Equity and Bond Valuation and Evaluation of Performance: formula sheet

Key formulas

Gordon (constant growth) model

P0 = D1 ÷ (ke − g) = D0 × (1 + g) ÷ (ke − g)

Valid only for ke > g and growth constant forever. D1 is next year's dividend.

Cost of equity from Gordon

ke = D1 ÷ P0 + g

Use when the price is given and you need the required return.

Growth from retention

g = b × r

b = retention ratio (1 − payout), r = return on equity (or on retained funds).

General DDM

P0 = Σ Dt ÷ (1 + ke)^t + Pn ÷ (1 + ke)^n

Pn is the expected price at the end of year n, often found by the Gordon model at that date.

Two-stage model

P0 = PV of dividends in high-growth years + [D(n+1) ÷ (ke − gn)] ÷ (1 + ke)^n

gn is the stable growth rate after year n.

H-model

P0 = [D0 × (1 + gL) + D0 × H × (gS − gL)] ÷ (ke − gL)

gS = initial high growth, gL = long-term growth, H = half the years over which growth declines linearly.

P/E valuation

Value per share = EPS × P/E ratio

Use the P/E of comparable firms. Check that EPS is the same basis (current or expected).

P/E from fundamentals

P/E = Payout ratio ÷ (ke − g)

Uses next year's EPS and payout for a constant-growth firm.

Free cash flow to equity

FCFE = PAT + Depreciation − Capex − Increase in working capital + Net borrowing

Value of equity = PV of FCFE at ke.

Price of a coupon bond

P = Σ C ÷ (1 + r)^t (t = 1 to n) + M ÷ (1 + r)^n = C × PVAF(r, n) + M × PVF(r, n)

C = coupon per period, M = maturity (redemption) value, r = required return per period, n = number of periods.

Price of a zero-coupon bond

P = M ÷ (1 + r)^n

No coupons. The YTM is r = (M ÷ P)^(1/n) − 1.

Current yield

Current yield = Annual coupon ÷ Current market price × 100

Ignores the gain or loss at redemption and the time value of money.

Approximate YTM

YTM ≈ [C + (M − P) ÷ n] ÷ [(M + P) ÷ 2]

Quick estimate. Use it as the starting rate for trial and error, or when the question asks for an approximation.

YTM by interpolation

YTM = r1 + [(P1 − P) ÷ (P1 − P2)] × (r2 − r1)

P1 and P2 are the bond values at the lower rate r1 and the higher rate r2. P is the market price, and it must lie between P1 and P2.

Yield to call

P = Σ C ÷ (1 + r)^t (t = 1 to k) + Call price ÷ (1 + r)^k

k = number of periods to the call date. Solve for r exactly as for YTM.

Realised compound yield

r = (Terminal value ÷ P)^(1/n) − 1

Terminal value = coupons compounded at the reinvestment rate to the end of the holding period + sale or redemption price.

Semi-annual bonds

Coupon per period = Annual coupon ÷ 2; periods = 2n; discount rate per period = annual rate ÷ 2

Follow the question's convention for annualising, usually multiplying the half-yearly rate by 2.

Macaulay duration

D = Σ [t × PV(CFt)] ÷ Σ PV(CFt) = Σ [t × PV(CFt)] ÷ P

t is the time of each cash flow in years. PV is discounted at the bond's yield to maturity. The denominator is the bond's price.

Modified duration

D* = D ÷ (1 + y ÷ m)

y is the annual yield and m is the number of coupon payments per year. For annual coupons, D* = D ÷ (1 + y).

Price change using duration

ΔP ÷ P ≈ − D* × Δy

Δy in decimals, so 1% = 0.01. The sign is negative because price and yield move in opposite directions.

Convexity (annual coupons)

C = [Σ t × (t + 1) × CFt ÷ (1 + y)^t] ÷ [P × (1 + y)²]

Use this form only when cash flows are annual. Check the convexity convention given in the question before using a stated value.

Price change using duration and convexity

ΔP ÷ P ≈ − D* × Δy + ½ × C × (Δy)²

The convexity term is always positive for an ordinary bond, so it adds to the price in both directions.

Zero-coupon bond duration

D = Maturity in years

Holds exactly for a bond with no interim cash flows.

Portfolio duration

Dp = Σ (wi × Di)

wi is the market-value weight of each bond. Use market values, not face values.

Price value of a basis point

PVBP ≈ D* × P × 0.0001

Approximate rupee change in price for a 0.01% change in yield.

Immunization conditions

Duration of assets = Duration of liabilities (investment horizon); PV of assets ≥ PV of liabilities

Rebalance periodically as time passes and yields change.

Implied forward rate (one period, annual compounding)

(1 + s₂)² = (1 + s₁) × (1 + ₁f₂), so ₁f₂ = (1 + s₂)² ÷ (1 + s₁) − 1

s₁ and s₂ are 1-year and 2-year spot rates. ₁f₂ is the 1-year rate starting at the end of year 1.

