly as the All-in YTM pricing mode does.

vs Industry Tools

Excel YIELD(): Uses Newton-Raphson; can differ from Strata's bisection at edge cases (deep discount/premium, near-maturity).

Bloomberg YLD: Generally matches for standard fixed-rate bonds.

↗Bond Pricing

Go deeper

CFA — Fixed-Income: Yield Measures

Fixed Income · conventions

Day-Count Conventions

Four rules for turning a span of calendar dates into a year fraction. The choice is market convention, and it changes accrued interest and the discount-time of every cashflow.

Approach	How it builds the distribution	Use when
ACT/365 Fixed◢ default	Count the real days, always divide by 365 — simple and predictable, ignoring leap years entirely.	pricing in Strata, UK gilts, and several Asian markets — Strata's engine default for all time fractions.
30/360 US	Pretend every month has 30 days and every year 360 — so each coupon period is exactly equal, the US corporate-bond convention.	quoting US corporate bonds, where equal idealised periods are the market standard.
ACT/ACT ISDA	Count real days, split at each year boundary, and divide each slice by that year's own length — the most calendar-accurate rule.	pricing government securities, where calendar-exact accrual is the convention.
ACT/360	Count real days, divide by 360 — the money-market convention that makes a year worth slightly more than 1.0.	money-market instruments and floating-rate notes, where ACT/360 is the standard.

Year fraction

t = \dfrac{\text{day-count numerator}}{\text{denominator}}

Each convention is a numerator (real days, or idealised 30-day months) over a denominator (360, 365 or the year's own length). The denominator is the quirk: 360 inflates, fixed-365 ignores leap years, ACT/ACT alone matches the calendar.

▸ Variables

t

Year fraction — the output of every convention

Y_1, Y_2

Start and end years (the 30/360 inputs)

M_1, M_2

Start and end months

D_1, D_2

Start and end days, after the 30/360 day adjustments

△ The insight

The numerator (real vs idealised days) and the denominator (360, fixed 365, or the calendar year) together set the year fraction — so the same dates accrue different interest under each convention. ACT/360 inflates it, 30/360 idealises it, ACT/ACT alone tracks the real calendar.

What Strata computes

Strata's pricing engine uses ACT/365 Fixed for every time fraction — it's simple, predictable and leap-year-stable, which keeps the discount times deterministic across the curve. The other three are offered because accrued interest must be quoted in the market's convention for the instrument (30/360 for US corporates, ACT/ACT for governments, ACT/360 for money markets).

Assumptions

•30/360 applies day adjustments: if $D_1=31$ set it to $30$ ; if $D_1\in\{30,31\}$ and $D_2=31$ set $D_2=30$ ; February month-ends map to $30$ .
•ACT/ACT splits the period at January 1 and divides each slice by that year's day count ( $365$ or $366$ ); full intermediate years count as exactly $1.0$ .
•ACT/360 and ACT/365 Fixed are simple actual-day counts over a constant denominator, with no leap-year adjustment.

Limitations

—The conventions are not interchangeable: the same dates give different accrued interest and discount times, so a bond must be priced in its market's convention.
—ACT/360 produces year fractions above $1.0$ for periods longer than 360 days — interest is overstated relative to a 365 basis.
—30/360's idealised months break around month-end and February, requiring the day-adjustment rules above.

vs Industry Tools

Excel YEARFRAC(): The basis argument selects the convention: basis=0 for 30/360 US, basis=1 for ACT/ACT, basis=2 for ACT/360, basis=3 for ACT/365 — one function, four conventions.

↗Accrued Interest ↗Bond Pricing

Go deeper

ISDA — 2006 Definitions, day-count fractions

Fixed Income · conventions

Accrued Interest

The slice of the next coupon the seller has earned but not yet been paid — added to the quoted price to get the cash that actually settles.

AI = \dfrac{C}{m}\times\dfrac{t_{\mathrm{accrued}}}{t_{\mathrm{period}}}

▸ Variables

AI

Accrued interest (per 100 par)

C

Annual coupon rate (decimal)

m

Coupon frequency per year

t_{\mathrm{accrued}}

Year fraction from the last coupon date to settlement (day-count dependent)

t_{\mathrm{period}}

Year fraction for the full coupon period

What Strata computes

Strata accrues pro-rata within the coupon period: the period coupon $C/m$ scaled by the fraction of the period elapsed at settlement. Both year fractions come from the bond's day-count convention, so changing the convention changes accrued interest — and the buyer always pays the seller this accrued amount on top of the clean price.

Assumptions

•Pro-rata accrual within the coupon period (straight-line, no compounding inside the period).
•Both year fractions use the bond's day-count convention — so accrued moves with the convention.
•The buyer pays the seller accrued interest at settlement; it is then recovered in the next full coupon.

Limitations

—The clean (quoted) price excludes accrued; the dirty (settlement) price is $P_{\mathrm{dirty}} = P_{\mathrm{clean}} + AI$ — quoting clean while settling dirty is a constant source of confusion.
—Accrued is convention-dependent: the same bond on the same date accrues differently under 30/360 vs ACT/ACT (see day-count).
—The straight-line pro-rata rule assumes a regular coupon period — odd first/last (long or short) coupons need the schedule's actual period length.

vs Industry Tools

Excel ACCRINT(): Matches when given the same day-count convention.

Bloomberg (clean vs dirty quote): Bonds quote on the clean price; the dirty price is clean + accrued, as here.

↗Bond Pricing ↗Day-Count Conventions

Go deeper

CFA — Fixed-Income: Prices and Yields

Fixed Income · holding-period return

Horizon Return

The total return from holding a bond to a future date, broken into four additive parts: carry, roll-down, the price move from a yield change, and reinvestment income. All four share one denominator so they sum to the whole.

R_{\text{total}} = R_{\text{carry}} + R_{\text{rolldown}} + R_{\text{price}} + R_{\text{reinvest}}

▸ Variables

R_{\text{total}}

Total horizon return, as a fraction of the initial investment:

R_{\text{total}} = \dfrac{\sum_{t_i < T} CF_i^{\text{coupon}} + (\text{reinvest income}) + P_T^{\text{new}} - P_0}{P_0}

R_{\text{carry}}

Carry from coupon income received before the horizon:

R_{\text{carry}} = \dfrac{\sum_{t_i < T} CF_i^{\text{coupon}}}{P_0}

R_{\text{rolldown}}

Price change from the bond aging along an unchanged curve:

R_{\text{rolldown}} = \dfrac{P_T^{\text{unchanged}} - P_0}{P_0}

(both prices dirty)

R_{\text{price}}

Price change from the assumed yield/spread move at the horizon:

R_{\text{price}} = \dfrac{P_T^{\text{new}} - P_T^{\text{unchanged}}}{P_0}

R_{\text{reinvest}}

Extra income from reinvesting coupons to the horizon:

R_{\text{reinvest}} = \dfrac{\sum_{t_i < T} CF_i \bigl((1+r)^{T-t_i} - 1\bigr)}{P_0}

P_0

Initial dirty price per 100 par — the single common denominator for all four legs (notional cancels in a return)

P_T^{\text{unchanged}},\ P_T^{\text{new}}

Dirty price at the horizon under the unchanged curve vs. under the assumed yield move

CF_i^{\text{coupon}},\ CF_i

Coupon paid at time

t_i

(per 100 par);

CF_i

also denotes the cashflow reinvested

T,\ t_i,\ r

Horizon date, time of cashflow

t_i

(years), and the annual reinvestment rate (default: current YTM)

y_T,\ m

Horizon yield (initial yield + assumed change) and coupon frequency, used to re-price at the horizon

What Strata computes

Strata decomposes horizon return into the four additive legs above and prices the horizon by discounting the remaining cashflows at the horizon yield: $P_T = \sum_{i:\,t_i > T} \dfrac{CF_i}{(1 + y_T/m)^{m(t_i - T)}}$ — the same pricing formula as the initial price, but settled at the horizon date so accrued interest sets the clean/dirty split. Every leg is divided by the **initial dirty price $P_0$ ** (per 100), the choice that makes the four parts sum exactly to the total. The default reinvestment rate is the current YTM; in Curve+Spread mode the rate and spread changes are additive.

Assumptions

•All four components share **one denominator — the initial dirty price $P_0$ per 100** — so they are additive (this is the audit fix; notional cancels in a return).
•Roll-down uses dirty prices for both $P_0$ and $P_T^{\text{unchanged}}$ — premium bonds roll down (pull to par), discount bonds roll up.
•The horizon date is before maturity; the bond is sold at the horizon at its market (dirty) price, and only cashflows after the horizon are discounted into $P_T$ .
•Only coupons received before the horizon contribute to carry and reinvestment; reinvestment compounds annually at rate $r$ .
•The yield change is a parallel shift specified in basis points; a yield fall gives a positive price return and a rise a negative one.

