# Finance Formula Sheet: [Course, exam or desk]

> A personal formula reference for core finance mathematics, written in
> KaTeX: time value of money, returns, statistics and risk, portfolio
> and performance measures, risk measures, fixed income, equity
> valuation, corporate finance and ratios, and derivatives basics. Each
> block gives the formula, the notation, when to use it and the most
> common mistake, followed by a table of worked checks you can
> reproduce on a calculator. The formulas are general financial
> mathematics found in many textbooks; the notation, wording and
> examples here are original, and no exam curriculum text is used.
> Replace every `[bracketed]` value and every row marked *Example*. This
> is a document format, not financial, investment, tax or accounting
> advice.

<!--
Guidance: How to use this template
- Fill section 1 (notation and conventions) first. Most errors in
  finance calculations come from mismatched conventions: an annual rate
  with monthly periods, a percentage typed as a whole number, a price
  quoted per 100 of face value used as a currency amount.
- Rewrite each block in your own notation if your course or exam uses
  different symbols. A sheet you have written yourself is easier to
  remember than one you have copied.
- Delete sections you do not need and add your own: the sheet should
  fit what you are studying or the desk you work on.
- Check every worked example in section 11 on your own calculator
  before you rely on the sheet. If your result differs, find out why.
- Related templates: Exam Study Plan (`finance/exam-study-plan`), Study
  Guide (`academic/study-guide`), Investment Appraisal
  (`finance/investment-appraisal`), Valuation Report
  (`finance/valuation-report`), Bond Credit Analysis
  (`finance/bond-credit-analysis`).
-->

## Document control

<!-- Guidance: Record which course, exam or desk the sheet supports and the conventions it assumes, so that anyone reusing it knows when a formula does not apply. -->

| Field | Value |
|---|---|
| Document ID | [FORM-001] |
| Course, exam or desk | [Course code / exam and level / desk] |
| Notation convention | [This sheet / course notation] |
| Rate convention | [Decimal rates, annual unless stated] |
| Compounding default | [Annual / semi-annual / continuous] |
| Day count default | [30/360 / Actual/365 / Actual/360 / Actual/Actual] |
| Calculator | [Model] |
| Version | [0.1] |
| Status | [Draft / In use / Archived] |
| Owner | [@owner] |
| Approvers/reviewers | [@study-partner or @desk-head] |
| Last updated | YYYY-MM-DD |
| Classification | [Personal / Internal] |

### Approvals

| Role | Name | Decision | Date |
|---|---|---|---|
| Owner | [@owner] | Checked all worked examples | YYYY-MM-DD |
| Reviewer (optional) | [@name] | [Reviewed / Corrections requested] | YYYY-MM-DD |

### Revision history

| Version | Date | Author | Change summary |
|---|---|---|---|
| 0.1 | YYYY-MM-DD | [@owner] | First draft |
| 1.0 | YYYY-MM-DD | [@owner] | *Example:* All worked checks reproduced on calculator |
| 1.1 | YYYY-MM-DD | [@owner] | *Example:* Added continuous compounding variants for derivatives |

## Contents

<!-- Guidance: Keep this list aligned with the headings. -->

1. [Notation and conventions](#1-notation-and-conventions)
2. [Time value of money](#2-time-value-of-money)
3. [Returns](#3-returns)
4. [Statistics and regression](#4-statistics-and-regression)
5. [Portfolio and performance measures](#5-portfolio-and-performance-measures)
6. [Risk measures](#6-risk-measures)
7. [Fixed income](#7-fixed-income)
8. [Equity valuation](#8-equity-valuation)
9. [Corporate finance and ratios](#9-corporate-finance-and-ratios)
10. [Derivatives basics](#10-derivatives-basics)
11. [Worked checks](#11-worked-checks)
12. [Calculator notes](#12-calculator-notes)
13. [Glossary and references](#13-glossary-and-references)
14. [Standard and licence note](#standard-and-licence-note)

