ACCA Applied Skills · Financial Management

The valuation of debt and other financial assets: formula sheet

Full chapter guide

Key formulas

Market value of irredeemable debt: P₀ = I ÷ Kd; I = annual interest (coupon rate × nominal value). Kd = investors' required pre-tax return. Interest is assumed paid annually and a year away from now, i.e. ex-interest.
Annual interest: I = coupon rate × nominal value; Always work from the nominal value, not the market price.
Market value of irredeemable preference shares: P₀ = D ÷ Kp; D = annual preference dividend (rate × nominal value). Kp = required return of preference investors.
Yield from a known price (rearranged): Kd = I ÷ P₀; Gives the investors' return. This is also the pre-tax cost of debt to the company.
Value with interest about to be paid (cum-interest): P cum-int = I ÷ Kd + I; Use only if the next payment is due immediately and the question says so.
Market value of redeemable debt: V₀ = I × AF(r%, n) + R × DF(r%, n); I = annual interest (coupon rate × nominal value). R = redemption value. r = investors' required yield. n = years to redemption. Assumes interest is paid annually in arrears.
Annuity factor: AF = [1 − (1 + r)^−n] ÷ r; Use it if the rate or year is not in the tables you are given. It gives the same value as the annuity table.
Discount factor: DF = 1 ÷ (1 + r)^n; Applies to the single redemption payment at the end of year n.
Interest per bond: I = coupon rate × nominal value; Use nominal (par) value, not market value. Coupon of 6% on $100 nominal is $6 a year.
Total market value of an issue: Total value = (V₀ per $100 nominal ÷ 100) × total nominal value in issue; Scale the price per $100 up to the whole issue.
Bond price from YTM: P0 = Σ [Interest ÷ (1 + r)^t] + Redemption value ÷ (1 + r)^n; r is the YTM. Use annuity factors for the interest and a single discount factor for redemption.
Interpolation for IRR / YTM: YTM = L + [NPV at L ÷ (NPV at L − NPV at H)] × (H − L); L is the lower rate and H the higher rate. NPV at L must be positive and NPV at H negative. Cash flows must include the price as a negative figure at time 0.
Coupon rate: Coupon rate = Annual interest ÷ Nominal value; Fixed by the bond terms. It does not change when the market price changes.
Current (running) yield: Current yield = Annual interest ÷ Market price; Ignores the gain or loss on redemption, so it differs from YTM for redeemable bonds.
Interest after tax relief: After-tax interest = Coupon × (1 − tax rate); Use for the company's cost of debt. Assume tax is paid in the same year unless told otherwise.
Irredeemable debt cost: Kd = Interest × (1 − T) ÷ P0; Use only if the debt is never redeemed. P0 is the ex-interest market price.
Starting-rate approximation: r ≈ [Interest + (Redemption − Price) ÷ n] ÷ [(Redemption + Price) ÷ 2]; Only a guide for choosing the first trial rate. Do not give it as the final answer.
Floor value: Floor value = Σ [interest ÷ (1 + r)^t] + redemption value ÷ (1 + r)^n; r is the pre-tax yield on similar straight debt, not the coupon rate. Use annuity and discount factor tables.
Conversion ratio: Conversion ratio = nominal value of bond ÷ conversion price; Often given directly, for example 30 shares per $100 bond.
Future share price: Future share price = current share price × (1 + g)^n; g is the expected annual share price growth. n is the years to the conversion date.
Conversion value (future): Conversion value = future share price × conversion ratio; Compare this with the cash redemption value at the same date. The holder takes the higher.
Current conversion value: Current conversion value = current share price × conversion ratio; Used as the base for the conversion premium.
Expected market value of the bond: Value = Σ [interest ÷ (1 + r)^t] + higher of (conversion value, redemption value) ÷ (1 + r)^n; Discount at the straight-debt yield. If the answer is below the floor value, the floor value is the better estimate.
Conversion premium: Premium per share = (market price of bond ÷ conversion ratio) − current share price; Per bond premium = market price − current conversion value. Percentage premium = premium ÷ current conversion value.
Yield on a corporate bond: Corporate yield = Risk-free yield (same maturity) + Credit spread; The spread rises as the credit rating falls and often differs by maturity.
Expectations theory (two-year bond, annual compounding): (1 + 2-year spot rate)² = (1 + 1-year spot rate) × (1 + forward rate for year 2); Use it to find the implied forward rate: forward = (1 + s₂)² ÷ (1 + s₁) − 1.
Liquidity preference theory: Long-term yield = Expected average short-term rate + Liquidity premium; The premium is normally larger for longer maturities.
Bond price from spot rates: Price = Σ [cash flow in year t ÷ (1 + spot rate for year t)ᵗ]; Each cash flow is discounted at the spot rate for its own maturity.
Irredeemable preference share value: P₀ = D ÷ k; D is the fixed annual dividend (rate × nominal value). k is the investor's required return. No growth.
Zero coupon bond value: P₀ = R ÷ (1 + r)^n; R is the redemption value, r the required annual yield, n the years to maturity.
Yield of a zero coupon bond: r = (R ÷ P₀)^(1/n) − 1; Use when price and redemption value are given. This is the pre-tax yield.
Deep discount bond value: P₀ = Σ [Coupon ÷ (1 + r)^t] + R ÷ (1 + r)^n; Use annuity factors for the coupon stream and a single discount factor for the redemption.
Redeemable preference share value: P₀ = Σ [D ÷ (1 + k)^t] + Redemption ÷ (1 + k)^n; Dividends form an annuity, redemption is a lump sum.