Limitations

—The price leg is a full re-pricing at the horizon, but the decomposition still assumes a parallel curve shift — it does not capture twist/steepener/flattener moves (use a scenario stress for those).
—Reinvestment risk is real: the realised reinvestment rate is unknown today, so the carry-plus-reinvest legs are only as good as the assumed $r$ .
—It is a point forecast for one horizon and one yield scenario — it says nothing about the distribution of outcomes (pair with VaR / Monte-Carlo for that).
—Roll-down depends on the shape of the curve assumed unchanged; a steep curve flatters roll-down that may not persist.

vs Industry Tools

Bloomberg HRZN / scenario horizon analysis: Same carry / roll / price / reinvestment decomposition; interpolation of the unchanged horizon curve may differ.

↗Duration ↗Bond Pricing

Go deeper

CFA — Fixed-Income: Sources of Return

Derivatives

Derivatives · options

Option Pricing

What an option is worth, and how that worth moves. One closed-form for European payoffs, one tree for early exercise, and the Greeks and implied vol read straight off the model.

Approach	How it builds the distribution	Use when
Black-Scholes-Merton◢ default	A closed-form price for a European option on a dividend-paying asset — instant, exact, and the reference every other method is checked against.	European payoffs and any time you need an instant analytic price or to seed the Greeks / implied vol.
CRR Binomial Tree◢ default	Discretise the price path into an up-down tree and price by backward induction, testing early exercise at every node — so American options are valued honestly.	American or Bermudan exercise, or whenever early-exercise value matters (e.g. deep-ITM puts, dividend-paying calls).
└Greeks◢ default	The partial derivatives of the BSM price — delta, gamma, vega, theta, rho — that tell you how the value moves and how to hedge it.	hedging or risk-managing an options book — sizing the delta hedge, watching gamma near expiry, or pricing a vol view through vega.
└Implied Volatility◢ default	Run BSM backwards: solve, by Newton-Raphson on vega, for the single $\sigma$ that reprices the option to its market price.	reading the market's volatility view out of a quoted price, or building the smile / surface from a chain of quotes.

Price

V = f(S, K, r, \delta, \sigma, T)

The fair value of the option from the model — call or put, European (BSM) or American (tree).

Implied vol

\sigma\ \text{s.t.}\ V^{\text{BSM}}(\sigma) = V^{\text{market}}

The model run in reverse: the volatility that reprices the option to the market — a measure read off any price, not a separate model.

▸ Variables

S,\ K

Spot price of the underlying, and the strike

r,\ \delta

Continuously compounded risk-free rate, and continuous dividend yield

\sigma,\ T

Annualised volatility of the underlying, and time to expiry in years

d_1,\ d_2

d_1 = \dfrac{\ln(S/K) + (r - \delta + \sigma^2/2)T}{\sigma\sqrt{T}}

and

d_2 = d_1 - \sigma\sqrt{T}

N(\cdot),\ n(\cdot)

Cumulative standard-normal distribution, and its density

C,\ P

Call and put price

u,\ d,\ p^*

Up factor

u = e^{\sigma\sqrt{\Delta t}}

, down

d = 1/u

, and the risk-neutral up-probability under carry

r - \delta

\Delta t,\ V_{n,j}

Tree step length

T/n

, and the option value at node

(n,j)

(

j

up-moves)

\Delta,\ \Gamma,\ \mathcal{V},\ \Theta,\ \rho

Delta, gamma, vega, theta and rho — the first-order Greeks (vega/rho per 1% move, theta per calendar day)

V^{\text{market}}

Observed market price of the option — the target the implied-vol solver inverts

△ The insight

The two pricing rows aren't rivals: the CRR tree **converges to BSM as the step count $n \to \infty$ **. So for a European option they agree, and the tree only earns its cost by pricing the early exercise BSM can't see.

What Strata computes

Strata prices European options analytically with BSM (instant, exact) and American options on a CRR tree (early exercise per node), then reports the Greeks and implied vol off the BSM engine. The tree's step count is set high enough internally for convergence; the 1–3-step view is for teaching only.

Assumptions

•Geometric Brownian motion for the underlying with constant volatility $\sigma$ and risk-free rate $r$ , continuous dividend yield $\delta$ , no transaction costs or short-sale limits.
•The CRR tree's risk-neutral probability uses cost-of-carry $r - \delta$ ; American early exercise is taken whenever intrinsic value exceeds continuation value.
•Greeks are scaled for practical reading: vega and rho per 1% move ( $\times 0.01$ ), theta per calendar day ( $\times 1/365$ ).
•Implied vol is solved by Newton-Raphson (max 50 iterations, tolerance $10^{-7}$ ) with an adaptive seed; it returns null if vega falls below $10^{-10}$ .

Limitations

—Constant volatility — BSM assumes a flat surface, so it misprices the smile / skew that real markets show; implied vol is how the market patches that.
—GBM has thinner tails than reality, so deep-OTM options and jump risk are under-valued.
—Discrete dividends need a PV-of-dividend adjustment; the continuous-yield $\delta$ is an approximation for dividend-paying single names.
—The tree converges only as $n$ grows and oscillates on the way — a low step count gives a visibly wrong American price.

vs Industry Tools

Bloomberg OVME: Prices on local-volatility or stochastic-vol (Heston) surfaces; BSM is the flat-vol special case Strata implements.

CFA Level II (Valuation of Contingent Claims): Standard two-period CRR tree formulation; Strata extends it to an n-period tree for convergence.

↗Linear Derivatives ↗CVA / DVA

Go deeper

Merton (1973) — Theory of Rational Option Pricing Cox, Ross & Rubinstein (1979) — Option Pricing: A Simplified Approach

Derivatives · forwards, FRAs & swaps

Linear Derivatives

Lock in a price, a rate, or a whole stream of rates. Three instruments, each one a longer version of the no-arbitrage forward below it.

01Forward Pricingfoundation

Carry the spot forward at the cost of money less any yield it throws off — the only price that admits no arbitrage against borrowing and holding the asset.

F_0(T) = \bigl(S_0 - \mathrm{PV}(\mathrm{Div})\bigr)\,e^{rT} \qquad F_0(T) = S_0\,e^{(r - \delta)T}

A single locked price at one date — it says nothing about a future borrowing rate or a stream of cashflows.

▲ why a step upA forward fixes a price for an asset; an FRA fixes a rate for one future period — the same no-arbitrage logic applied to borrowing instead of to a spot asset.

02FRA Settlementworking

Lock a single future interest rate: at fixing, exchange the PV of the gap between the realised rate

L

and the contracted rate, discounted back at

L

\text{Settlement} = \dfrac{(L - \mathrm{FRA}) \times N \times \text{days}/\text{basis}}{1 + L \times \text{days}/\text{basis}}

Covers one period only — a multi-year hedge needs a whole strip of FRAs, which is exactly the next rung.

▲ why a step upAn FRA locks one period's rate; a swap is a strip of forward rates — the par fixed rate

\mathrm{SFR}

is the single number that prices the entire strip to zero net value.

03Interest-Rate Swapadvanced◢ default

Find the one fixed rate that makes a whole schedule of floating payments net to zero PV at inception — an at-market swap costs nothing to enter.

\mathrm{SFR} = \dfrac{1 - DF_N}{\sum_{i=1}^{N} DF_i\,\delta_i}

Single-curve construction (one zero curve both discounts and forwards) on stylised period times — the CFA convention, not a modern multi-curve OIS-discount desk model.

▸ Variables

F_0(T)

Forward price agreed today for delivery at

T

S_0

Current spot price of the underlying

\mathrm{PV}(\mathrm{Div})

Present value of discrete dividends paid before

T

(subtracted before compounding)

r,\ \delta

Continuously compounded risk-free rate, and continuous dividend / carry yield

L,\ \mathrm{FRA}

Realised settlement reference rate (e.g. SOFR), and the contracted FRA rate

N

Notional principal

\text{days}/\text{basis}

Accrual fraction for the FRA period (e.g. Act/360 for USD)

\mathrm{SFR}

Par (at-market) swap fixed rate — set so the fixed and floating legs have equal PV

DF_i,\ DF_N

Discount factor at payment date

i

, and at final date

N

, from the bootstrapped zero curve

\delta_i

Day-count fraction for swap period

i

△ The insight

There's only one idea here, stretched across more periods: replicate the payoff with cash and the underlying, and no-arbitrage pins the price. A swap is just a strip of forwards — which is why the same $DF_i$ curve prices all three.

What Strata computes

Strata's headline linear-derivatives output is the par swap rate (the most-asked desk number), with forward and FRA pricing as the rungs you climb to understand it. The swap uses a single bootstrapped SOFR zero curve to both discount and imply the floating leg ( $V_{\text{float}} = N$ at each reset), with continuous compounding and stylised period times.