General forward rate

(1 + sₙ)ⁿ = (1 + sₘ)ᵐ × (1 + ₘfₙ)ⁿ⁻ᵐ, so ₘfₙ = [(1 + sₙ)ⁿ ÷ (1 + sₘ)ᵐ]^(1 ÷ (n−m)) − 1

Gives the annualised forward rate between year m and year n.

Spot rate from a bond price

Price of zero-coupon bond = Face value ÷ (1 + sₙ)ⁿ

Rearrange to find sₙ = (Face value ÷ Price)^(1/n) − 1.

Pure expectations theory

(1 + sₙ)ⁿ = (1 + s₁)(1 + E[₁f₂])(1 + E[₂f₃])...

Forward rate equals expected future spot rate. Long rate is a geometric average of expected short rates.

Liquidity preference theory

Forward rate = Expected future spot rate + Liquidity premium

The premium is positive and generally rises with maturity. So the curve is higher than under pure expectations.

Sharpe ratio

Sharpe = (Rp − Rf) ÷ σp

Rp is portfolio return, Rf risk-free rate, σp standard deviation of the portfolio. Higher is better.

Treynor ratio

Treynor = (Rp − Rf) ÷ βp

Uses beta. Meaningful for comparison when beta is positive. Higher is better.

Jensen's alpha

α = Rp − [Rf + βp × (Rm − Rf)]

Positive alpha means outperformance relative to CAPM.

M-squared

M² = Rf + Sharpe of portfolio × σm; M² difference = M² − Rm

σm is the market (benchmark) standard deviation. Positive difference means beating the market on a risk-matched basis.

Information ratio

IR = (Rp − Rb) ÷ Tracking error

Rb is benchmark return. Tracking error is the standard deviation of (Rp − Rb).

Market Sharpe and Treynor benchmarks

Market Sharpe = (Rm − Rf) ÷ σm; Market Treynor = Rm − Rf

Market beta is 1, so the market Treynor equals its excess return. Compare portfolio ratios with these.

Quick revision

• Value of a share is the present value of expected future dividends at the required return.
• 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.
• Bond price is the present value of coupons plus the present value of the redemption amount.
• Bond price moves inversely to yield; a bond sells at par when yield equals coupon rate.
• Current yield = annual coupon ÷ current market price.
• Yield to maturity is the discount rate that equates the bond price to the present value of its cash flows.
• Other things equal, a higher coupon rate and a shorter maturity give a lower Macaulay duration. A zero-coupon bond's Macaulay duration equals its maturity.
• Modified duration = Macaulay duration ÷ (1 + yield per period).
• Immunization matches the duration of assets with that of the liability.
• Sharpe ratio = (Rp − Rf) ÷ σp; Treynor ratio = (Rp − Rf) ÷ βp.
• Jensen's alpha = Rp − [Rf + β × (Rm − Rf)]; a positive value means the portfolio beat its required return.

Common mistakes

• Using D0 in the Gordon formula without growing it. Fix: Check the wording. 'Just paid' or 'last year' means D0, so multiply by (1 + g). 'Expected next year' means D1.
• Applying the Gordon model when g is greater than or equal to ke. Fix: The constant-growth model works only if ke > g. Use a multi-stage model for temporary high growth.
• Calculating the coupon as coupon rate × market price. Fix: The coupon is always coupon rate × face value, unless the question says otherwise. Compute it in rupees on the first line.
• Using the same rate and years for half-yearly bonds without adjusting. Fix: For semi-annual payments, halve the coupon and the rate and double the periods. Then annualise as the question requires.
• Dividing the t × PV total by the face value instead of the bond price. Fix: The denominator is always the sum of the present values, which is the current price. Write 'Price = ΣPV' under the table.
• Forgetting the face value in the final year's cash flow. Fix: Always write the final cash flow as coupon + redemption value. Tick it off before discounting.
• Taking the forward rate as the simple average or the difference of spot rates, for example 7% − 6% = 1%. Fix: Always divide growth factors: (1 + s₂)² ÷ (1 + s₁) − 1.
• Using yield to maturity of coupon bonds as if it were the spot rate. Fix: Use spot rates for forward calculations. If only coupon bond data is given, derive spot rates first by discounting each cash flow, unless the question says to use the yields directly.
• Dividing the portfolio return by risk without subtracting the risk-free rate. Fix: Always write Rp − Rf as the first line of working.
• Using beta in the Sharpe ratio or standard deviation in the Treynor ratio. Fix: Remember: Sharpe goes with Standard deviation (both start with S); Treynor goes with beta.

Exam tips

• Write whether the given dividend is D0 or D1 before you calculate. Many marks are lost here.
• For two-stage problems, show the dividend table, terminal value and discount factors clearly so partial marks are easy to give.
• Always end with a recommendation comparing intrinsic value to market price.
• In MCQs, check ke > g first. If not, the constant-growth model does not apply.
• If ke is not given, look for beta, risk-free rate and market return and use CAPM.
• In MCQs, use the price relationship to eliminate options: a bond at a discount has YTM above its coupon rate, and a premium bond has YTM below it.
• For YTM questions in the written section, show the approximate YTM, the two trial values and the interpolation. Marks are given for method.
• Read the redemption terms carefully. Redemption at a premium, or a call date earlier than maturity, changes M and n.