28 Cost of Capital and Time Value of Money
This topic combines two of the most-tested ideas in business finance: the time value of money — the foundation of every valuation in finance — and the cost of capital — the discount rate at which those valuations are made.
29 Part A — Time Value of Money
29.1 Why Money Has Time Value
A rupee today is worth more than a rupee a year from now. Four reasons stand behind this central premise of finance (pandey2021?; khan2022?; chandra2023?):
- Preference for present consumption. People prefer goods today over identical goods later.
- Productivity / opportunity cost. Money invested today can earn a return.
- Inflation. Prices generally rise; the real value of money falls.
- Risk and uncertainty. Future cash flows are not certain.
The first three are deterministic reasons; the fourth is stochastic.
29.2 Compounding — Future Value
Compounding converts a present amount into its future value (FV) at a given interest rate.
| Cash flow | Formula | Notes |
|---|---|---|
| Single sum | \(FV_n = PV \cdot (1+r)^n\) | Lump-sum at \(t=0\) |
| Multiple compounding per year | \(FV_n = PV \cdot \left(1 + \frac{r}{m}\right)^{m \cdot n}\) | \(m\) = compounding periods per year |
| Continuous compounding | \(FV_n = PV \cdot e^{r \cdot n}\) | \(m \to \infty\) |
| Future Value of Annuity (ordinary) | \(FVA = A \cdot \dfrac{(1+r)^n - 1}{r}\) | Equal annual payment \(A\) at end of period |
| FVA — annuity due | \(FVA_{\text{due}} = FVA \cdot (1+r)\) | Payment at the beginning of period |
29.3 Discounting — Present Value
Discounting converts a future amount into its present value (PV).
| Cash flow | Formula |
|---|---|
| Single sum at \(t = n\) | \(PV = \dfrac{FV_n}{(1+r)^n}\) |
| Annuity (ordinary) | \(PVA = A \cdot \dfrac{1 - (1+r)^{-n}}{r}\) |
| Annuity due | \(PVA_{\text{due}} = PVA \cdot (1+r)\) |
| Perpetuity | \(PV = \dfrac{A}{r}\) |
| Growing perpetuity (Gordon) | \(PV = \dfrac{A}{r - g}, \; r > g\) |
The present value of an asset is the sum of the present values of its expected future cash flows. This is the master rule of valuation in finance.
29.4 Effective vs Nominal Interest Rate
When interest is compounded more than once a year, the effective annual rate (EAR) is greater than the nominal (stated) rate.
\[ \text{EAR} = \left(1 + \frac{r}{m}\right)^m - 1 \]
A 12 per cent nominal rate compounded monthly works out to \((1 + 0.12/12)^{12} - 1 = 12.68\%\) EAR.
29.5 Doubling Period — Rules of 72 and 69
Two thumb-rules estimate how many years it takes for a sum to double at compound rate \(r\) (in per cent per period):
| Rule | Formula | Use |
|---|---|---|
| Rule of 72 | \(n \approx \dfrac{72}{r}\) | Quick mental estimate |
| Rule of 69 | \(n \approx 0.35 + \dfrac{69}{r}\) | Slightly more accurate |
At 8 per cent, the Rule of 72 gives 9 years; the Rule of 69 gives 0.35 + 69/8 = 8.98 years; the exact answer is \(\ln 2 / \ln 1.08 = 9.006\) years.
29.6 Worked Numerical
A father invests ₹1,00,000 today at 10 per cent compounded annually for 5 years.
- $FV_5 = 1,00,000 (1.10)^5 = 1,00,000 = $ ₹1,61,051.
- The same ₹1,00,000 today is the present value of ₹1,61,051 receivable in 5 years at 10 per cent.
- Annuity check: ₹10,000 paid at the end of each year for 5 years at 10 per cent has $FVA = 10,000 (1.6105 − 1)/0.10 = $ ₹61,051.