Assumptions

•No-arbitrage throughout: continuous compounding, no bid-ask spread, no transaction costs, and free capital flows for the FX-forward (CIRP) case.
•The floating leg of the swap resets to par at each coupon date (the floating-rate-bond analogy), so its value equals notional at reset.
•Discount factors come from a single bootstrapped SOFR zero curve; the same curve discounts and forwards (CFA single-curve convention).
•Swap period times are stylised ( $T_i = i/\text{frequency}$ ) with a constant day-count fraction — methodologically transparent but not calendar-accurate.

Limitations

—Single-curve pricing: real desks discount on OIS and forward on a separate index curve, so Strata's swap value diverges from a multi-curve mark.
—Stylised conventions (idealised period times, constant day-count) plus an estimated USD swap curve mean values are directionally meaningful, not trade-grade.
—FRA settlement assumes the payment is discounted at the prevailing reference rate $L$ , the market convention but an approximation.
—Linear payoffs only — these instruments carry no optionality, so duration/DV01 fully describe their rate risk (see CVA for the credit overlay).

vs Industry Tools

CFA Level II curriculum (Swap Valuation): Standard fixed-rate-bond / floating-rate-bond decomposition and the par-swap-rate formula; Strata follows the single-curve convention.

Bloomberg SWPM: Full multi-curve (OIS-discount + index forwarding), calendar-accurate schedules and live par-swap curves; Strata is an indicative single-curve estimate.

↗CVA / DVA

Go deeper

Hull — Options, Futures, and Other Derivatives CFA — Derivatives: Pricing and Valuation of Forward Commitments

Derivatives · counterparty risk

CVA / DVA

The price you knock off an OTC derivative for the chance your counterparty defaults (CVA), plus the symmetric credit for your own (DVA). BCVA is the net.

\begin{aligned} \mathrm{CVA} &= -\sum_{i} \mathrm{LGD}_{\text{cp}} \cdot DF_i \cdot \mathrm{EE}(t_i) \cdot \Delta\mathrm{PD}_{\text{cp},i} \\ \mathrm{DVA} &= +\sum_{i} \mathrm{LGD}_{\text{own}} \cdot DF_i \cdot \mathrm{NEE}(t_i) \cdot \Delta\mathrm{PD}_{\text{own},i} \\ \mathrm{BCVA} &= \mathrm{CVA} + \mathrm{DVA} \end{aligned}

▸ Variables

\mathrm{CVA},\ \mathrm{DVA}

Credit / debit valuation adjustment — expected loss from the counterparty's / your own default (CVA stored as a negative number)

\mathrm{BCVA}

Bilateral CVA, the net adjustment:

\mathrm{CVA} + \mathrm{DVA}

\lambda

Hazard (default-intensity) rate,

\lambda = \mathrm{Spread}_{\text{cp}} / (10000 \cdot \mathrm{LGD}_{\text{cp}})

; cumulative default

\mathrm{PD}(t) = 1 - e^{-\lambda t}

\mathrm{Spread}_{\text{cp}}

Counterparty CDS / credit spread in basis points

\mathrm{LGD}_{\text{cp}},\ \mathrm{LGD}_{\text{own}}

Loss given default for the counterparty / for you, each

= 1 - \text{Recovery Rate}

\mathrm{EE}(t)

Expected (positive) exposure at time

t

— convex DV01 decay

\times

the half-normal mean

\varphi(0) = 1/\sqrt{2\pi}

\mathrm{NEE}(t)

Negative expected exposure; the IRS profile is symmetric so

\mathrm{NEE}(t) = \mathrm{EE}(t)

\mathrm{DV01}

Dollar value of a 1bp rate move on the swap — the scale of the exposure

\sigma_r

Assumed short-rate volatility — the diffusion that grows the swap's mark-to-market exposure over time

T,\ t,\ t_i

Swap maturity, a generic time, and the

i

-th of

N

exposure time steps

DF_i,\ \Delta\mathrm{PD}(t_i)

Risk-free discount factor at

t_i

, and the marginal default probability over step

i

What Strata computes

Strata reports a screening-grade BCVA: expected exposure as a convex DV01 decay $\mathrm{EE}(t) \propto \sqrt{(T-t)/T}\cdot\sqrt{t}$ (peaks mid-life), the exposure integral discretised over $N = 20$ left-edge steps, hazard rates backed out from each party's spread, and $\mathrm{BCVA} = \mathrm{CVA} + \mathrm{DVA}$ (not minus). It is an indicative number, not a desk's IMM model.

Assumptions

•Expected exposure uses a convex DV01 decay $\mathrm{EE}(t) \propto \sqrt{(T-t)/T}\times\sqrt{t}$ — a stated modelling choice (the textbook profile is linear $(T-t)/T$ ) that peaks mid-life.
• $\mathrm{NEE} = \mathrm{EE}$ : the swap's exposure is symmetric, so the credit asymmetry between the two parties enters only through $\lambda_{\text{own}}$ vs $\lambda_{\text{cp}}$ , not through a weight on the exposure.
•The exposure integral is discretised over $N = 20$ left-edge time steps; CVA is stored as a negative number and $\mathrm{BCVA} = \mathrm{CVA} + \mathrm{DVA}$ .
•Hazard rates are constant and backed out of each party's credit spread at a flat LGD; discounting is at the risk-free curve.

Limitations

—A screening estimate only — production CVA simulates the full trade NPV path with netting sets and collateral, which can move the number materially.
—No wrong-way risk: exposure and default probability are assumed independent, so a counterparty whose default correlates with the exposure is under-charged.
—Single-trade, no netting or CSA — a real counterparty's CVA is computed across the whole portfolio of trades, not one swap in isolation.
—DVA is controversial: booking a gain when your own credit deteriorates is real under accounting but un-hedgeable and excluded from Basel CVA capital.

vs Industry Tools

Bloomberg CVA function: Full Monte-Carlo simulation with netting sets, collateral and IMM-approved models (Basel III); Strata's is a closed-form indicative estimate.

↗Linear Derivatives ↗Credit Spread

Go deeper

Gregory — The xVA Challenge (Counterparty Credit Risk)Basel III — CVA risk framework

Equity

Equity · discount rate

WACC

The blended after-tax cost of a firm's capital, the discount rate for unlevered free cash flow. One accepted formula; the lesson is how you source each input.

\begin{aligned} \mathrm{WACC} &= w_E\,k_E + w_D\,k_D\,(1 - t) \\ k_E &= R_f + \beta\,\mathrm{ERP} + \mathrm{CRP} + \mathrm{SP} \\ k_D &= \dfrac{|\text{Interest Expense}|}{\text{Total Debt}} \\ \beta_L &= \beta_U\,\bigl[1 + (1 - t)\,D/E\bigr] \end{aligned}

▸ Variables

w_E,\ w_D

Market-value weights of equity and debt (

w_E + w_D = 1

)

k_E

Cost of equity, from the four-component CAPM

k_D

Pre-tax cost of debt, implied from the income statement as

|\text{Interest Expense}| / \text{Total Debt}

t

Effective tax rate, the source of the debt tax shield (applied once, in the WACC sum)

R_f

Risk-free rate (10-year UST; FRED DGS10, fallback

4.25\%

)

\beta

Equity (levered) beta used in CAPM (FMP live; falls back to a re-levered Damodaran industry median)

\mathrm{ERP}

Equity risk premium (Damodaran implied; fallback

4.60\%

)

\mathrm{CRP}

Country risk premium (Damodaran;

0

for the US)

\mathrm{SP}

Size premium (user-specified; default

0

)

\beta_L,\ \beta_U

Levered (observed) and unlevered (asset) beta, linked by the Hamada equation

D/E

Debt-to-equity ratio at market values (the Hamada relevering input)

What Strata computes

Strata computes one WACC: market-value-weighted, four-component CAPM cost of equity, and a pre-tax cost of debt implied from the income statement (not a yield-to-maturity on outstanding bonds). When only an unlevered industry-median beta is available, it is re-levered via Hamada to the firm's own $D/E$ before entering CAPM. The tax shield is applied once, in the $w_D\,k_D(1-t)$ term.

Assumptions

•Beta is re-levered via the Hamada equation when an unlevered industry-median is used; debt is assumed risk-free and the tax shield permanent (no Miles-Ezzell / Harris-Pringle adjustment).
•Cost of debt is derived from the income statement ( $|\text{Interest Expense}| / \text{Total Debt}$ ), not yield-to-maturity on traded bonds.
•The risk-free rate falls back to $4.25\%$ when FRED is unavailable; the tax rate to a Damodaran sector median, fallback $21\%$ .
•Capital weights are at market value, so $w_E$ uses market cap, not book equity.