## 1. Notation and conventions

<!-- Guidance: Define every symbol once. State the defaults (rates in decimals, annual periods, end-of-period cash flows) and flag every formula that departs from them. Pitfall: using one symbol for two things, such as r for both a discount rate and a return. -->

| Symbol | Meaning |
|---|---|
| $PV$, $FV$ | Present value, future value |
| $C_t$ or $CF_t$ | Cash flow at the end of period $t$ |
| $r$ | Discount rate or required return per period (decimal) |
| $n$, $T$ | Number of periods; time in years |
| $m$ | Compounding periods per year |
| $g$ | Constant growth rate |
| $R_p$, $R_f$, $R_m$, $R_b$ | Portfolio, risk-free, market and benchmark returns |
| $\sigma$, $\sigma^2$ | Standard deviation, variance |
| $\rho$ | Correlation coefficient |
| $\beta$ | Beta (sensitivity to the market or a factor) |
| $w_i$ | Portfolio weight of asset $i$ |
| $y$ | Yield to maturity per period |
| $t_c$ | Corporate tax rate |
| $S$, $K$ | Spot price of the underlying, strike or delivery price |
| $c$, $p$ | European call and put prices |
| $q$ | Continuous dividend or income yield (forward pricing) |
| $\pi$, $\pi_u$ | Inflation rate; risk-neutral probability of the up move (binomial model) |
| $r^{\log}$ | Log (continuously compounded) return |
| $D_t$; $D$; $D_{\text{Mac}}$, $D_{\text{mod}}$, $D_{\text{eff}}$ | Dividend in period $t$; market value of debt; Macaulay, modified and effective duration |
| $E$ | Market value of equity |

Conventions used throughout unless a block says otherwise:

- Rates are decimals: 5% is written 0.05.
- Cash flows occur at the end of each period (ordinary annuity).
- The rate and the period must match: with monthly periods, use the monthly rate.
- Money amounts are written with the currency code, for example USD 1,000 or EUR 1,000.
- Bond prices are per 100 of face value.
- Continuous compounding uses $e$, the base of natural logarithms.

## 2. Time value of money

<!-- Guidance: These are the building blocks for every valuation. Learn the annuity and perpetuity forms by heart; most other formulas are sums of them. Pitfall: mixing annual rates with monthly or semi-annual periods. -->

**Future and present value of a single sum.** Use for one cash flow moved through time.

$$
FV = PV \times (1 + r)^n \qquad PV = \frac{FV}{(1 + r)^n}
$$

**Effective annual rate.** Use to compare rates quoted with different compounding. A stated annual rate $r_s$ compounded $m$ times a year, or continuously:

$$
EAR = \left(1 + \frac{r_s}{m}\right)^m - 1 \qquad EAR_{\text{cont}} = e^{r_s} - 1
$$

Pitfall: comparing a monthly-compounded loan rate with an annually compounded deposit rate without converting both to an effective rate.

**Ordinary annuity.** Equal payments $PMT$ at the end of each of $n$ periods:

$$
PV = PMT \times \frac{1 - (1 + r)^{-n}}{r} \qquad FV = PMT \times \frac{(1 + r)^n - 1}{r}
$$

**Annuity due.** Payments at the start of each period: multiply the ordinary annuity value by $(1 + r)$.

**Loan payment.** Solve the annuity formula for the payment; use for mortgages and amortising loans:

$$
PMT = \frac{PV \times r}{1 - (1 + r)^{-n}}
$$

**Perpetuity and growing perpetuity.** First payment at the end of period 1, growth $g < r$:

$$
PV = \frac{C}{r} \qquad PV = \frac{C_1}{r - g}
$$

**Growing annuity.** $n$ payments starting with $C_1$ and growing at $g$, with $g \neq r$:

$$
PV = \frac{C_1}{r - g}\left[1 - \left(\frac{1 + g}{1 + r}\right)^n\right]
$$

**Net present value and internal rate of return.** NPV discounts every cash flow, including the initial outlay at $t = 0$. IRR is the rate that sets NPV to zero; it has no closed form and is solved by iteration or a calculator.

$$
NPV = \sum_{t=0}^{n} \frac{C_t}{(1 + r)^t} \qquad 0 = \sum_{t=0}^{n} \frac{C_t}{(1 + IRR)^t}
$$

Pitfalls: a project with sign changes in its cash flows after the start can have more than one IRR or none; when two projects are mutually exclusive, choose by NPV, not IRR, because IRR ignores scale.

## 3. Returns

<!-- Guidance: Know which return measure answers which question: holding period return for one period, geometric mean for compound growth, time-weighted return to judge a manager, money-weighted return to judge an investor's own result. Pitfall: averaging annual returns arithmetically to describe multi-year growth. -->

**Holding period return.** Price change plus income, divided by the starting price:

$$
HPR = \frac{P_1 - P_0 + D_1}{P_0}
$$

**Arithmetic and geometric mean.** For returns $R_1, \dots, R_n$:

$$
\bar{R}_A = \frac{1}{n}\sum_{t=1}^{n} R_t \qquad \bar{R}_G = \left[\prod_{t=1}^{n}(1 + R_t)\right]^{1/n} - 1
$$

The geometric mean is never above the arithmetic mean; the gap grows with volatility. Use the arithmetic mean as an estimate of a single future period's expected return, and the geometric mean to describe past compound growth.