30 Part B — Cost of Capital
30.1 Meaning
The cost of capital is the minimum rate of return the firm must earn on its investments in order to satisfy the providers of funds without depressing the value of the firm. Solomon Ezra’s classic definition: the cost of capital is “the rate of return that must be earned on a project investment to keep unchanged the market price of the firm’s stock” (solomon1963?; pandey2021?).
Three working ideas:
- It is the opportunity cost of investing in the firm rather than in alternatives of comparable risk.
- It is future-oriented — based on expected returns, not past returns.
- It is the discount rate the firm uses in NPV calculations and the hurdle rate for new investments.
30.2 Significance
| Use | Working content |
|---|---|
| Capital-budgeting decisions | Discount rate in NPV; cut-off in IRR |
| Capital-structure decisions | Trade-off between cheaper debt and costlier equity |
| Working-capital decisions | Inventory, receivables financing |
| Performance evaluation | EVA = NOPAT − WACC × Capital Employed |
| Acquisition pricing | DCF valuation of target |
30.3 Components — Specific Costs
The firm’s cost of capital is the weighted average of the costs of its individual capital components.
30.3.1 Cost of debt — \(K_d\)
For a debenture issued at par with coupon \(i\) and tax rate \(t\):
\[ K_d = i \cdot (1 - t) \]
The (1 − t) factor recognises that interest is tax-deductible, so the firm’s effective cost is below the coupon. For a debenture issued at a discount \(D\) with face \(F\), redeemable at premium \(P\) in \(n\) years:
\[ K_d = \dfrac{i (1 - t) + (F + P - D)/n}{(F + D)/2} \]
30.3.3 Cost of equity — \(K_e\)
Five approaches recur in textbooks (pandey2021?; chandra2023?).
| Approach | Formula | Note |
|---|---|---|
| Dividend Yield (no growth) | \(K_e = D_1 / P_0\) | Constant dividend forever |
| Gordon’s constant-growth model | \(K_e = D_1 / P_0 + g\) | Dividend grows at constant rate \(g\) |
| Earnings yield | \(K_e = E_1 / P_0\) | Uses earnings instead of dividend |
| CAPM | \(K_e = R_f + \beta (R_m - R_f)\) | Risk-based; market premium |
| Bond Yield + Risk Premium | \(K_e = K_d + \text{equity risk premium}\) | Empirical add-on |
| Realised yield approach | \(K_e\) = historical IRR on equity investment | Backward-looking |
The Capital Asset Pricing Model (CAPM) is the dominant approach in modern textbooks. The risk-free rate is typically the yield on a long-dated government bond; beta measures the stock’s sensitivity to market returns; (R_m − R_f) is the equity risk premium.
30.3.4 Cost of retained earnings — \(K_r\)
Retained earnings are the opportunity cost of profits not paid out as dividend. The shareholder forgoes the dividend; the firm reinvests on her behalf. In the simplest formulation:
\[ K_r = K_e \]
A more refined version subtracts the personal tax (and brokerage) the shareholder would have paid on a received dividend:
\[ K_r = K_e \cdot (1 - t_p) \cdot (1 - b) \]
where \(t_p\) is the personal-tax rate and \(b\) is brokerage. For many exam problems, \(K_r = K_e\) is acceptable.
30.4 Weighted Average Cost of Capital (WACC)
The firm’s overall cost of capital is the weighted average of the costs of all sources, weighted by their proportions in the capital structure:
\[ \text{WACC} = w_e K_e + w_p K_p + w_d K_d (1 - t) + w_r K_r \]
The weights may be on a book value (historical) or market value basis. The market-value weights are theoretically preferred because they reflect current opportunity cost.
| Source | Amount (₹ lakh) | Weight | Cost (after tax) | Weighted cost |
|---|---|---|---|---|
| Equity | 600 | 0.60 | 16 % | 9.60 % |
| Preference | 100 | 0.10 | 12 % | 1.20 % |
| Debt | 300 | 0.30 | 8 % (after tax) | 2.40 % |
| Total | 1,000 | 1.00 | WACC = 13.20 % |
30.5 Marginal Cost of Capital
When the firm raises additional capital, the marginal cost of capital (MCC) becomes relevant — the cost of the next rupee of capital, which may differ from the average. As the firm raises more debt, its capital structure may shift, raising the cost of equity (Modigliani-Miller proposition II).