Limitations

—It is a single blended rate, so it implicitly assumes a constant capital structure over the whole forecast, true for few firms mid-deleveraging.
—Income-statement $k_D$ lags the market: a firm whose credit has just widened still shows its old average coupon, understating the true marginal cost of debt.
—WACC is the right discount rate for unlevered (free-cash-flow-to-firm) flows only, discount FCFE at $k_E$ instead.
—Every input is a convention choice (beta source, ERP, $R_f$ basis), the same firm can carry a materially different WACC under defensible inputs.

vs Industry Tools

Bloomberg WACC: Uses a market-implied beta and a YTM-based cost of debt, more precise for large-cap issuers with liquid bonds; Strata's

k_D

is income-statement-implied (transparent, but backward-looking).

↗Discounted Cash Flow ↗Dividend Discount Model

Go deeper

Damodaran — Cost of Capital Hamada (1972) — The Effect of the Firm's Capital Structure

Equity · intrinsic valuation

Discounted Cash Flow

Value a firm as the present value of the cash it will generate. Forecast the flows, discount them, and bridge to a per-share price.

01Two-Stageworking◢ default

Discount an explicit FCF forecast at WACC, add a Gordon terminal value, then bridge enterprise value to a per-share price by netting out debt.

V_0 = \sum_{t=1}^{N} \dfrac{\mathrm{FCF}_t}{(1+\mathrm{WACC})^{\,t - 0.5\,\mathbf{1}_{\mathrm{mid}}}} + \dfrac{\mathrm{TV}}{(1+\mathrm{WACC})^{N}}, \qquad \mathrm{TV} = \dfrac{\mathrm{FCF}_N(1+g_T)}{\mathrm{WACC} - g_T}, \qquad P_0 = \dfrac{V_0 - \text{Net Debt}}{\text{Shares}}

A single jump from

g_1

g_T

at year

N

, no glide path; the terminal value usually dominates and is undefined when

\mathrm{WACC} \le g_T

└Reverse DCFvariant

Invert the model: hold every other input fixed and solve (bisection, bracket-expanding) for the Stage-1 growth

g^{*}

that makes the DCF price equal today's market price, the growth the market is paying for.

g^{*} = \arg\min_{g}\,\bigl|\,P_0^{\mathrm{DCF}}(g) - P_0^{\mathrm{mkt}}\,\bigr|

Returns one number conditional on all other inputs, a different WACC or terminal growth implies a different

g^{*}

; returns null if no solution sits in the search range.

└FCFEvariant

Discount cash available to equity directly at the cost of equity

k_E

, so there is no enterprise-to-equity bridge, debt service is already inside the cash flow.

\mathrm{FCFE}_t = \mathrm{NI} + \mathrm{D\&A} - \mathrm{CapEx} - \Delta\mathrm{WC} + \text{Net Borrowing}, \qquad P_0 = \sum_{t=1}^{N} \dfrac{\mathrm{FCFE}_t}{(1+k_E)^{t}} + \dfrac{\mathrm{TV}}{(1+k_E)^{N}}

Net borrowing is hard to forecast and swings FCFE; the equity-side result is sensitive to capital-structure assumptions the firm-side FCFF model sidesteps.

▲ why a step upTwo-stage snaps from high growth to terminal in one year; three-stage interpolates across

N_2

years, which is more realistic for a maturing firm and softens the terminal-value dependence.

02Three-Stageadvanced

Insert a transition stage where growth fades linearly from

g_1

down to

g_T

before the perpetuity, so the step-down is gradual rather than a cliff.

g_t^{(2)} = g_1 + (g_T - g_1)\,\dfrac{t}{N_2}, \qquad V_0 = \sum_{t=1}^{N_1+N_2} \dfrac{\mathrm{FCF}_t}{(1+\mathrm{WACC})^{t}} + \dfrac{\mathrm{TV}}{(1+\mathrm{WACC})^{N_1+N_2}}

The linear fade is itself an assumption, real growth rarely declines in a straight line, and the extra stage adds inputs without removing terminal-value sensitivity.

▸ Variables

V_0

Enterprise value today, the discounted sum of forecast cash flows plus terminal value

P_0

Implied value per share after the enterprise-to-equity bridge

\mathrm{FCF}_t

Free cash flow to the firm in year

t

g_1,\ g_T

Stage-1 (explicit) growth rate and the terminal (long-run) growth rate

\mathrm{WACC}

Weighted average cost of capital, the discount rate for firm cash flows

N,\ N_1,\ N_2

Length of the explicit forecast; and (3-stage) the Stage-1 and Stage-2 lengths in years

\mathbf{1}_{\mathrm{mid}}

Mid-year convention flag:

1

if active (cash arrives mid-period), else

0

\mathrm{TV}

Terminal value, via the Gordon perpetuity or an exit EV/EBITDA multiple

g_t^{(2)}

Stage-2 growth in year

t

, fading linearly from

g_1

g_T

(3-stage only)

\text{Net Debt}

Total debt minus cash, subtracted at the enterprise-to-equity bridge

\text{Shares}

Diluted shares outstanding, the divisor in the per-share bridge

g^{*}

Reverse-DCF: the Stage-1 growth that makes the model value equal the market price

P_0^{\mathrm{DCF}}(g)

Reverse-DCF: the two-stage DCF per-share value as a function of the Stage-1 growth

g

P_0^{\mathrm{mkt}}

Reverse-DCF: the observed market price the inversion solves against

\mathrm{FCFE}_t

Free cash flow to equity in year

t

, cash left after debt service

\mathrm{NI}

Net income (the FCFE starting point)

\mathrm{D\&A}

Depreciation and amortisation, added back (non-cash)

\mathrm{CapEx}

Capital expenditure, subtracted

\Delta\mathrm{WC}

Change in net working capital, subtracted

\text{Net Borrowing}

New debt issuance minus repayment, added in FCFE

k_E

Cost of equity, the discount rate for equity cash flows in FCFE

△ The insight

Two-stage to three-stage removes the growth cliff; reverse-DCF inverts the same machine to read the market's expectation; FCFE re-routes it down the equity side at $k_E$ . Same discounting engine, four lenses on it.

What Strata computes

Strata defaults to the two-stage DCF (explicit FCF forecast + Gordon terminal value, mid-year convention optional) and bridges to a per-share price. Three-stage is offered for firms with a long, visible glide path; reverse-DCF and FCFE are the inversion and equity-side cuts you switch to when the question changes.

Assumptions

•Mid-year convention applies only to explicit-period FCFs; the terminal value is always discounted at end of year $N$ .
•The Gordon terminal value is undefined and returns null when $\mathrm{WACC} \le g_T$ .
•The enterprise-to-equity bridge subtracts net debt; FCFE skips the bridge because its cash flows are already post-debt-service.
•Reverse-DCF holds WACC, terminal growth and forecast length fixed and solves only for Stage-1 growth.

Limitations

—The output is dominated by the terminal value and the WACC-minus- $g_T$ denominator, small input moves swing the answer.
—Three-stage's linear fade and FCFE's net-borrowing path are modelling conventions, not observed quantities.
—Every variant inherits the forecast's quality, garbage forecasts produce confident-looking but wrong intrinsic values.

vs Industry Tools

Bloomberg DSCF: Pulls consensus estimates for the explicit period; same Gordon terminal-value formula. Strata uses your forecast or SEC-derived actuals and exposes the reverse and FCFE cuts in the same panel.

↗WACC ↗Comparable Company Analysis ↗Dividend Discount Model

Go deeper

Damodaran — Investment Valuation CFA — Discounted Dividend & Free Cash Flow Valuation

Equity · intrinsic valuation

Dividend Discount Model

Value a share as the present value of its future dividends. Two rungs, differing only in how the dividend growth rate is allowed to change.

01Gordon Growthfoundation◢ default

A perpetuity: next year's dividend divided by the spread between the required return and a single growth rate held constant forever.

P_0 = \dfrac{D_0\,(1 + g)}{r - g}, \qquad r > g

One growth rate forever fits only a mature, steady-state payer; it is undefined (returns null) when

r \le g

and is hypersensitive to

g

as it approaches

r

▲ why a step upGordon assumes a single growth rate forever; the H-model adds a second term that prices a temporary high-growth phase decaying to the terminal rate, no need to project each year explicitly.

02H-Modelworking

A Gordon base on the terminal rate

g_L

, plus a closed-form premium for the extra growth that fades linearly from

g_S

down to

g_L

over the transition.

P_0 = \dfrac{D_0\,(1 + g_L)}{r - g_L} + \dfrac{D_0\,H\,(g_S - g_L)}{r - g_L}, \qquad H = \dfrac{\text{high-growth years}}{2}

The linear decay is an approximation, real transitions rarely follow a straight line, and it still requires

r > g_L

for the base term to converge.

