Compounding and discounting

Money has a time value because a sum held today can be invested to earn a return. Compounding carries a present sum forward; discounting brings a future sum back to today. Capital budgeting, bond pricing and share valuation all rest on these two operations.

Future value: FV = PV(1 + r)ⁿ. Rs 10,000 at 10% for 3 years grows to 10,000 × 1.331 = Rs 13,310.

Present value: PV = FV / (1 + r)ⁿ. Rs 1,00,000 due in 2 years, discounted at 10%, is worth 1,00,000 / 1.21 = Rs 82,645 today. The factor 1/(1 + r)ⁿ is called the discount factor, or present value interest factor (PVIF).

Annuities and perpetuities

An annuity is a series of equal payments for a fixed number of periods. In an ordinary annuity payments fall at the end of each period; in an annuity due they fall at the beginning, as with rent or insurance premiums.

C₁ is the cash flow expected one period ahead. Using the current flow without growing it, or growing a figure already stated for next year, are the two standard errors.

Nominal and effective rates

A nominal rate r compounded m times a year has an effective annual rate (EAR) of (1 + r/m)^m − 1. More frequent compounding raises the effective rate, but by progressively smaller amounts. 12% nominal is 12.36% effective when compounded semi-annually, 12.55% quarterly and 12.68% monthly.

Continuous compounding is the limit as m grows without bound: EAR = e^r − 1, and FV = PV × e^rn. 12% compounded continuously is 12.75% effective. Rs 10,000 at 10% for 3 years compounded continuously grows to 10,000 × e^0.3 = Rs 13,499, against Rs 13,310 with annual compounding.

The Rule of 72 approximates doubling time as 72 divided by the rate in per cent. At 8%, money doubles in about 9 years (the exact figure is 9.01); at 12%, in about 6 (exactly 6.12).

Walter model

P = [D + (r/k_e)(E − D)] / k_e, where E is EPS, D is DPS, r the return on retained earnings and k_e the cost of equity.

• Growth firm (r > k_e): value is maximised at zero payout.
• Declining firm (r < k_e): value is maximised at 100% payout.
• Normal firm (r = k_e): payout is irrelevant.

Worked case (growth firm): E = Rs 10, k_e = 10%, r = 15%. At D = Rs 4, P = (4 + 1.5 × 6) / 0.10 = Rs 130. At zero payout, P = (1.5 × 10) / 0.10 = Rs 150; at full payout, P = 10 / 0.10 = Rs 100.

Same firm with r = 10% (normal firm): P = Rs 100 at every payout, so payout is irrelevant. With r = 8% (declining firm): P = Rs 80 at zero payout and Rs 100 at full payout.

The model assumes retained earnings are the only source of finance, that r and k_e stay constant, that EPS and DPS do not change, and that the firm has an infinite life.

Gordon model

P₀ = E(1 − b) / (k_e − br), where b is the retention ratio and g = br is the growth rate. It requires k_e > br.

Worked case: E = Rs 10, b = 0.4, r = 15%, k_e = 12%. Then g = 0.4 × 15% = 6%, D = 10 × 0.6 = Rs 6 and P₀ = 6 / (0.12 − 0.06) = Rs 100. At zero retention, P₀ = 10 / 0.12 = Rs 83.33, so with r > k_e retention adds value. If r = k_e = 12%, the price is Rs 83.33 at both b = 0.4 and b = 0, so payout is irrelevant. Strictly, E is the earnings expected for next year; textbook questions use the given EPS.

The basic conclusions mirror Walter. Gordon’s refinement under uncertainty, the bird-in-hand argument, holds that investors discount distant gains more heavily than current dividends, so k_e rises with retention and payout affects value even where r = k_e.

Modigliani–Miller irrelevance

Modigliani and Miller (1961) argued that dividend policy does not affect value under these assumptions: perfect capital markets with rational investors, no taxes (or no tax difference between dividends and capital gains), no flotation or transaction costs, a fixed investment policy, and (in the basic model) perfect certainty about future profits. Value is determined by the earning power of the firm’s assets and its investment policy, not by how earnings are split between dividends and retention.

P₀ = (D₁ + P₁) / (1 + k_e), so P₁ = P₀(1 + k_e) − D₁. With P₀ = Rs 100 and k_e = 10%, a Rs 5 dividend leaves P₁ at 110 − 5 = Rs 105, while no dividend gives Rs 110. Either way, the shareholder’s total return is Rs 10, or 10%. Any shortfall in investment funds is met by issuing new shares: m = (I − (E − nD₁)) / P₁.

Dividend policy in practice

The irrelevance result weakens once its assumptions are relaxed. Taxes on dividends and capital gains differ, new issues carry flotation costs, and dividend changes convey information about management’s view of future earnings, which is the signalling or information-content effect.

Lintner’s 1956 study found that firms set a long-run target payout ratio and move towards it gradually, avoiding cuts. This explains the stable dividend policy most listed companies follow.

Key points to remember

Frequently asked questions

What is the difference between an ordinary annuity and an annuity due?

In an ordinary annuity, payments come at the end of each period; in an annuity due, at the beginning. An annuity due is worth the ordinary annuity value × (1 + r). Rs 10,000 for 3 years at 10% is Rs 24,869 as an ordinary annuity and Rs 27,355 as an annuity due.

How is the effective annual rate calculated?

EAR = (1 + r/m)^m − 1 for m compounding periods a year. 12% compounded quarterly is 12.55% effective. With continuous compounding, EAR = e^r − 1, which is 12.75% at 12%.

What is the value of a growing perpetuity?

PV = C1 / (r − g), where C1 is next period’s cash flow and r > g. Rs 10 next year growing at 4%, discounted at 12%, is worth Rs 125.

What is the optimal dividend payout under the Walter model?

Zero payout when r > ke (growth firm), 100% payout when r < ke (declining firm), and any payout when r = ke (normal firm), since payout is then irrelevant.

Why do Modigliani and Miller say dividends are irrelevant?

In perfect capital markets with no taxes, no flotation costs and a fixed investment policy, a dividend lowers the ex-dividend share price by the same amount. The shareholder’s total return is unchanged, and any funds paid out are replaced by issuing new shares.

Practice: 10 MCQs

Attempt all ten questions, then press Submit. The correct answer and a short explanation appear under every question, followed by your score analysis.

Verification notes

• Every worked figure on this page was recomputed by program: FV 13,310; PV 82,645; PVIFA 2.4869 → 24,869; annuity due 27,355; FVIFA 3.31 → 33,100 and 36,410; perpetuity 62,500; growing perpetuity 125; EAR 12.36/12.55/12.68/12.75%; continuous FV 13,499; Walter 130/150/100, 100 and 80/100; Gordon 100 and 83.33; MM 105/110.
• Sources: Walter, J. E. (1956), “Dividend Policies and Common Stock Prices”, Journal of Finance. Gordon, M. J. (1962), The Investment, Financing and Valuation of the Corporation. Miller & Modigliani (1961), “Dividend Policy, Growth, and the Valuation of Shares”, Journal of Business. Lintner (1956), American Economic Review. Also standard texts: Prasanna Chandra; I. M. Pandey.