30.6 CAPM in Detail
The Capital Asset Pricing Model — Sharpe (1964), Lintner (1965), Mossin (1966) — links risk and return (sharpe1964?).
\[ K_e = R_f + \beta_e \left( R_m - R_f \right) \]
| Symbol | Meaning |
|---|---|
| \(R_f\) | Risk-free rate; usually long-dated government-bond yield |
| \(R_m\) | Expected return on the market portfolio |
| \(R_m - R_f\) | Equity risk premium |
| \(\beta_e\) | Beta of the firm’s equity; sensitivity to market |
A beta of 1 means the stock moves with the market; a beta of 1.5 means it moves 50 per cent more than the market; a beta of 0.6 means it moves 40 per cent less.
30.6.1 Worked CAPM example
Risk-free rate 7 per cent, expected market return 14 per cent, beta 1.2:
\[ K_e = 7 + 1.2 \times (14 - 7) = 7 + 8.4 = 15.4\% \]
30.7 Determinants of Cost of Capital
| Family | Factors |
|---|---|
| External | Risk-free rate; equity market risk premium; corporate tax rate; market sentiment |
| Operational | Industry risk, business cycle, operating leverage |
| Financial | Capital structure, dividend policy, financial leverage |
| Firm-specific | Size, age, growth prospects, credit rating |
30.8 Exam-Pattern MCQs
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| Formula | Cash flow | ||
| (i) | $A / r$ | (a) | Future value of a lump sum |
| (ii) | $PV \cdot (1+r)^n$ | (b) | Perpetuity |
| (iii) | $A \cdot \dfrac{1 - (1+r)^{-n}}{r}$ | (c) | Growing perpetuity |
| (iv) | $A / (r - g)$ | (d) | Present value of an ordinary annuity |
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| Model | Formula | ||
| (i) | Dividend yield (no growth) | (a) | $D_1 / P_0 + g$ |
| (ii) | Gordon's constant-growth | (b) | $D_1 / P_0$ |
| (iii) | CAPM | (c) | $E_1 / P_0$ |
| (iv) | Earnings yield | (d) | $R_f + \beta (R_m - R_f)$ |
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- Money has time value: preference for now, opportunity cost, inflation, risk.
- Compounding turns PV into FV; discounting turns FV into PV. PV of an asset = sum of PVs of its future cash flows.
- Annuity formulas: \(FVA = A \cdot \frac{(1+r)^n - 1}{r}\); \(PVA = A \cdot \frac{1 - (1+r)^{-n}}{r}\). Annuity due = ordinary \(\times (1+r)\).
- Perpetuity: \(PV = A/r\). Growing perpetuity: \(PV = A/(r - g)\) (Gordon).
- EAR = \((1 + r/m)^m - 1\).
- Doubling: Rule of 72 (\(n ≈ 72/r\)); Rule of 69 (\(n ≈ 0.35 + 69/r\)).
- Cost of capital = minimum return that keeps firm value unchanged (Solomon).
- Specific costs: \(K_d = i(1 − t)\); \(K_p = D_p / P_0\) (no tax adj.); \(K_e\) via Dividend Yield, Gordon \(D_1/P_0 + g\), CAPM \(R_f + \beta(R_m - R_f)\), Earnings Yield, Bond Yield + RP.
- \(K_r = K_e\) (with optional personal-tax adjustment).
- WACC = \(w_e K_e + w_p K_p + w_d K_d (1-t)\), market-value weights preferred.
- CAPM authors: Sharpe (1964), Lintner (1965), Mossin (1966).
- Beta = 1 → moves with market; > 1 → more volatile; < 1 → less volatile.