**Annualising.** A return $R$ earned over $n$ years, or over $d$ days of a 365-day year:

$$
R_{\text{annual}} = (1 + R)^{1/n} - 1 \qquad R_{\text{annual}} = (1 + R)^{365/d} - 1
$$

Pitfall: annualising a return over a few days or weeks produces misleading numbers; many performance reports do not annualise periods shorter than one year.

**Log (continuously compounded) return.** Written $r^{\log}$ so that it is not confused with the discount rate $r$. Log returns add up over time:

$$
r^{\log} = \ln(1 + R) = \ln\frac{P_1 + D_1}{P_0}
$$

**Real return (Fisher relation).** Nominal return $R$, inflation $\pi$:

$$
1 + R_{\text{real}} = \frac{1 + R}{1 + \pi} \qquad R_{\text{real}} \approx R - \pi
$$

**Time-weighted return (TWR).** Split the period at each external cash flow, compute each sub-period return $R_i$ (ending value before the flow divided by starting value after the previous flow, minus one) and chain-link them. TWR removes the effect of the timing of client contributions and withdrawals, so it is the standard measure of a manager's skill.

$$
TWR = \prod_{i=1}^{k}(1 + R_i) - 1
$$

**Money-weighted return (MWR).** The internal rate of return of the investor's cash flows, with the ending value as the final inflow. It reflects the investor's timing decisions, so it can differ a lot from TWR when large flows happen before strong or weak periods. From the investor's side, contributions (including the starting value) are negative and withdrawals and the ending value are positive:

$$
0 = \sum_{t=0}^{n}\frac{CF_t}{(1 + MWR)^t}
$$

## 4. Statistics and regression

<!-- Guidance: These formulas underpin risk, beta and most empirical work. Know when to use the sample (n - 1) or population (n) version. Pitfall: annualising a standard deviation by multiplying by the number of periods instead of by its square root. -->

**Sample variance and standard deviation.** For $n$ observations with mean $\bar{X}$:

$$
s^2 = \frac{1}{n - 1}\sum_{i=1}^{n}(X_i - \bar{X})^2 \qquad s = \sqrt{s^2}
$$

**Covariance and correlation.**

$$
\operatorname{Cov}(X, Y) = \frac{1}{n - 1}\sum_{i=1}^{n}(X_i - \bar{X})(Y_i - \bar{Y}) \qquad \rho_{XY} = \frac{\operatorname{Cov}(X, Y)}{\sigma_X \sigma_Y}
$$

**Annualising volatility.** If returns are independent over time, variance scales with time, so standard deviation scales with the square root of time:

$$
\sigma_{\text{annual}} = \sigma_{\text{monthly}}\sqrt{12} \qquad \sigma_{\text{annual}} = \sigma_{\text{daily}}\sqrt{252}
$$

Use the number of trading days your source uses (252 is common for equities).

**Simple linear regression (ordinary least squares).** For $Y_i = b_0 + b_1 X_i + \varepsilon_i$:

$$
\hat{b}_1 = \frac{\operatorname{Cov}(X, Y)}{\operatorname{Var}(X)} \qquad \hat{b}_0 = \bar{Y} - \hat{b}_1\bar{X}
$$

**Coefficient of determination and t-statistic.**

$$
R^2 = 1 - \frac{SSE}{SST} \qquad t = \frac{\hat{b}_1 - b_1^{H_0}}{s_{\hat{b}_1}}
$$

where $SSE$ is the sum of squared residuals and $SST$ the total sum of squares around the mean of $Y$. In a simple regression, $R^2$ equals the squared correlation between $X$ and $Y$.

**Standardised value (z-score).** Distance from the mean in standard deviations: $z = (X - \mu)/\sigma$.

## 5. Portfolio and performance measures

<!-- Guidance: Portfolio risk depends on correlations, not only on the risk of each asset. Performance ratios are only comparable when measured over the same period, frequency and risk-free rate. Pitfall: comparing a Sharpe ratio from monthly data with one from annual data. -->

**Two-asset portfolio.** Expected return and variance:

$$
E(R_p) = w_1 E(R_1) + w_2 E(R_2)
$$

$$
\sigma_p^2 = w_1^2\sigma_1^2 + w_2^2\sigma_2^2 + 2 w_1 w_2 \rho_{12}\sigma_1\sigma_2
$$

**Many assets (matrix form).** With weight vector $\mathbf{w}$ and covariance matrix $\Sigma$: $\sigma_p^2 = \mathbf{w}^\top \Sigma\, \mathbf{w}$.