▸ Variables

P_0

Intrinsic value per share today

D_0

Most recent annual dividend per share

g

Constant long-term dividend growth rate (single-stage)

g_S,\ g_L

Short-term (initial) and long-term (terminal) dividend growth rates (H-model)

r

Required rate of return on equity (cost of equity), the discount rate

H

Half the high-growth period in years,

H = (\text{high-growth years}) / 2

△ The insight

Both rungs are the same perpetuity: the H-model is Gordon-on- $g_L$ plus one extra term for a linearly-fading head start. When $g_S = g_L$ the premium vanishes and the H-model collapses back to Gordon.

What Strata computes

Strata defaults to the Gordon Growth model for steady-state dividend payers, it is the cleanest single-input read. The H-model is offered for firms in transition (a high payout growth that is visibly maturing), where a single constant $g$ would mis-state value.

Assumptions

•Both models are undefined and return null when the discount rate does not exceed the relevant terminal growth ( $r \le g$ for Gordon, $r \le g_L$ for the H-model).
•The H-model permits a temporary $g_S > r$ ; only a terminal $g_L \ge r$ makes the perpetuity blow up.
•Dividends are the right cash-flow proxy, which holds for established payers, not for non- or low-payout firms.

Limitations

—Useless for firms that pay no dividends or whose payout is unrelated to earning power, use a free-cash-flow model instead.
—Both rungs are dominated by the $r - g_L$ denominator, the value explodes as growth nears the required return.
—The H-model's linear-fade shape is assumed, not estimated, two analysts can pick different transition lengths.

vs Industry Tools

CFA curriculum (Equity Valuation): Standard single-stage Gordon and two-stage H-model formulations; the H-model is the textbook tool for emerging-market firms with a transitional growth profile.

↗Discounted Cash Flow ↗WACC

Go deeper

Gordon (1959) — Dividends, Earnings, and Stock Prices CFA — Discounted Dividend Valuation

Equity · relative valuation

Comparable Company Analysis

Value a firm by what the market pays for its peers. Apply a peer-group multiple to the firm's own fundamental, several multiples, one implied value each.

Approach	How it builds the distribution	Use when
P/E◢ default	Apply the peer price-to-earnings multiple straight to net income, an equity-level multiple, so no debt bridge is needed.	comparing profitable firms with similar capital structures and clean earnings.
P/S	Price-to-sales, anchored on revenue, which stays positive and is harder to manipulate than earnings.	the firm is pre-profit or earnings are too noisy to trust (early-stage, cyclical trough).
EV/EBITDA	An enterprise-level multiple on pre-financing, pre-tax operating cash, so it is capital-structure-neutral, then bridge EV to equity.	comparing firms with different leverage or tax regimes, the cross-capital-structure default.
EV/FCF	An enterprise multiple on actual free cash flow, the cash left after capex, which EV/EBITDA overlooks.	capex intensity differs across the peer set and cash conversion is the real question.

Implied value

P_0 = \dfrac{\text{Implied equity value}}{\text{Shares}}

Each multiple yields one implied price; the football-field chart shows the range (min / median / mean / max) across the peer set rather than a single point estimate.

▸ Variables

m

The peer-group multiple (median or mean of the comparable set)

\text{NI}

Net income, the P/E denominator fundamental

\text{Rev}

Revenue (sales), the P/S denominator fundamental

\mathrm{EBITDA}

Earnings before interest, tax, depreciation and amortisation, the EV/EBITDA fundamental

\mathrm{FCF}

Free cash flow, the EV/FCF fundamental

\mathrm{EV}

Implied enterprise value (multiple applied to an EV-level fundamental)

\text{Net Debt}

Total debt minus cash, the enterprise-to-equity bridge term

\text{Shares}

Diluted shares outstanding, the per-share divisor

P_0

Implied value per share after the bridge

△ The insight

Comps measures relative value, what the firm is worth if peers are fairly priced, while DCF measures intrinsic value from the firm's own cash flows. When a whole sector is mispriced, comps inherits the error; the two methods are a cross-check, not substitutes.

What Strata computes

Strata leads with P/E for profitable firms, the most quoted multiple, and triangulates with P/S, EV/EBITDA and EV/FCF so leverage, profitability and capex intensity each get a vote. The football-field chart presents the spread of implied values, not a false-precision single number.

Assumptions

•The peer group comes from the live `/api/equity/peers` endpoint; the user can add or remove peers.
•Multiples are sourced from FMP; the firm's fundamentals from SEC XBRL trailing-twelve-month actuals.
•Negative or zero multiples are excluded from the peer statistics so a loss-maker does not poison the median.
•EV-level multiples (EV/EBITDA, EV/FCF) net out debt at the bridge; equity-level multiples (P/E, P/S) do not.

Limitations

—Comps is only as honest as the peer set, a sector-wide bubble or trough is baked straight into the implied value.
—Each multiple is blind to something, P/E to leverage, P/S to margins, EV/EBITDA to capex, no single row is sufficient.
—Point-in-time multiples ignore differences in growth and risk between the firm and its peers.

vs Industry Tools

Bloomberg COMP: Uses real-time consensus estimates; Strata uses trailing-twelve-month actuals from SEC filings, more transparent, but backward-looking.

↗Discounted Cash Flow

Go deeper

Damodaran — Relative Valuation CFA — Market-Based Valuation: Price & Enterprise Value Multiples

Portfolio

Portfolio · optimization

Efficient Frontier

The lowest-variance portfolio for each level of expected return. One quadratic program, swept across a target return to trace the curve.

\min_{\mathbf{w}}\ \tfrac{1}{2}\,\mathbf{w}^{\mathsf T}\mathbf{\Sigma}\,\mathbf{w} \quad \text{s.t.}\quad \mathbf{1}^{\mathsf T}\mathbf{w}=1,\;\; \boldsymbol{\mu}^{\mathsf T}\mathbf{w}=t,\;\; L \le w_i \le U

▸ Variables

\mathbf{w}

Asset weight vector being optimised

\mathbf{\Sigma}

Covariance matrix of asset returns (annualised, positive definite)

\boldsymbol{\mu}

Vector of expected (annualised) asset returns

t

Target portfolio return swept from the minimum-variance point up to the maximum feasible return

L,\ U

Per-asset weight box (long-only when

L = 0

)

\mathbf{1}

Vector of ones — the fully-invested budget constraint

\mathbf{1}^{\mathsf T}\mathbf{w}=1

r_f

Risk-free rate — fixes the tangency (max-Sharpe) point where the capital market line touches the frontier

What Strata computes

Strata solves the exact convex QP with a primal active-set method (not a grid search), long-only box constraints ( $0 \le w_i \le \text{maxWeight}$ ) on the public endpoint, then sweeps $t$ from the global minimum-variance (GMV) portfolio to the max feasible return. The tangency portfolio is the frontier point with the steepest line from $r_f$ ; the capital market line is that line extended through it.

Assumptions

•Exact convex QP (primal active-set), not a grid-search approximation.
•The covariance matrix is positive definite (no ridge regularisation is added).
•Expected returns come from the historical mean — the usual mean-variance input sensitivity applies.
•GMV $\Sigma^{-1}\mathbf{1}/(\mathbf{1}^{\mathsf T}\Sigma^{-1}\mathbf{1})$ and tangency $\Sigma^{-1}(\mu - r_f\mathbf{1})/(\mathbf{1}^{\mathsf T}\Sigma^{-1}(\mu - r_f\mathbf{1}))$ recover the unconstrained closed forms when the box does not bind.

Limitations

—Mean-variance is extremely sensitive to the expected-return inputs — small changes in $\boldsymbol{\mu}$ swing the weights, the classic estimation-error problem.
—It only sees mean and variance, so skew and fat tails are invisible to it (the case for pairing with $\mathrm{VaR}$ / $\mathrm{ES}$ ).
—Sample covariance is noisy for many assets; institutions often add shrinkage or a factor model that Strata's public endpoint does not.
—Long-only box constraints mean the constrained frontier can sit inside the unconstrained one.

vs Industry Tools

Bloomberg PORT: Also solves an exact mean-variance QP; may add covariance shrinkage and a richer constraint set.

Excel Solver: GRG Nonlinear / Simplex reaches the same optimum on the same convex QP, one target at a time.

↗Portfolio Variance ↗Sharpe Ratio

Go deeper

Markowitz (1952) — Portfolio Selection

Portfolio · risk

Portfolio Variance

How much a whole portfolio's return wobbles. It's not the average of each asset's risk — the cross-terms between assets are where diversification lives.

\sigma_p^2 = \mathbf{w}^{\mathsf T}\mathbf{\Sigma}\,\mathbf{w} = \sum_{i=1}^{n}\sum_{j=1}^{n} w_i\,w_j\,\sigma_{ij}

▸ Variables

\sigma_p^2

Portfolio variance (its square root

\sigma_p

is the portfolio volatility)

\mathbf{w}

Vector of asset weights, summing to 1 (fully invested)

\mathbf{\Sigma}

Covariance matrix of asset returns

\sigma_{ij}

Covariance between assets

i

and

j

(the diagonal

\sigma_{ii}

is asset

i

's own variance)

n

Number of assets in the portfolio

What Strata computes

Strata computes the full quadratic form $\mathbf{w}^{\mathsf T}\mathbf{\Sigma}\mathbf{w}$ from the sample covariance matrix — every pairwise term is included, so the diversification benefit from imperfectly-correlated assets is captured exactly. Volatility $\sigma_p$ is then the square root, annualised consistently with the covariance inputs.

