Present value and discounting
Present value works backwards from a future sum to its worth today: divide by (1 + i)ⁿ. A positive NPV means a project earns more than the required rate.
Why a rupee today is worth more
A rupee today can be deposited to earn interest, so it grows to more than a rupee later. To compare money at different dates, bring every amount to one date, usually today. Moving money back in time is called discounting.
Present value formulas
Discount factors at 10% a year
| Years (n) | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| 1 ÷ 1.1ⁿ | 0.9091 | 0.8264 | 0.7513 | 0.6830 | 0.6209 |
Present value of a single sum
You will receive Rs. 500,000 in 4 years. At 10% a year compounded yearly, what is it worth today?
- PV = A ÷ (1 + i)ⁿ = 500,000 ÷ 1.1⁴.
- 1.1⁴ = 1.4641.
- PV = 500,000 ÷ 1.4641 = 341,506.73.
- Check: 341,506.73 × 1.4641 = 500,000 (to the nearest rupee). ✓
Rs. 341,506.73
Comparing two offers
A buyer offers Rs. 1,000,000 now, or Rs. 1,250,000 in 3 years, for a plot of land in Kurunegala. Money can earn 8% a year compounded yearly. Which offer is better?
- Bring the later offer to today: PV = 1,250,000 ÷ 1.08³.
- 1.08³ = 1.259712.
- PV = 1,250,000 ÷ 1.259712 = 992,290.30.
- Compare: Rs. 1,000,000 now is more than Rs. 992,290.30.
Take Rs. 1,000,000 now; it is worth Rs. 7,709.70 more in today's money.
Net present value of a project
A Galle hotel can spend Rs. 2,000,000 on a renovation that brings extra cash of Rs. 800,000, Rs. 900,000 and Rs. 1,000,000 at the end of years 1, 2 and 3. The required rate is 12%. Should it go ahead?
- Year 1: 800,000 ÷ 1.12 = 714,285.71.
- Year 2: 900,000 ÷ 1.2544 = 717,474.49.
- Year 3: 1,000,000 ÷ 1.404928 = 711,780.25.
- Total PV of inflows = 714,285.71 + 717,474.49 + 711,780.25 = 2,143,540.45.
- NPV = 2,143,540.45 − 2,000,000 = 143,540.45.
NPV = +Rs. 143,540.45, so the renovation is worth doing at 12%.
When a Sri Lankan group such as John Keells weighs a new hotel or plant, the standard method is to compare the present value of future cash flows with today's cost. A positive NPV supports the investment.
Short answer: “Find the present value of Rs. 121,000 due in 2 years at 10% compounded yearly.” 121,000 ÷ 1.21 = Rs. 100,000.
Structured question: “Rs. 1,000,000 now or Rs. 1,250,000 in 3 years at 8%? Advise.” PV of the later sum = Rs. 992,290.30, so take Rs. 1,000,000 now.
Essay: “Find the NPV at 12% of a Rs. 2,000,000 project returning Rs. 800,000, Rs. 900,000 and Rs. 1,000,000 over 3 years, and advise.” PVs total Rs. 2,143,540.45; NPV = +Rs. 143,540.45; accept.
Future → today: divide by (1 + i)ⁿ. Today → future: multiply. Higher rate or longer wait means smaller present value.