Quick revision

Value of any financial asset = present value of its future cash flows at the required return.
Irredeemable debt value = annual interest ÷ required return.
Redeemable debt value = PV of interest annuity + PV of redemption value.
Interest is based on the nominal value, not the market price.
Yield to maturity is the rate at which PV of cash flows equals the current market price.
Estimate YTM by interpolation using two trial rates on either side of the answer.
Convertible debt: compare its value as debt with its conversion value and take the higher as the floor.
Conversion value = number of shares per bond × expected future share price.
An upward-sloping yield curve is the normal shape for debt of increasing maturity.
Liquidity preference theory says investors require extra yield for longer maturities.
Check whether interest is paid in arrears and whether you use ex-interest or cum-interest prices.
Show workings clearly in Section C, as method marks are given even if the final figure is wrong.

Common mistakes

Calculating interest on the market price instead of the nominal value. Fix: Interest always comes from coupon rate × nominal value. The market price is what you are solving for.
Deducting tax from the interest when valuing for investors. Fix: Use the given required return and the gross interest unless the question says investors' return is after personal tax. Tax relief belongs in the company's cost of debt only.
Discounting at the coupon rate Fix: The coupon only gives the cash interest. Always discount at the yield or required return.
Using the after-tax cost of debt as the discount rate Fix: Market value reflects what investors earn before the company's tax. Use the pre-tax yield.
Treating the coupon rate as the cost of debt or the YTM. Fix: The coupon is fixed on nominal value. YTM depends on the market price. Use the market price at time 0 and calculate the IRR.
Using the nominal value instead of the market price as the time 0 flow. Fix: Time 0 is always the current market price. Nominal value is used only to calculate interest and, if redeemed at par, the redemption.
Discounting the interest and redemption at the coupon rate to get the floor value. Fix: Use the yield on similar straight debt. Read the question for a market yield or the cost of the company's non-convertible debt.
Using the current share price to compute the conversion value at a future date. Fix: Always ask: at which date? For a future conversion value, grow the price by (1 + g)^n.
Saying liquidity preference theory explains an inverted curve on its own. Fix: Say an inverted curve needs expected falls in short-term rates that outweigh the premium.
Mixing up the three theories. Fix: Link each to one idea: expectations = future rates, liquidity = premium, segmentation = separate markets.

Exam tips

In OT questions, the answer options often include the trap values: interest on market price, or a redeemable-style answer. Do the two-line calculation and match exactly.
Check the wording on cum- or ex-interest before dividing. It changes the answer by one full payment.
In constructed response questions, show the formula, the interest line and the division. Method marks are available even if the final number is wrong.
If a question gives the price and asks for the return, rearrange to Kd = I ÷ P₀. Remember this is the pre-tax cost of debt, and apply tax relief only afterwards for the company's cost.
Add a one-line comment when asked to discuss: a price below nominal value means investors require more than the coupon rate.
Read the question for the rate to use. Phrases like 'investors require' or 'yield' mean the discount rate. Ignore the coupon for discounting.
Check whether redemption is at par or at a premium before you start. This is a common trap in OT questions.
Use the factors given in the exam tables. Keep three decimal places and do not worry about small rounding differences.