**Beta.** Sensitivity of an asset to the market:

$$
\beta_i = \frac{\operatorname{Cov}(R_i, R_m)}{\sigma_m^2} = \rho_{im}\frac{\sigma_i}{\sigma_m}
$$

**Capital asset pricing model (CAPM).** Required return given beta:

$$
E(R_i) = R_f + \beta_i\left[E(R_m) - R_f\right]
$$

**Sharpe ratio and Treynor ratio.** Excess return per unit of total risk and per unit of systematic risk:

$$
S_p = \frac{R_p - R_f}{\sigma_p} \qquad T_p = \frac{R_p - R_f}{\beta_p}
$$

**Jensen's alpha.** Return above the CAPM prediction:

$$
\alpha_p = R_p - \left[R_f + \beta_p(R_m - R_f)\right]
$$

**Tracking error and information ratio.** Active return over its volatility:

$$
TE = \sigma(R_p - R_b) \qquad IR = \frac{\bar{R}_p - \bar{R}_b}{TE}
$$

**Sortino ratio.** Excess return over a target $R_{\text{target}}$ per unit of downside deviation $\sigma_d$ (volatility of returns below the target only):

$$
\text{Sortino} = \frac{R_p - R_{\text{target}}}{\sigma_d}
$$

**M-squared.** Portfolio return scaled to the market's volatility, comparable directly with the market return:

$$
M^2 = R_f + S_p\,\sigma_m
$$

Pitfall: a high Sharpe ratio built on strategies with rare large losses (selling options, for example) can hide tail risk that standard deviation does not capture; pair it with the measures in section 6.

## 6. Risk measures

<!-- Guidance: Value at risk answers "how much could we lose at a given confidence over a given horizon, in normal conditions". It says nothing about the size of losses beyond that level; expected shortfall does. Pitfall: scaling a one-day VaR to longer horizons with the square root of time when returns are autocorrelated or positions change. -->

**Parametric (variance-covariance) VaR.** For a position of value $V$, return volatility $\sigma$ over the horizon, expected return assumed zero, and the standard normal quantile $z_\alpha$ (1.645 at 95%, 2.326 at 99%):

$$
\text{VaR}_\alpha = z_\alpha \,\sigma\, V
$$

**Scaling to a longer horizon.** Under independent, identically distributed returns:

$$
\text{VaR}_{h\text{-day}} = \text{VaR}_{1\text{-day}}\sqrt{h}
$$

**Expected shortfall (normal distribution).** The average loss in the tail beyond the VaR level, with $\phi$ the standard normal density:

$$
ES_\alpha = \sigma V \frac{\phi(z_\alpha)}{1 - \alpha}
$$

**Historical simulation VaR.** Sort past portfolio profit and loss outcomes and read the loss at the chosen percentile. It needs no distribution assumption but depends entirely on the chosen look-back window.

**Maximum drawdown.** The largest fall from a running peak to a later trough, as a percentage of the peak:

$$
MDD = \max_{t}\left(\frac{\max_{s \le t} V_s - V_t}{\max_{s \le t} V_s}\right)
$$

Backtesting VaR (counting days when the loss exceeded VaR) is covered in VaR Backtesting Report (`finance/var-backtesting-report`).

## 7. Fixed income

<!-- Guidance: Bond formulas assume a flat yield and a known schedule of cash flows unless stated. Duration measures price sensitivity to small parallel yield changes; convexity corrects for larger changes. Pitfall: using Macaulay duration (a time measure) directly as a price sensitivity without dividing by one plus the periodic yield. -->

**Bond price.** Coupon $C$ per period, face value $F$, $n$ periods, yield $y$ per period:

$$
P = \sum_{t=1}^{n}\frac{C}{(1 + y)^t} + \frac{F}{(1 + y)^n} = C\,\frac{1 - (1 + y)^{-n}}{y} + \frac{F}{(1 + y)^n}
$$

For semi-annual coupons, use half the annual coupon, half the annual yield and twice the number of years.

**Current yield.** Annual coupon divided by clean price. It ignores the pull to par and is not a total return measure.

**Accrued interest and full price.** Accrued interest is the coupon times the fraction of the coupon period elapsed, using the bond's day count convention. The full (dirty) price is the clean price plus accrued interest.

