Time value of money, valuation and risk-return
A rupee today beats a rupee tomorrow. A bond or share is worth the present value of its future cash flows, discounted at a rate that rises with risk.
Formulas to know
| What | Formula | Note |
|---|---|---|
| Future value (lump sum) | FV = PV × (1 + r)^n | (1 + r)^n is the compound value factor |
| Present value (lump sum) | PV = FV ÷ (1 + r)^n | 1 ÷ (1 + r)^n is the PV factor (PVIF) |
| FV of an annuity | A × [(1 + r)^n − 1] ÷ r | Equal payments at each year-end |
| PV of an annuity | A × [1 − (1 + r)^−n] ÷ r | This factor is the PVIFA |
| Annuity due | Ordinary annuity value × (1 + r) | Payments at the start of each year |
| Perpetuity | A ÷ r | A constant payment forever |
| Growing perpetuity | A1 ÷ (r − g) | The basis of the Gordon model |
| Effective annual rate | (1 + r/m)^m − 1 | m = compounding periods a year |
| Rule of 72 | Years to double ≈ 72 ÷ interest rate (%) | At 8%, about 9 years |
Worked example 1: compounding, discounting and annuities
Interest is 10% a year. Find (a) the future value of ₹10,000 after 3 years, (b) the present value of ₹5,000 received at the end of each year for 3 years, and (c) the effective annual rate of 12% compounded quarterly.
- (a) FV = 10,000 × (1.10)^3 = 10,000 × 1.331 = ₹13,310.
- (b) PVIFA(10%, 3) = (1 − 1 ÷ 1.331) ÷ 0.10 = 2.48685.
- PV = 5,000 × 2.48685 = ₹12,434 (to the nearest rupee).
- (c) EAR = (1 + 0.12 ÷ 4)^4 − 1 = 1.03^4 − 1 = 1.1255 − 1 = 12.55%.
(a) ₹13,310 (b) ₹12,434 (c) 12.55%. More frequent compounding always makes the effective rate higher than the nominal rate.
Worked example 2: value of a bond
A ₹1,000 bond pays a 10% annual coupon and matures in 3 years. Investors require 12%. Find its value.
- Annual coupon = 10% × 1,000 = ₹100; the ₹1,000 face value is repaid at the end of year 3.
- PVIF(12%, 3) = 1 ÷ 1.12^3 = 1 ÷ 1.404928 = 0.71178.
- PVIFA(12%, 3) = (1 − 0.71178) ÷ 0.12 = 2.40183.
- PV of coupons = 100 × 2.40183 = ₹240.18. PV of face value = 1,000 × 0.71178 = ₹711.78.
- Value = 240.18 + 711.78 = ₹951.96.
₹951.96. It sells below par because the coupon (10%) is less than the required yield (12%).
Worked example 3: approximate yield to maturity
A ₹1,000 bond with an 8% annual coupon has 5 years left and trades at ₹950. Estimate its YTM.
- Approximate YTM = [C + (F − P) ÷ n] ÷ [(F + P) ÷ 2].
- Numerator = 80 + (1,000 − 950) ÷ 5 = 80 + 10 = 90.
- Denominator = (1,000 + 950) ÷ 2 = 975.
- YTM ≈ 90 ÷ 975 = 9.23%.
- Current yield = 80 ÷ 950 = 8.42%. For a discount bond, YTM > current yield > coupon rate (8%).
- Trial and error gives an exact YTM of about 9.30%. Some books use 0.4F + 0.6P as the denominator: 90 ÷ 970 = 9.28%.
YTM ≈ 9.23% by the approximation formula (exact ≈ 9.30%).
Worked example 4: Gordon constant-growth model
A company has just paid a dividend of ₹4 a share. Dividends will grow at 6% a year forever. Shareholders require 14%. Find the value of the share.
- D1 = D0 × (1 + g) = 4 × 1.06 = ₹4.24.
- P0 = D1 ÷ (ke − g) = 4.24 ÷ (0.14 − 0.06) = 4.24 ÷ 0.08 = ₹53.
- Trap: using D0 gives 4 ÷ 0.08 = ₹50, which is wrong.
- With zero growth the share is a perpetuity: P0 = D ÷ ke = 4 ÷ 0.14 = ₹28.57.
₹53. The model breaks down if ke is not greater than g.
Worked example 5: expected return and risk
A share may return 30% in a boom (probability 0.3), 15% in a normal year (0.5) and −5% in a recession (0.2). Find its expected return, standard deviation and coefficient of variation.
- E(R) = 0.3 × 30 + 0.5 × 15 + 0.2 × (−5) = 9 + 7.5 − 1 = 15.5%.
- Deviations from 15.5: +14.5, −0.5 and −20.5.
- Variance = 0.3 × 210.25 + 0.5 × 0.25 + 0.2 × 420.25 = 63.075 + 0.125 + 84.05 = 147.25.
- Standard deviation = √147.25 = 12.13%.
- CV = 12.13 ÷ 15.5 = 0.78, the risk per unit of return.
E(R) = 15.5%, σ = 12.13%, CV = 0.78.
Bond and share rules NET tests
- Coupon rate = YTM: the bond sells at par. Coupon < YTM: at a discount. Coupon > YTM: at a premium.
- Bond prices move opposite to interest rates. Longer maturity and lower coupon mean bigger price swings.
- Current yield = annual coupon ÷ market price. YTM also counts the gain or loss up to maturity.
- Duration (Macaulay, 1938) is the weighted average time to receive a bond's cash flows; higher duration means more interest-rate risk.
- An irredeemable preference share is a perpetuity: value = fixed dividend ÷ required return.
- Equity can also be valued by multiples: price = EPS × a suitable P/E ratio.
Systematic vs unsystematic risk
Total risk (variance) = systematic risk + unsystematic risk.
NET gives one-step numericals: future or present value, effective rate, bond value, approximate YTM and the Gordon price (use D1, not D0). It also tests the bond price-yield link and systematic vs unsystematic risk in statement form.
Typical MCQ: “A bond's coupon rate is 8% and its YTM is 10%. Will it sell at a premium, at par or at a discount?” At a discount: the market wants more than the coupon pays, so the price falls below face value. Trap: confusing the coupon rate with the yield.
Typical MCQ: “D0 = ₹2, g = 5% and ke = 15%. Find the share price.” D1 = 2 × 1.05 = ₹2.10, so P0 = 2.10 ÷ 0.10 = ₹21. Trap: 2 ÷ 0.10 = ₹20 uses D0.
Typical MCQ: “Which risk does diversification remove?” Only unsystematic (firm-specific) risk. Systematic risk stays, which is why beta matters.
Discount down, compound up. Bond price and yield sit on a see-saw: yield up, price down. Gordon needs next year's dividend: D1 over (ke − g).