Assumptions

•Weights sum to 1 (fully invested) and are market-value weights in the display currency.
•The covariance matrix is estimated from a historical window and is treated as stable going forward.
•Returns are approximately multivariate normal, so variance summarises the risk.

Limitations

—Variance treats upside and downside swings identically — it can't see skew or fat tails.
—It rests entirely on the estimated covariances; in a crisis correlations jump toward 1 and the diversification it credits evaporates.
—For $n$ assets there are $n(n-1)/2$ covariances to estimate — the matrix gets noisy fast without shrinkage or a factor model.

vs Industry Tools

Bloomberg PORT: Same quadratic form; may apply shrinkage to the covariance matrix for stability.

BARRA / Axioma: Use a factor-model covariance instead of the raw sample matrix to reduce estimation error.

↗Covariance Matrix ↗Efficient Frontier

Go deeper

Markowitz (1952) — Portfolio Selection

Portfolio · estimation

Covariance Matrix

The table of how every pair of assets moves together — the raw material that portfolio variance and the efficient frontier both consume.

\hat{\sigma}_{ij} = \dfrac{1}{T-1}\sum_{t=1}^{T}\bigl(r_{i,t} - \bar{r}_i\bigr)\bigl(r_{j,t} - \bar{r}_j\bigr)

▸ Variables

\hat{\sigma}_{ij}

Estimated covariance between assets

i

and

j

(the matrix entry)

r_{i,t}

Return of asset

i

in period

t

\bar{r}_i

Mean return of asset

i

over the window

T

Number of observations (default 252 trading days = 1 year)

T-1

Bessel's correction — the unbiased sample denominator

What Strata computes

Strata uses the plain sample covariance with the $T-1$ (Bessel-corrected) denominator over the lookback window — no shrinkage, no factor model. It's the transparent estimator: every entry is computable by hand from the return series, which matches Strata's self-compute positioning. Downstream, the efficient-frontier QP requires this matrix to be positive definite.

Assumptions

•Returns are stationary — the covariance structure is stable over the window.
•The sample is large enough relative to the number of assets for the estimates to be reliable.
•No structural break in the correlation regime inside the window.
•Simple sample estimator — no Ledoit-Wolf shrinkage or factor decomposition.

Limitations

—With $n$ assets there are $n(n+1)/2$ entries to estimate — for many assets and a short window the matrix is noisy and can become near-singular.
—Sample covariance gives equal weight to every observation, so it's slow to react to a regime change.
—Correlations are not constant — they spike toward 1 in a crisis, exactly when diversification is most needed.

vs Industry Tools

Bloomberg: Often applies Ledoit-Wolf shrinkage toward a structured target for more stable estimates.

BARRA / Axioma: Replace the raw sample matrix with a factor-model covariance to cut estimation error.

↗Portfolio Variance ↗Efficient Frontier ↗Monte-Carlo Simulation

Go deeper

Ledoit & Wolf (2004) — Honey, I Shrunk the Sample Covariance Matrix

Portfolio · risk-adjusted return

Risk-Adjusted Return

Return earned per unit of risk taken. One family, but each ratio divides by a different risk — and that choice decides which risk it rewards.

Approach	How it builds the distribution	Use when
Sharpe◢ default	Excess return over total volatility — reward per unit of all variability, up or down.	the return stream is roughly symmetric and you want the one universal benchmark.
Sortino◢ default	Excess return over downside deviation only — upside swings stop counting as risk.	returns are skewed and you only care about volatility that loses money.
Treynor	Excess return over systematic risk (beta) — reward per unit of market exposure, ignoring diversifiable risk.	the portfolio is a diversified sleeve of a larger book, so only market risk is priced.
Information Ratio	Active return over tracking error — skill per unit of deviation from the benchmark.	the mandate is to beat a benchmark and active risk is the budget that matters.
Jensen's Alpha	Return above what CAPM predicts for the portfolio's beta — value added net of market exposure.	you want the absolute outperformance attributable to skill rather than a ranking ratio.
Calmar	Return over the worst drawdown — reward per unit of the deepest pain an investor actually felt.	capital-preservation matters and the path of losses, not just their variance, is the risk.
└M² (Modigliani)	Sharpe restated in return units — what the portfolio would have earned levered to the benchmark's volatility $\sigma_b$ .	you need Sharpe expressed as a return an investor can compare to the benchmark directly.

Ratio

\dfrac{\text{excess return}}{\text{chosen risk measure}}

Every member is the same shape — an excess return over a denominator. The denominator is the whole lesson: it picks which risk gets rewarded.

▸ Variables

R_p,\ R_f

Portfolio return and the risk-free rate, on the same annualised basis

R_b

Benchmark return (the active/tracking reference)

\sigma_p

Total volatility — standard deviation of all returns (the Sharpe denominator)

\sigma_d

Downside deviation — standard deviation of returns below a target (the Sortino denominator)

\beta_p

Portfolio beta — sensitivity to the market,

\mathrm{cov}(R_p, R_m)/\sigma_m^2

(the Treynor / Jensen denominator)

R_m

Market (or benchmark) return used to fit beta

\mathrm{TE}

Tracking error — standard deviation of the active return

R_p - R_b

(the Information Ratio denominator)

\mathrm{MDD}

Maximum drawdown — the largest peak-to-trough decline over the window (the Calmar denominator)

\alpha_J

Jensen's alpha — return above the CAPM-predicted return

\sigma_b

Benchmark volatility — the standard deviation M² levers the portfolio's Sharpe ratio up to

△ The insight

Each ratio is excess return over a different risk — and the denominator is the opinion. Total vol (Sharpe) treats up and down alike; downside deviation (Sortino) forgives upside; beta (Treynor) prices only market risk; tracking error (Information) prices only benchmark deviation; max drawdown (Calmar) prices the worst path. Pick the denominator that matches the risk you actually fear.

What Strata computes

Strata leads with Sharpe (the universal cross-comparison) and Sortino (its skew-aware twin), then shows Treynor, Information Ratio, Jensen's alpha and Calmar alongside so the same stream can be judged on systematic, active and drawdown risk. M² is reported as Sharpe in return units. No single ratio is right — that's why it's a family, not one number.

Assumptions

•Returns and the risk-free / benchmark series are on the same annualised basis before any ratio is formed.
•Sharpe and Sortino assume volatility (total or downside) summarises risk; Treynor and Jensen assume a single-factor CAPM beta.
•Beta and tracking error are estimated from the historical window and are treated as stable over it.

Limitations

—All ratios are backward-looking — a calm window inflates every one of them.
—Ratios built on volatility miss fat tails; pair them with $\mathrm{VaR}$ / $\mathrm{ES}$ for the tail.
—Drawdown-based (Calmar) and benchmark-relative (Information Ratio) measures are especially sensitive to the chosen window and reference.
—A higher ratio is not automatically better risk-adjusted performance if the denominator misprices the portfolio's real risk.

vs Industry Tools

Morningstar / Bloomberg PORT: Report the same family; conventions differ (downside-target for Sortino, the benchmark and window for Information Ratio / Calmar).

↗Value at Risk ↗Sharpe Ratio

Go deeper

Sharpe (1994) — The Sharpe Ratio Modigliani & Modigliani (1997) — Risk-Adjusted Performance

Portfolio · risk-adjusted return

Sharpe Ratio

Excess return per unit of volatility. One accepted formula, the lesson is how you build each input.

\mathrm{SR} = \dfrac{\mathbb{E}[R_p] - R_f}{\sigma_p}

▸ Variables

\mathbb{E}[R_p]

Portfolio return, annualised (realised mean for ex-post, expected for ex-ante)

R_f

Risk-free rate on the same annualised basis (your chosen reference)

\sigma_p

Portfolio volatility, the annualised standard deviation of returns

What Strata computes

Strata reports the ex-post Sharpe: realised daily mean excess return annualised by $\times\sqrt{252}$ , with the risk-free basis you set (default $4\%$ ). The $\sqrt{252}$ scaling assumes returns are independent across days.

Assumptions

•Returns are independent and identically distributed across periods (what justifies the $\sqrt{\text{time}}$ annualisation).
•Returns are approximately normal, so mean and standard deviation summarise the whole distribution.
•The risk-free rate is on the same period/annualisation basis as the return.

