CA Final · Advanced Financial Management

Security Valuation: formula sheet

Full chapter guide

Key formulas

Value of a bond: V = Σ [C ÷ (1 + kd)^t] + M ÷ (1 + kd)^n, t = 1 to n; C is annual coupon (face value × coupon rate), M is redemption value, n is years to maturity, kd is required return.
Annuity-factor form: V = C × PVIFA(kd, n) + M × PVIF(kd, n); Use the table factors given in the question. This is the usual exam layout.
Zero coupon bond: V = M ÷ (1 + kd)^n; No coupons, so only one cash flow.
Perpetual (irredeemable) bond: V = C ÷ kd; Use only when the bond never matures.
Current yield: Current yield = Annual coupon ÷ Current market price × 100; Ignores redemption gain or loss and the time value of money.
Approximate YTM: YTM ≈ [C + (M − P) ÷ n] ÷ [(M + P) ÷ 2]; P is current price. A quick estimate and a good starting rate for the exact method.
Exact YTM by interpolation: YTM = L + [(PV at L − P) ÷ (PV at L − PV at H)] × (H − L); L is the lower trial rate and H the higher one. PV at L must be above P and PV at H below P.
Yield to call: Find the rate where C × PVIFA(r, nc) + Call price × PVIF(r, nc) = P; nc is years to the call date. Use the call price instead of M.
Semi-annual coupons: Coupon = C ÷ 2; periods = 2n; rate per period = kd ÷ 2; Effective annual yield = (1 + y ÷ 2)² − 1, where y is the annual nominal yield.
Macaulay duration: D = Σ [t × PV(CFt)] ÷ Σ PV(CFt) = Σ [t × PV(CFt)] ÷ Bond price; PV at the yield to maturity. t is in years if cash flows are annual.
Modified duration: MD = D ÷ (1 + y); For m payments a year, use MD = D ÷ (1 + y/m), with D and y on the same basis.
Price change using duration: ΔP ÷ P ≈ − MD × Δy; Δy is in decimals: 1% = 0.01. The sign is negative, so yield up means price down.
Convexity: C = Σ [CFt × t × (t + 1) ÷ (1 + y)^t] ÷ [P × (1 + y)²]; This is the standard form for annual cash flows. Check which form the question gives and use that one.
Price change using duration and convexity: ΔP ÷ P ≈ − MD × Δy + ½ × C × (Δy)²; The convexity term is always positive for a plain bond. It reduces a fall and increases a rise.
Zero-coupon bond: D = maturity in years; Coupon bonds always have Macaulay duration below maturity.
Spot rate to forward rate (one period ahead): (1 + Sₙ)ⁿ = (1 + Sₙ₋₁)ⁿ⁻¹ × (1 + fₙ₋₁,ₙ); Gives the one-year forward rate for year n. So fₙ₋₁,ₙ = (1 + Sₙ)ⁿ ÷ (1 + Sₙ₋₁)ⁿ⁻¹ − 1.
General forward rate between years m and n: (1 + Sₙ)ⁿ = (1 + Sₘ)ᵐ × (1 + fₘ,ₙ)ⁿ⁻ᵐ; Use for a multi-year forward rate, such as the 2-year rate starting after year 1. Annual compounding assumed.
Bond value using spot rates: P = Σ [Cₜ ÷ (1 + Sₜ)ᵗ] + M ÷ (1 + Sₙ)ⁿ; Cₜ is coupon in year t, M is redemption value, Sₜ is the spot rate for year t.
Liquidity preference theory: Forward rate = Expected future spot rate + Liquidity premium; Premium is positive and usually grows with maturity. Under pure expectations the premium is zero.
Spot rate from discount factors: Sₙ = (1 ÷ DFₙ)^(1/n) − 1; DFₙ is the price today of ₹1 received in year n.
General dividend discount model: P0 = Σ Dt ÷ (1 + Ke)^t, for t = 1 to ∞; Value today is the present value of all future dividends.
Zero growth: P0 = D ÷ Ke; Dividend stays the same every year forever.
Constant growth (Gordon model): P0 = D1 ÷ (Ke − g) = D0 × (1 + g) ÷ (Ke − g); Valid only when Ke > g. D1 is next year's dividend, not the last one paid.