**Macaulay and modified duration.** Weighted average time to the cash flows, then converted to price sensitivity:

$$
D_{\text{Mac}} = \frac{1}{P}\sum_{t=1}^{n}\frac{t \cdot CF_t}{(1 + y)^t} \qquad D_{\text{mod}} = \frac{D_{\text{Mac}}}{1 + y}
$$

With semi-annual periods, $D_{\text{Mac}}$ comes out in half-years; divide by 2 for years.

**Effective duration.** For bonds with embedded options or when cash flows change with rates, reprice with yields down and up by $\Delta y$:

$$
D_{\text{eff}} = \frac{P_{-} - P_{+}}{2 P_0 \,\Delta y}
$$

**Convexity.** For annual periods:

$$
\text{Convexity} = \frac{1}{P(1 + y)^2}\sum_{t=1}^{n}\frac{t(t + 1)\,CF_t}{(1 + y)^t}
$$

With semi-annual periods ($y$ the semi-annual yield, $t$ in half-years), divide the result by 4 ($m^2$) to express convexity in years.

**Price change approximation.**

$$
\frac{\Delta P}{P} \approx -D_{\text{mod}}\,\Delta y + \frac{1}{2}\,\text{Convexity}\,(\Delta y)^2
$$

**Price value of a basis point (PVBP or DV01).** The price change for a one basis point change in yield: $\text{PVBP} \approx D_{\text{mod}} \times P \times 0.0001$.

**Spot and forward rates.** With annual spot rates $z_1$ and $z_2$, the one-year rate one year forward, $f_{1,1}$, satisfies:

$$
(1 + z_2)^2 = (1 + z_1)(1 + f_{1,1})
$$

Bootstrapping derives spot rates one maturity at a time from coupon bond prices, using the spot rates already found for earlier cash flows.

Credit analysis of bond issuers is covered in Bond Credit Analysis (`finance/bond-credit-analysis`).

## 8. Equity valuation

<!-- Guidance: Every equity model is a present value of expected cash flows to a claim holder, discounted at the rate that matches that claim. Free cash flow to the firm goes with the weighted average cost of capital; free cash flow to equity goes with the cost of equity. Pitfall: discounting free cash flow to the firm at the cost of equity, or the reverse. -->

**Dividend discount model (constant growth).** Next year's dividend $D_1$, required return $r$, growth $g < r$:

$$
V_0 = \frac{D_1}{r - g}
$$

**Two-stage dividend discount model.** High growth for $n$ years, then constant growth $g_L$:

$$
V_0 = \sum_{t=1}^{n}\frac{D_t}{(1 + r)^t} + \frac{1}{(1 + r)^n}\cdot\frac{D_{n+1}}{r - g_L}
$$

**Sustainable growth rate.** Retention ratio $b$ times return on equity:

$$
g = b \times ROE
$$

**Justified price to earnings ratio.** From the constant growth model, with payout ratio $(1 - b)$:

$$
\frac{P_0}{E_1} = \frac{1 - b}{r - g} \qquad \frac{P_0}{E_0} = \frac{(1 - b)(1 + g)}{r - g}
$$

**Free cash flow to the firm (FCFF) and to equity (FCFE).**

$$
FCFF = EBIT(1 - t_c) + D\&A - \text{Capex} - \Delta NWC
$$

$$
FCFE = FCFF - \text{Interest}(1 - t_c) + \text{Net borrowing}
$$

**Firm value and terminal value.** Discount FCFF at the weighted average cost of capital; the terminal value uses the constant growth form at the end of the explicit forecast:

$$
V_{\text{firm}} = \sum_{t=1}^{n}\frac{FCFF_t}{(1 + WACC)^t} + \frac{TV_n}{(1 + WACC)^n} \qquad TV_n = \frac{FCFF_{n+1}}{WACC - g}
$$

Equity value is firm value minus net debt and other non-equity claims (preferred stock, minority interests, pension deficits where treated as debt).

**Enterprise value.** Market capitalisation plus net debt and other claims; enterprise value multiples (EV/EBITDA, EV/Sales) are compared with capital-structure-neutral earnings.

**Residual income model.** Book value plus the present value of earnings above the equity charge:

$$
V_0 = B_0 + \sum_{t=1}^{\infty}\frac{E_t - r B_{t-1}}{(1 + r)^t}
$$

Pitfall: the terminal value often makes up most of a discounted cash flow value; always show the share it represents and a sensitivity to $WACC$ and $g$. A full valuation write-up belongs in Valuation Report (`finance/valuation-report`).

