Foundations · F4 · 9 min
Time value of money and day counts
Work out what future money is worth today, judge an investment by its NPV and IRR, and count interest days correctly.
In the news: When interest rates rise, money promised far in the future is worth less today. This lesson shows why.
The essentials
- Time value of money
- Money today is worth more than the same sum later, because it can earn interest meanwhile. At 5%, £100 grows to £105 in a year and £110.25 in two, as interest earns interest: compounding.
- Present value
- What a future sum is worth today, after discounting at the interest rate for each year of waiting. The further away and the higher the rate, the less it is worth: at 5%, £1 due in ten years is worth about 61p.
- Net present value (NPV)
- The present value of everything an investment pays, minus its cost today. Positive means it beats the discount rate: paying £1,000 for £400 a year over three years has an NPV of about +£31 at 8%.
- Internal rate of return (IRR)
- The discount rate at which NPV is exactly zero: the return the investment earns. £1,000 for £400 a year over three years has an IRR of about 9.7%, so it clears an 8% hurdle.
- Day counts
- Money-market interest is paid for the days a loan runs. Sterling counts a 365-day year, dollars and euros 360, so 4% for 91 days pays £99,726 on £10m but $101,111 on $10m.
Present value
PV = CF ÷ (1 + r)ⁿ
- CF
- the cash received in n years
- r
- the interest rate per year, as a decimal (5% = 0.05)
- n
- the number of years you wait
How it works
1. £100 now or £100 in a year
Offered £100 now or £100 in a year, take it now: put it in a savings account at 5% and you have £105 in a year. Money today is worth more than the same money later, because it can earn interest in between.
2. Interest on interest
Leave it a second year and the 5% is paid on £105, not £100: you end with £110.25. Interest earning interest is compounding. After n years, £100 grows to £100 × 1.05 multiplied by itself n times, written 1.05ⁿ.
3. Working backwards
Turn it round. If £105 in a year is worth £100 today at 5%, a future amount is worth that amount divided by 1.05 for each year you wait. That is its : £110.25 due in two years is worth £110.25 ÷ 1.05² = £100 today.
4. Discount factors
The number you multiply a future amount by, 1 ÷ (1 + r)ⁿ, is its . At 5%, £1 in one year is worth about 95p today, and £1 in ten years about 61p. The further away the money and the higher the rate, the less it is worth now.
5. A stream of equal payments
Many deals pay the same amount every year, such as a loan or a bond’s interest. Add up the present value of each payment and you have the value of the stream. The does it in one step: £1,000 a year for five years at 4% is worth about £4,452 today, not £5,000.
6. Net present value
To judge an investment, add the present values of everything it pays, then subtract what it costs today. That is its , or NPV. Pay £1,000 now for £400 a year for three years: at 8% the payments are worth £1,030.84, so NPV is +£30.84. Positive NPV means it earns more than 8%.
7. IRR: the break-even rate
Raise the discount rate and NPV falls. The rate at which NPV hits exactly zero is the , or IRR: the return the investment earns. For the project above it is about 9.7%. So NPV is positive at any rate below the IRR, and negative above it.
8. Counting the days
Short loans pay interest for the days they run: amount × rate × days ÷ days in a year. Sterling markets count a 365-day year; dollar and euro markets count 360. This rule is the . So 4% for 91 days pays £99,726 on £10m, but $101,111 on $10m.
Try it
Move the interest rate. Hollow bars are the payments, filled bars what each is worth today. The lower line is NPV; it crosses zero at the IRR.
£1,000 now, then £400 a year for three years
Year 3: £400.00 × discount factor 0.7938 = £317.53 today. Tap or hover a year.
- PV of the income
- £1,030.84
- NPV
- +£30.84
- IRR
- 9.70%
| Year | Cash flow | Discount factor | Present value |
|---|---|---|---|
| 0 | −£1,000.00 | 1.0000 | −£1,000.00 |
| 1 | £400.00 | 0.9259 | £370.37 |
| 2 | £400.00 | 0.8573 | £342.94 |
| 3 | £400.00 | 0.7938 | £317.53 |
Worked example
A project costs £1,000 today and pays £400 at the end of each of the next three years. Your cost of borrowing is 8%.
- Year 1: £400 ÷ 1.08 = £370.37.
- Year 2: £400 ÷ 1.08² = £342.94.
- Year 3: £400 ÷ 1.08³ = £317.53.
- Add them: £370.37 + £342.94 + £317.53 = £1,030.84.
- NPV = £1,030.84 − £1,000 = +£30.84, so the IRR is above 8%.
- Trying 9.7% gives an NPV of about zero: the IRR is about 9.7%.
NPV ≈ +£31 at 8% and IRR ≈ 9.7%: the project clears the 8% hurdle, narrowly.
Check yourself
8 quick questions on the explanation above.
Practice
New numbers every set. Misses and guesses go into your review deck.
In the interview
Answer out loud in under a minute, then compare.
£100 today or £110 in a year: when does your answer change?
What’s IRR, and when can it mislead?
Why do money markets use ACT/360, and what difference does it make?
Remember
PV = CF ÷ (1 + r)^n
The higher the rate and the further away the cash, the less it is worth today; NPV is positive when the IRR beats the discount rate.