Limitations

—Penalises upside volatility exactly like downside — for skewed return streams that understates a good manager (the case for $\textrm{Sortino}$ ).
—Under-states risk for fat-tailed / non-normal returns, where standard deviation misses the tail (the case for a $\textrm{Cornish-Fisher}$ adjustment).
— $\sqrt{252}$ annualisation breaks when returns are autocorrelated (e.g. monthly, illiquid, or smoothed series).
—The excess basis is a choice: the same stream scores differently against $R_f = 0\%$ vs $4\%$ .

vs Industry Tools — n/a · same formula everywhere, only conventions differ

↗Value at Risk ↗Efficient Frontier

Go deeper

Sharpe (1994) — The Sharpe Ratio

Portfolio · tail risk

Value at Risk

The loss not exceeded at a confidence level over a horizon. Several ways to estimate the distribution, two ways to measure the tail.

Approach	How it builds the distribution	Use when
Parametric	Assume returns are normal; read the loss straight off the $\alpha$ quantile.	fast checks on roughly linear, roughly normal books.
└Cornish-Fisher	Bend the normal quantile for skew $S$ and excess kurtosis $K$ — fat tails without a full simulation.	a cheap fat-tailed cross-check on the parametric number.
Historical◢ default	Rank the realised outcomes and take the empirical $(1-\alpha)$ percentile — no distribution assumed.	you have a clean lookback window and non-linear payoffs.
Monte-Carlo◢ default	Simulate correlated forward paths (GBM + Cholesky) and take the percentile of terminal values.	forward-looking paths or path-dependent positions.

VaR (95%)

\mathrm{VaR}_\alpha = V_0 - P_{1-\alpha}

A threshold: losses won't exceed this $\alpha$ of the time. Silent on how bad the other $(1-\alpha)$ gets.

ES / CVaR (97.5%)

\mathrm{ES}_\alpha = \mathbb{E}\bigl[L \mid L > \mathrm{VaR}_\alpha\bigr]

The tail average — mean loss beyond VaR, always $\ge$ VaR. A measure, not a method: compute it on any row.

▸ Variables

V_0

Initial portfolio value

\alpha

Confidence level (e.g. 95%, 97.5%)

z_\alpha

Standard-normal quantile at

\alpha

(

z_{95}=1.645

)

\mu,\ \sigma

Mean and standard deviation of P&L

P_{1-\alpha}

The

(1-\alpha)

percentile of the outcome distribution

L

Loss, as a positive number

S,\ K

Skewness and excess kurtosis (the Cornish-Fisher inputs)

△ The insight

VaR is not sub-additive — a diversified book can post a higher VaR than the sum of its parts, which is both wrong and gameable. ES is coherent (sub-additive), which is why Basel/FRTB moved the trading book to Expected Shortfall at 97.5%.

What Strata computes

Strata computes Historical + Monte-Carlo (one grounded in data, one in dynamics), each reported with ES alongside VaR. Cornish-Fisher is a fast fat-tailed cross-check; plain parametric is taught, then retired. No single row wins — that's why it's a grid.

Assumptions

•Every cell uses the same horizon and confidence — the only thing that varies is the distribution each approach assumes.
•Parametric/Cornish-Fisher assume a (corrected) normal; Historical assumes the window represents the future; Monte-Carlo assumes its simulation model.

Limitations

—Each approach is only as good as its inputs — a calm lookback understates risk for all of them.
—VaR ignores the shape of the tail beyond the cutoff — the reason ES exists.
—Parametric/Monte-Carlo inherit thin-tailed assumptions; Historical can't see un-sampled events.

vs Industry Tools

RiskMetrics / Basel FRTB: RiskMetrics popularised parametric VaR; Basel FRTB now mandates Expected Shortfall (97.5%) for the trading book — Strata reports both.

↗Sharpe Ratio ↗Monte-Carlo Simulation

Go deeper

Jorion — Value at Risk Basel — Minimum capital requirements for market risk (FRTB)

Portfolio · simulation

Monte-Carlo Simulation

Roll the portfolio forward thousands of times along random-but-realistic paths, then read the distribution of outcomes. The engine behind forward-looking VaR and projection fans.

S_{t+\Delta t} = S_t\,\exp\!\Bigl[\bigl(\mu - \tfrac{1}{2}\sigma^2\bigr)\Delta t + \sigma\sqrt{\Delta t}\,\mathbf{X}\Bigr],\qquad \mathbf{X} = \mathbf{L}\,\mathbf{Z},\;\; \mathbf{\Sigma} = \mathbf{L}\mathbf{L}^{\mathsf T}

▸ Variables

S_t

Asset (or portfolio) value at time

t

\mu

Expected return (drift)

\sigma

Volatility of the asset

\Delta t

Time step of the simulation (Strata uses monthly steps for efficiency)

\mathbf{Z}

Vector of independent standard-normal draws (one per asset, via Box-Muller)

\mathbf{X}

The correlated shock vector that actually drives each path

\mathbf{L}

Lower-triangular Cholesky factor of

\mathbf{\Sigma}

— turns independent normals into correlated ones

\mathbf{\Sigma}

Target covariance matrix the paths must reproduce

What Strata computes

Strata models each asset as geometric Brownian motion (log-normal, the Black-Scholes assumption) and injects cross-asset correlation through the Cholesky factor $\mathbf{L}$ of the covariance matrix: independent normals $\mathbf{Z}$ become correlated shocks $\mathbf{X} = \mathbf{L}\mathbf{Z}$ . Monthly time steps keep it fast enough for the live VaR endpoint (capped at 5,000 sims on the public API). A tiny $10^{-10}$ ridge is added before factoring for numerical stability.

Assumptions

•Returns are log-normally distributed (geometric Brownian motion, no jumps).
•Volatility and the correlation structure are constant over the simulation horizon.
•The covariance matrix is positive semi-definite so its Cholesky factor exists (a $10^{-10}$ regularisation is added for safety).
•Independent normals are generated via the Box-Muller transform, then correlated by $\mathbf{L}$ .

Limitations

—GBM's tails are thinner than reality — it under-states the chance of extreme moves, so MC VaR can look reassuring.
—Constant volatility and correlation ignore clustering and the crisis regime shift toward correlation 1.
—Output is only as good as the estimated $\mu$ and $\mathbf{\Sigma}$ inputs — garbage in, confident-looking garbage out.
—Monthly steps trade path resolution for speed; path-dependent payoffs need finer steps.

vs Industry Tools

RiskMetrics: Similar simulation approach; may use GARCH for time-varying volatility instead of constant

\sigma

Bloomberg MARS: Uses richer stochastic models for derivatives; Cholesky correlation is the universal standard either way.

↗Value at Risk ↗Covariance Matrix

Go deeper

Glasserman — Monte Carlo Methods in Financial Engineering

Portfolio · aggregation

Market-Value Weighted Metrics

How a per-holding number becomes a portfolio number: weight each holding by its market value. Bigger positions pull the average toward them.

X_{\text{port}} = \dfrac{\sum_i w_i\,X_i}{\sum_i w_i}

▸ Variables

X_{\text{port}}

Portfolio-level metric (e.g. duration, yield, dividend yield)

X_i

The same metric for holding

i

w_i

Market value of holding

i

, in the display currency

What Strata computes

Strata weights by market value in the display currency — every holding is converted via the FX cross-rate first, then weighted, so a USD and a EUR position are compared on one basis. This is the right roll-up for linear, value-additive metrics (duration, yield); metrics that aren't linear in value (e.g. volatility, which needs the covariance matrix) are computed at the portfolio level instead, not weighted.

Assumptions

•All holding values are converted to the display currency before weighting.
•Weights are positive market values (a long-only roll-up).
•The metric being aggregated is approximately linear in market value, so a simple weighted average is valid.

Limitations

—Only valid for value-additive metrics — averaging volatilities or betas this way ignores correlation and overstates risk.
—Stale or mismatched FX rates distort the weights across currencies.
—Equal-weighting or risk-weighting would give a different portfolio number for the same holdings.

vs Industry Tools — n/a · same formula everywhere, only conventions differ

↗Portfolio Variance ↗FX Cross-Rate

Go deeper

CFA — Portfolio Management: Return & Risk

Portfolio · currency

FX Cross-Rate

Get the price of one currency in another when every quote is against a common base. Go through the base: divide out the source, multiply in the target.

\mathrm{rate}_{\text{FROM}\to\text{TO}} = \dfrac{\mathrm{rates}[\text{TO}]}{\mathrm{rates}[\text{FROM}]}

▸ Variables

\text{FROM}

Source currency being converted out of

\text{TO}

Target currency being converted into

\mathrm{rates}[\cdot]

Quote map: units of that currency per 1 unit of the base (USD), so

\mathrm{rates}[\text{USD}] = 1

What Strata computes

Strata holds every rate as units per 1 USD and triangulates: $\text{FROM}\to\text{TO}$ is $\mathrm{rates}[\text{TO}]/\mathrm{rates}[\text{FROM}]$ (equivalently, USD-out then target-in). ECB is the primary source — EUR-based quotes are re-expressed onto the USD base on ingest. Every portfolio monetary calc routes through this so a multi-currency book is valued on one consistent basis.