## 9. Corporate finance and ratios

<!-- Guidance: Ratios are only meaningful against a benchmark: the same company over time, peers, or a covenant threshold. State whether balance sheet items are year-end or average values. Pitfall: mixing a full-year income statement figure with a single balance sheet date in a busy season. -->

**Weighted average cost of capital.** Market-value weights of equity $E$ and debt $D$, cost of equity $r_E$, pre-tax cost of debt $r_D$:

$$
WACC = \frac{E}{D + E} r_E + \frac{D}{D + E} r_D (1 - t_c)
$$

**Unlevering and relevering beta.** Assuming debt carries no market risk:

$$
\beta_U = \frac{\beta_L}{1 + (1 - t_c)\dfrac{D}{E}} \qquad \beta_L = \beta_U\left[1 + (1 - t_c)\frac{D}{E}\right]
$$

**DuPont analysis.** Three-step and five-step decomposition of return on equity:

$$
ROE = \frac{NI}{\text{Sales}} \times \frac{\text{Sales}}{\text{Assets}} \times \frac{\text{Assets}}{\text{Equity}}
$$

$$
ROE = \frac{NI}{EBT} \times \frac{EBT}{EBIT} \times \frac{EBIT}{\text{Sales}} \times \frac{\text{Sales}}{\text{Assets}} \times \frac{\text{Assets}}{\text{Equity}}
$$

**Liquidity, leverage and coverage.**

| Ratio | Definition |
|---|---|
| Current ratio | Current assets / current liabilities |
| Quick ratio | (Cash + short-term investments + receivables) / current liabilities |
| Debt to equity | Total debt / total equity |
| Net debt to EBITDA | (Debt - cash) / EBITDA |
| Interest coverage | EBIT / interest expense |
| Debt service coverage ratio (DSCR) | Cash flow available for debt service / (interest + scheduled principal) |

**Working capital cycle.** With days in the year $N$ (365 unless stated):

$$
DSO = \frac{\text{Receivables}}{\text{Revenue}} N \qquad DIO = \frac{\text{Inventory}}{\text{COGS}} N \qquad DPO = \frac{\text{Payables}}{\text{COGS}} N
$$

$$
CCC = DSO + DIO - DPO
$$

**Degree of operating leverage.** Percentage change in EBIT per percentage change in sales, equal to contribution margin over EBIT at a given level of sales:

$$
DOL = \frac{\%\Delta EBIT}{\%\Delta \text{Sales}} = \frac{\text{Sales} - \text{Variable costs}}{EBIT}
$$

**Capital budgeting extras.** Profitability index (PI) is the present value of future cash flows divided by the initial investment; accept when PI is above 1, which matches a positive NPV. Payback is the time until cumulative undiscounted cash flows turn positive; discounted payback uses discounted cash flows. The equivalent annual annuity compares projects with different lives:

$$
EAA = NPV \times \frac{r}{1 - (1 + r)^{-n}}
$$

Covenant definitions in loan agreements often differ from textbook ratios; use the agreement's definitions in Covenant Compliance Certificate (`finance/covenant-compliance-certificate`).

## 10. Derivatives basics

<!-- Guidance: Prices in this section come from no-arbitrage arguments: if the formula does not hold, a riskless profit exists. Be clear whether rates are continuously compounded or discrete. Pitfall: confusing the forward price (set at inception so the contract is worth zero) with the value of an existing forward contract. -->

**Forward price (cost of carry).** Continuous compounding, risk-free rate $r$, continuous income or dividend yield $q$:

$$
F_0 = S_0\, e^{(r - q)T}
$$

With discrete compounding and no income: $F_0 = S_0 (1 + r)^T$. With known cash income, subtract its present value from $S_0$ first.

**Value of an existing long forward** with delivery price $K$ at time $t$:

$$
V_t = (F_t - K)\,e^{-r(T - t)}
$$

**Option payoffs at expiry.**

$$
\text{Call} = \max(S_T - K, 0) \qquad \text{Put} = \max(K - S_T, 0)
$$

Profit equals payoff minus the premium paid (for the buyer).