Implied cost of equity: Ke = (D1 ÷ P0) + g; Use when market price is given and you need the return expected.
Growth rate from retention: g = b × r, where b = 1 − payout ratio; r is return on equity. Assumes constant b and r.
Gordon value with retention: P0 = E1 × (1 − b) ÷ (Ke − b × r); This is Gordon's model. E1 is next year's EPS. Same as Gordon with D1 = E1 × (1 − b) and g = b × r.
Two-stage model: P0 = Σ Dt ÷ (1 + Ke)^t (t = 1 to n) + [Dn+1 ÷ (Ke − gn)] ÷ (1 + Ke)^n; gn is the stable growth rate after year n. Terminal value is at end of year n.
Holding period model: P0 = Σ Dt ÷ (1 + Ke)^t (t = 1 to n) + Pn ÷ (1 + Ke)^n; Use when the expected selling price Pn at year n is given.
P/E ratio: P/E = Market price per share ÷ EPS; Use EPS of the same period (trailing or forward) for the target and the comparables.
Value using P/E: Value per share = EPS × Industry P/E; Equity value = Net profit after tax (available to equity) × P/E.
Earnings yield: Earnings yield = EPS ÷ Market price × 100 = 1 ÷ P/E × 100; It is the inverse of P/E. Value = EPS ÷ Earnings yield.
Price-to-book ratio: P/B = Market price per share ÷ Book value per share; Book value per share = Net worth ÷ Number of equity shares.
Value using P/B: Value per share = Book value per share × Comparable P/B; Use net worth available to equity shareholders only, excluding preference capital.
Enterprise value: EV = Market capitalisation + Debt + Preference capital − Cash and cash equivalents; Add minority interest if given. Net debt = Debt − Cash.
Value using EV/EBITDA: EV of target = EBITDA × Comparable EV/EBITDA; Equity value = EV − Net debt; Divide equity value by number of shares for value per share.
P/E from growth model: P/E = Payout ratio × (1 + g) ÷ (ke − g), or with next year's EPS, P/E = Payout ratio ÷ (ke − g); Use only if the question gives a constant growth rate g below ke.
Price-to-sales: P/S = Market price per share ÷ Sales per share; Used when earnings are negative or unstable.
FCFF from EBIT: FCFF = EBIT(1 – t) + Depreciation – Capex – Increase in NWC; EBIT(1 – t) is NOPAT. Use the increase in net working capital; a decrease is added back.
FCFF from operating cash flow: FCFF = CFO + Interest(1 – t) – Capex; Use when cash flow from operations is given after interest. Add back interest net of tax.
FCFE from FCFF: FCFE = FCFF – Interest(1 – t) + Net borrowing; Net borrowing = new debt raised – debt repaid.
FCFE from PAT: FCFE = PAT + Depreciation – Capex – Increase in NWC + Net borrowing; If preference dividend is paid, subtract it too.
Firm value (FCFF): Firm value = Σ FCFFt ÷ (1 + WACC)^t + TV ÷ (1 + WACC)^n; Discount at WACC.
Terminal value: TV at year n = FCFF(n+1) ÷ (WACC – g) = FCFF(n) × (1 + g) ÷ (WACC – g); Valid only when g < WACC. For FCFE, use Ke instead of WACC.
Equity value from FCFF: Equity value = Firm value – Market value of debt (+ non-operating cash); Divide by the number of shares for value per share.
Equity value (FCFE): Equity value = Σ FCFEt ÷ (1 + Ke)^t + TV(equity) ÷ (1 + Ke)^n; Debt is not subtracted, since FCFE is already after debt flows.
EVA: EVA = NOPAT – (WACC × Capital employed) = EBIT(1 – t) – WACC × Capital employed; Use the capital employed the question specifies (opening, if stated). Cost of capital is the after-tax WACC.
MVA: MVA = Market value of firm (equity + debt) – Capital invested; Also equals the present value of all future EVAs.

