Time value: one date, then compare
Introduced| English | Meaning |
|---|---|
| annuity/əˈnjuːɪti/ | In financial mathematics, a series of equal payments made at regular time intervals. |
| perpetuity | equal payments continuing for ever, worth the payment divided by the rate |
| discounting/ˈdɪskaʊntɪŋ/ | Reducing future money to its value today because of the time value of money. |
| compounding | earning returns on earlier returns as money moves forward in time |
A pound today is not a pound tomorrow
- The second site costs 60,000 now and pays back over five years. Until every pound sits at one common date, "it pays back more than it costs" is not a sentence.
- The whole machine is four movements at one rate: compound, discount, annuity, perpetuity.
The four movements at 8%
- Discounting 折现 back: $10{,}000/1.08^3 = 7{,}938$ ✓. Compounding 复利 forward: $6{,}000 \times 1.08^2 = 6{,}998$ ✓.
- Annuity 年金: $4{,}000 \times [1-1.08^{-5}]/0.08 = 15{,}971$ ✓. Perpetuity 永续年金: $2{,}400/0.08 = 30{,}000$ ✓.
Which movement is this?
Route each situation to discounting, compounding, annuity or perpetuity - and catch the effective-rate traps.
Discount 10,000 at 8% for 3 years, to the nearest pound?
Difficulty 2/5 · 140 coins + a small bonus on first completion today
$10{,}000/1.08^3 = £7{,}938$ - three divisions, nothing more.
Compound 6,000 at 8% for 2 years, to the nearest pound?
Difficulty 2/5 · 140 coins + a small bonus on first completion today
$6{,}000 \times 1.08^2 = £6{,}998$.
A 4,000 five-year annuity at 8%, to the nearest pound?
Difficulty 2/5 · 140 coins + a small bonus on first completion today
$4{,}000 \times (1-1.08^{-5})/0.08 = £15{,}971$.
A 2,400 perpetuity at 8%?
Difficulty 2/5 · 140 coins + a small bonus on first completion today
$C/r = 2{,}400/0.08 = £30{,}000$.
Match each stream to its tool.
Difficulty 2/5 · 140 coins + a small bonus on first completion today
Four shapes, four tools - the shape names the formula.
Nominal is not effective
- 8% nominal with monthly rests = 0.667% a month = $(1+0.08/12)^{12}-1 = 8.30\%$ effective ✓.
- On 60,000 over ten years, quoting the flat 8% understates the true cost by roughly 180 a year. Always ask: how many rests, and which rate was quoted?
8% nominal compounded monthly - the effective annual rate, in % (two decimals)?
Difficulty 2/5 · 140 coins + a small bonus on first completion today
$(1+0.08/12)^{12}-1 = 8.30\%$ - the rests do the work.
10,000 in year 3 vs 6,000 today + 4,000 in year 2. Which wins, and by how much?
Difficulty 1/5 · 100 coins + a small bonus on first completion today
Today values: 7,938 vs $6{,}000+3{,}430=9{,}430$ - a 1,492 gap, visible only because both sit at one date.
A monthly-payment loan should be discounted at the quoted annual rate.
Difficulty 1/5 · 100 coins + a small bonus on first completion today
Monthly cash needs the monthly rate (nominal/12) - or convert the rate to effective annual. Mixing frequencies is the classic error.
One date, then compare
- 10,000 in year 3 against 6,000 today plus 4,000 in year 2: move both to today — 7,938 against 9,430. The second wins by 1,492 in one line.
Three time-value slips
- Comparing cash at different dates. Move one side first.
- Nominal treated as effective. Rests compound.
- Monthly cash discounted at the annual rate. Match the rate to the frequency.
Recap
- $PV = FV/(1+r)^n$; annuities and perpetuities are discounting pre-assembled.
- Nominal 8% monthly = 8.30% effective.
- One date, then compare.