**Put-call parity (European options, no dividends).**

$$
c + K e^{-rT} = p + S_0
$$

**One-period binomial model.** Up factor $u$, down factor $d$, risk-neutral probability $\pi_u$ of the up move (not the dividend yield $q$), discrete rate $r$ per period:

$$
\pi_u = \frac{(1 + r) - d}{u - d} \qquad c_0 = \frac{\pi_u\,c_u + (1 - \pi_u)\,c_d}{1 + r}
$$

**Black-Scholes-Merton (European options, no dividends).** With $N(\cdot)$ the standard normal cumulative distribution:

$$
c = S_0 N(d_1) - K e^{-rT} N(d_2) \qquad p = K e^{-rT} N(-d_2) - S_0 N(-d_1)
$$

$$
d_1 = \frac{\ln(S_0/K) + (r + \sigma^2/2)\,T}{\sigma\sqrt{T}} \qquad d_2 = d_1 - \sigma\sqrt{T}
$$

**Covered interest rate parity.** Spot rate $S$ and forward rate $F$ quoted as units of the price currency per one unit of the base currency, with interest rates $i_{\text{price}}$ and $i_{\text{base}}$ over the same period:

$$
F = S \times \frac{1 + i_{\text{price}}}{1 + i_{\text{base}}}
$$

**Plain vanilla interest rate swap fixed rate.** With discount factors $Z_i$ for each payment date and equal accrual periods:

$$
\text{Swap rate (per period)} = \frac{1 - Z_n}{\sum_{i=1}^{n} Z_i}
$$

Hedging policy and hedge accounting documentation are covered in FX and Interest Rate Hedging Policy (`finance/fx-interest-rate-hedging-policy`) and Hedge Designation Documentation (`finance/hedge-designation-documentation`).

## 11. Worked checks

<!-- Guidance: Reproduce each example on your calculator before you trust the sheet. Every value below has been recomputed; results are rounded for display. Pitfall: changing the inputs and forgetting to change the rounding or the period count. -->

| ID | Formula | Inputs *(Example)* | Result *(Example)* |
|---|---|---|---|
| WC-01 | Future value | USD 1,000, 5% a year, 10 years | USD 1,628.89 |
| WC-02 | Annuity present value | USD 100 a year, 6%, 10 years | USD 736.01 |
| WC-03 | Effective annual rate | 12% stated, monthly compounding | 12.68% (continuous: 12.75%) |
| WC-04 | Loan payment | USD 200,000, 6% a year paid monthly (0.5% a month), 360 months | USD 1,199.10 a month |
| WC-05 | NPV and IRR | -1,000 at t0; 300, 400, 500 at t1 to t3; r = 8% | NPV 17.63; IRR 8.90% |
| WC-06 | Real return | Nominal 7%, inflation 2.5% | 4.39% (approximation 4.5%) |
| WC-07 | TWR and MWR | Start 100; value 110 at end of period 1, then deposit 50; value 152 at end of period 2 | TWR 4.50% (10% then -5%); MWR 0.80% per period |
| WC-08 | Two-asset volatility | Weights 60/40; volatilities 20% and 10%; correlation 0.3 | Variance 0.01888; volatility 13.74% |
| WC-09 | CAPM | Risk-free 3%, beta 1.2, market risk premium 5% | 9.0% |
| WC-10 | Sharpe ratio | Return 8%, risk-free 3%, volatility 12.5% | 0.40 |
| WC-11 | Treynor and Jensen | Return 8%, risk-free 3%, beta 1.1, market 7% | Treynor 4.55%; alpha 0.60% |
| WC-12 | Information ratio | Active return 1.5%, tracking error 3% | 0.50 |
| WC-13 | Parametric VaR | USD 10,000,000, daily volatility 1.2%, 99% (z = 2.3263) | 1-day USD 279,156; 10-day USD 882,769 |
| WC-14 | Expected shortfall | Same inputs, 99% | 1-day USD 319,861 |
| WC-15 | Bond price | 5-year annual 4% coupon, yield 5%, per 100 face | 95.6705 |
| WC-16 | Duration and convexity | Same bond | Macaulay 4.6203 years; modified 4.4003; convexity 24.47 |
| WC-17 | Price change estimate | Same bond, yield +1% | Estimate -4.278% (duration -4.400%, convexity +0.122%); full repricing -4.281% |
| WC-18 | PVBP | Same bond | 0.0421 per 100 face |
| WC-19 | Forward rate | One-year spot 3%, two-year spot 4% | 5.01% |
| WC-20 | Constant growth DDM | Next dividend USD 2.00, r = 9%, g = 4% | USD 40.00 |
| WC-21 | WACC | Equity 60% at 9%, debt 40% at 5%, tax 25% | 6.90% |
| WC-22 | Unlevered beta | Levered beta 1.2, debt to equity 0.40/0.60, tax 25% | 0.80 |
| WC-23 | Forward price | Spot 50, r = 5% continuous, no income, 6 months | 51.27 |
| WC-24 | Put-call parity | Call 4.50, spot 50, strike 50, r = 5% continuous, 1 year | Put 2.06 |
| WC-25 | Binomial call | Spot 50, u = 1.2, d = 0.8, r = 5% per period, strike 50 | $\pi_u$ = 0.625; call 5.95 |
| WC-26 | Covered interest parity | Spot 1.10 USD per EUR; USD rate 4%, EUR rate 2%, one year | Forward 1.1216 USD per EUR |