Quick revision

Bond value = Σ coupon ÷ (1 + r)^t + redemption value ÷ (1 + r)^n, discounting at the required yield.
If yield > coupon rate, the bond trades below par; if yield < coupon rate, it trades above par.
Current yield = annual coupon ÷ market price. It ignores capital gain or loss.
YTM is the rate that makes the present value of cash flows equal to the market price.
Macaulay duration = Σ (t × PV of cash flow) ÷ bond price. Modified duration = Macaulay duration ÷ (1 + y) for annual compounding.
Approximate price change % ≈ −modified duration × change in yield. Convexity improves this estimate for larger yield changes.
Forward rate from spot rates: (1 + s2)² = (1 + s1) × (1 + f), where f is the one-year rate one year ahead.
Constant growth model: P0 = D1 ÷ (ke − g), valid only when ke > g.
D1 = D0 × (1 + g). Do not use D0 in the numerator by mistake.
Earnings-based value = EPS × P/E multiple. Earnings yield = EPS ÷ price.
Enterprise value from FCFF discounted at WACC; equity value = enterprise value − debt (adjusted for cash where given).
In multi-stage models, discount the terminal value back from the year it is calculated.

Common mistakes

Discounting cash flows at the coupon rate Fix: Discount at the required return or market yield. Use the coupon rate only to calculate the coupon in rupees.
Leaving out the redemption value Fix: Always write two lines: coupons × PVIFA and redemption × PVIF. Check the redemption value, which may be at a premium.
Dividing the total of PV × t by face value instead of the bond price. Fix: Always divide by the sum of present values. That sum is the price at the given yield.
Using the undiscounted cash flows when weighting time. Fix: Compute present values first. Weights come from PVs, not raw cash flows.
Taking the forward rate as the simple difference of spot rates, such as S₂ − S₁. Fix: Always use the ratio of compounded factors, (1 + S₂)² ÷ (1 + S₁) − 1.
Discounting all bond cash flows at one rate, usually the YTM, when the question gives spot rates. Fix: If spot rates are given, discount each year's cash flow at that year's spot rate.
Using D0 in the Gordon formula instead of D1. Fix: Check wording. If the dividend is 'just paid' or 'last year', multiply by (1 + g) to get D1. If 'expected next year', use it as D1.
Discounting the terminal value by the wrong number of years. Fix: The Gordon formula gives value at end of year n. Discount TV by (1 + Ke)^n, the same as the last high-growth dividend.
Using total profit instead of profit available to equity shareholders. Fix: Always compute EPS as (PAT − preference dividend) ÷ number of equity shares.
Treating earnings yield as the same number as P/E. Fix: Remember that earnings yield = 1 ÷ P/E. A yield of 8% means P/E of 12.5.

Exam tips

Write the cash flow line first (coupon, years, redemption). Examiners award marks for correct inputs even if the arithmetic slips.
Read the redemption terms carefully. Redemption at a premium or discount changes M and the whole answer.
If the question says 'approximate YTM', use the short formula and do not waste time on interpolation.
Add a one-line interpretation: undervalued or overvalued, buy or sell, premium or discount. Case-scenario MCQs often test only this.
Check whether coupons are paid annually or half-yearly, and whether the required return is a nominal annual rate.
Show the table every time, even for a short question. Step marks are given for the PV, PV × t and divisor.
Read the question for the yield basis. If coupons are half-yearly, state clearly whether your duration is in half-years or years.
In theory-plus-numerical questions, add one line of interpretation: for example, 'a 1% rise in yield reduces the price by about 2.49%'.