Assumptions

•USD is the common base; all quotes are stored as 1 USD = $X$ of each currency.
•Cross rates are derived by triangulation through USD, not quoted directly.
•A single mid rate is used (no separate bid/ask).

Limitations

—Triangulation uses mid rates, so it ignores the bid-ask spread a real conversion would pay.
—The result is only as fresh as the underlying quotes — stale rates misprice the book.
—Two conversions (out of base, into target) compound any rounding in each leg.

vs Industry Tools

ECB reference rates: Strata's primary source; EUR-based, re-based to USD before triangulation.

↗Market-Value Weighted Metrics ↗Portfolio Variance

Go deeper

ECB — Euro foreign exchange reference rates

Portfolio · return estimation

Bond Return Proxy

When a bond has no usable price history, estimate its daily return from carry plus a duration-scaled move in the matched Treasury yield. A first-order approximation: it sees rates, not credit.

r_{\text{bond}} \approx \dfrac{c}{252} - D_{\text{mod}}\,\Delta y

▸ Variables

r_{\text{bond}}

Estimated daily bond return

c

Annual coupon rate (decimal);

c/252

is the daily carry

D_{\text{mod}}

Modified duration — the price sensitivity to a yield move

\Delta y

Daily change in the duration-matched benchmark UST yield

What Strata computes

Strata picks the benchmark UST by duration — ^IRX (3M), ^FVX (5Y), ^TNX (10Y), ^TYX (30Y) — applies the bond's modified duration to that day's yield change, and adds the daily carry $c/252$ . The credit spread is held constant, so this is a rates-only proxy: it fills the return series so portfolio analytics (covariance, VaR, Monte-Carlo) can run when no clean price history exists.

Assumptions

•The benchmark UST is matched to the bond by duration: ^IRX (3M), ^FVX (5Y), ^TNX (10Y), ^TYX (30Y).
•The credit spread is held constant — only the rates component drives the proxied return.
•Linear duration approximation: convexity is ignored.
•Daily carry is the annual coupon spread evenly across the year, $c/252$ .

Limitations

—Rates-only — it cannot see credit-spread moves, which dominate the return of lower-rated corporates (cross-link: credit-spread, DV01/CS01).
—First-order in duration: it ignores convexity, so it overstates losses and understates gains for large yield moves.
—A single benchmark tenor is a coarse match — a bond between key tenors inherits the wrong sensitivity to curve reshaping.
—Carry as $c/252$ ignores the actual accrual schedule and pull-to-par, so the level of return is approximate even when rates don't move.

vs Industry Tools

Bloomberg BRTN: Uses actual price changes; materially more accurate for corporate bonds, where credit-spread moves matter.

↗Duration ↗Monte-Carlo Simulation

Go deeper

CFA — Fixed-Income Risk & Return

Portfolio · rate stress

Portfolio Shock Analysis

Estimate the P&L hit from a parallel rate or spread shock across the book using duration. A fast first-order read — the same sensitivity that DV01 and duration express, applied as a finite shock.

\Delta P \approx -D_{\text{mod}}\,P\,\Delta y

▸ Variables

\Delta P

Estimated change in portfolio value from the shock

D_{\text{mod}}

Modified duration of the position (or book-weighted duration)

P

Current value of the position / portfolio

\Delta y

The applied yield (or spread) shock, in decimal

What Strata computes

Strata applies the book's modified duration to a user-set parallel shock $\Delta y$ to get a quick P&L estimate across all rate-sensitive holdings. This is the same first-order sensitivity as DV01 and duration, just expressed as a finite shock rather than a per-1bp or per-1% rate: $\Delta P \approx -\,\text{DV01} \times (\Delta y \text{ in bp})$ . Use it for a fast scan; for an exact figure on large moves, re-price (the duration measures already cross-link here).

Assumptions

•Linear, first-order approximation — the price/yield relationship is treated as a straight line over the shock.
•A single parallel shift is applied across all holdings (one $\Delta y$ for the whole book).
•Each position's modified duration is taken as fixed over the shock (no re-pricing).

Limitations

—Ignores convexity, so it overestimates losses and underestimates gains — the error grows with the size of the shock.
—Parallel-only: it cannot represent a steepener, flattener, or twist — only a uniform shift (use scenario/curve stress for those).
—It is the duration/DV01 sensitivity re-applied at a finite shock — it adds no new risk information beyond those, so read it alongside DV01 and duration, not instead of them.
—Spread and rate shocks are treated identically through one duration; it does not separate the credit (CS01) from the rate (DV01) channel.

vs Industry Tools — n/a · same formula everywhere, only conventions differ

↗DV01 ↗Duration ↗Portfolio Variance

Go deeper

CFA — Fixed-Income Risk & Return

Data Sources

External data feeds and refresh schedules

UST Yield Curve

US Treasury par yields for benchmark pricing

Daily (market close)

•Source: treasury.gov XML feed
•Tenors: 1M, 2M, 3M, 6M, 1Y, 2Y, 3Y, 5Y, 7Y, 10Y, 20Y, 30Y
•Interpolated for non-standard maturities

FX Exchange Rates

Foreign exchange rates for currency conversion

Daily (ECB reference rate)

•Primary: European Central Bank (ECB)
•Fallback providers for ECB downtime
•Supports G10 + RUB, THB

Equity Market Data

Stock prices, fundamentals, and historical data

Intraday (delayed)

•Price history: Alpha Vantage
•Fundamentals: Financial Modeling Prep (FMP)
•Some fundamental metrics may be unavailable

SEC EDGAR XBRL Financials

Standardised financial statement data sourced from SEC XBRL inline filings (10-K and 10-Q)

As filed (SEC updates within hours of submission)

•Source: SEC EDGAR XBRL API (free, no key required)
•60+ standardised metrics: revenue, operating income, net income, EPS, assets, liabilities, equity, capex, D&A, working capital
•Up to 20 years of annual history where available
•Rate limit: 10 req/sec; User-Agent: Project Strata/1.0
•Bank/financial sector requires alternate GAAP line items (e.g., InterestAndDividendIncomeOperating)
•Used in: Equity Lab financials tab, DCF inputs auto-population, comps fundamentals

FRED — Federal Reserve Economic Data

US macroeconomic and rates data from the St. Louis Federal Reserve

Daily (market close for rates series)

•Series used: DGS10 (10Y UST yield, risk-free rate for WACC/CAPM)
•SOFR compounded averages (SOFR30DAYAVG, SOFR90DAYAVG, SOFR180DAYAVG) for short end of USD swap curve
•DGS2, DGS5, DGS10, DGS30 for market context panel
•FRED FEDFUNDS for federal funds rate display
•ICE swap rate series (ICERATES*) and H.15 DSWP series discontinued — not available
•Fallback: DGS10 defaults to 4.25% when FRED is unreachable

Damodaran Online — Valuation Datasets

Annual datasets from Prof. Aswath Damodaran (NYU Stern) used for market-implied equity risk premium, industry betas, and default spreads

Annual (updated each January)

•Implied equity risk premium (ERP): used as fallback when live ERP is unavailable (default 4.60%)
•Industry unlevered betas: used when FMP live beta is outside [0.1, 5.0] or unavailable
•Country risk premiums (CRP) by nation for international WACC estimation
•Default spreads by credit rating and seniority for cost-of-debt estimation
•Hosted at: pages.stern.nyu.edu/~adamodar/

Project Strata is for educational purposes. Calculations are illustrative and should not be used for actual trading decisions.

Methodology

Fixed Income

Duration

△ The insight

What Strata computes

Assumptions

Limitations

vs Industry Tools

Related

Convexity

What Strata computes

Assumptions

Limitations

vs Industry Tools

Related

DV01

What Strata computes

Assumptions

Limitations

vs Industry Tools

Related

CS01

What Strata computes

Assumptions

Limitations

vs Industry Tools

Related

Credit Spread

What Strata computes

Assumptions

Limitations

vs Industry Tools

Related

Bond Pricing

Dirty price

△ The insight

What Strata computes

Assumptions

Limitations

vs Industry Tools

Related

YTM Solver

What Strata computes

Assumptions

Limitations

vs Industry Tools

Related

Day-Count Conventions

Year fraction

△ The insight

What Strata computes

Assumptions

Limitations

vs Industry Tools

Related

Accrued Interest

What Strata computes

Assumptions

Limitations

vs Industry Tools

Related

Horizon Return

What Strata computes

Assumptions

Limitations

vs Industry Tools

Related

Derivatives

Option Pricing

Price

Implied vol

△ The insight

What Strata computes

Assumptions

Limitations

vs Industry Tools

Related

Linear Derivatives

△ The insight

What Strata computes