Notes on the checks: in WC-07 the MWR is low because the extra 50 was invested just before the losing period; TWR ignores that timing. In WC-13 the ten-day figure assumes independent daily returns and an unchanged position. In WC-17 the approximation is close because the yield change is small; it becomes less accurate for larger moves or bonds with embedded options.

## 12. Calculator notes

<!-- Guidance: Write down the exact settings and key sequence for your own calculator model; generic function names are given here. Pitfall: a calculator left in "payments at beginning" mode, or with payments per year set to 12, from an earlier problem. -->

| Task | Generic steps |
|---|---|
| Clear memory | Clear the time value of money registers and the cash flow worksheet before each new problem |
| Payments per year | Set to 1 and enter the periodic rate and number of periods yourself, unless you deliberately use the built-in conversion |
| Timing mode | END for ordinary annuities and bonds; BGN only for annuities due |
| Sign convention | Money paid out is negative, money received is positive; a PV and FV of the same sign usually produce an error |
| Time value keys | N (periods), I/Y or i (rate per period, entered as a percentage), PV, PMT, FV; compute the unknown |
| Uneven cash flows | Enter the initial flow (CF0 or CFo), then each flow with its frequency (C01/F01 or CFj/Nj); compute NPV with I, or IRR |
| Bonds | Price = PV with N = periods, I/Y = periodic yield, PMT = coupon per period, FV = 100 |
| Statistics | Use the two-variable statistics mode for mean, standard deviation, correlation and regression slope |
| Display | Set at least four decimal places for intermediate results |

| Setting check | My calculator |
|---|---|
| Model | [Model] |
| Decimal places | [n] |
| Payments per year default | [1] |
| Timing mode default | [END] |
| Known quirks | [Notes] |

## 13. Glossary and references

<!-- Guidance: Define terms you use but have not explained above. Cite textbooks as background reading; do not copy their worked examples or problem sets. -->

| Term | Meaning |
|---|---|
| Basis point | One hundredth of one percent (0.01%) |
| Clean and full price | Bond price without and with accrued interest |
| Cost of carry | Net cost of holding the underlying until delivery: financing cost minus income |
| Day count convention | Rule for counting days in an interest period and in a year (for example Actual/365, 30/360) |
| Discount factor | Present value of one unit received at a future date |
| Duration | Sensitivity of a bond's price to a parallel change in yield |
| No-arbitrage | Condition under which no riskless profit can be made without investment |
| Risk-neutral probability | Probability under which the expected return on every asset equals the risk-free rate, used for pricing |
| Terminal value | Value at the end of the explicit forecast period of all later cash flows |
| Yield to maturity | Single discount rate that sets the present value of a bond's cash flows equal to its price |

Background reading (cite; do not copy text, examples or problems):

- Standard university textbooks on corporate finance, investments, fixed income and derivatives, in their current editions.
- Your course notes or the official curriculum of your exam, read directly from the exam body or your provider.
- Related templates: Exam Study Plan (`finance/exam-study-plan`), Study Guide (`academic/study-guide`), Investment Appraisal (`finance/investment-appraisal`), Valuation Report (`finance/valuation-report`), Bond Credit Analysis (`finance/bond-credit-analysis`), VaR Backtesting Report (`finance/var-backtesting-report`), Covenant Compliance Certificate (`finance/covenant-compliance-certificate`).

## Standard and licence note

Based on general financial mathematics (time value of money, portfolio theory, the capital asset pricing model, bond mathematics, discounted cash flow valuation and no-arbitrage derivative pricing) as taught across university courses and professional programmes; no single standard governs it. The notation, explanations, worked examples and layout are original; no exam curriculum, textbook text, worked example or problem set is reproduced. Check your own course or exam materials for the notation and conventions they require, and the current official version of any exam rules on permitted calculators. Template licence: MIT.