Finance-Theory · Unit 2 · Video 4 · Interactive Practice
An Asset Is Its Cashflows, Dated From Today Forward
IKey Formulas
Formula
Name
What it says
Assett≡{CFt,CFt+1,CFt+2,…}
An asset is a sequence of cashflows
The asset at date t is its cashflows from t forward, running on without end
Assett−10≡{CFt−10,CFt−9,…}
The date is part of the asset
No past cashflow appears; a different date gives a different sequence, so a different asset
CFs∈R
Entries are real numbers
Positive or negative (a negative entry is money leaving you), known or not
{0,0,0,…}
The zero asset
Admissible: a patent that will never produce one additional dollar is worth nothing
Key Insight: The definition lists the entries in date order and never adds them together — nothing has been summed yet. The concept is simple; the hard part is working out what the entries are.
IIWhere You Stand Decides the Asset
Same shares, different dates: the sequence starts where you stand, and no past cashflow appears in it.
IIIFrom an English Description to a Sequence
Boeing's regional jet, as dated entries in $MM: development money out first, then a level run.
IVOne Unknown Entry, No Ceiling
HP's option grant is a run of zeros and one year-10 entry: never negative, with no ceiling at the past high.
💡 A sequence written out like this is the thing that gets valued — putting a single number on it is what the next videos build.
VQuiz Questions
Problem 1 · Coca-Cola, a Century From Now
Given: today is date t, and the asset is Coca-Cola a hundred years from now. Which sequence is Assett+100?
✅ Correct! The sequence opens at the date the asset is dated and runs forward without end; nothing dated before t+100 appears in it.
❌ Not quite. Standing a century from now, CFt through CFt+99 are the past, and no past cashflow appears in the sequence. Nor does it stop at t+100 — it runs forward from there.
❌ Not quite. Those are the right entries, but added up. The definition lists the entries in date order and never adds them together.
❌ Close, but…CFt+100 is only the first entry. The sequence continues with CFt+101,CFt+102,… without end.
Show solution
The first subscript dates the asset: Assets≡{CFs,CFs+1,CFs+2,…}. Setting s=t+100:
Assett+100≡{CFt+100,CFt+101,CFt+102,…}
Three properties of the definition rule out the other options: the sequence contains no past cashflow (not a), it runs forward without end (not d), and it is a list, not a sum (not c).
Set against Assett≡{CFt,CFt+1,…} for Coca-Cola today, the entries differ — so Coca-Cola today and Coca-Cola a century from now are two different assets.
Problem 2 · Which One Is Not an Asset?
Given: four candidate sequences, each dated from today. Which one fails the definition Assett≡{CFt,CFt+1,CFt+2,…}?
✅ Correct! Every entry must be a real number, CFs∈R — there is no imaginary component to money changing hands.
❌ Not quite. An all-negative sequence is something you would not want to own — in ordinary language, a liability — but its entries are real numbers, so it is still a sequence of cashflows. The sign does not matter yet.
❌ Not quite. Zero, zero, zero, for ever, is a perfectly good sequence of cashflows: the zero asset is admissible, and it is worth nothing — the right answer for that patent.
❌ Not quite. Entries need not be known. Writing them as symbols is deliberate: you can have no idea what next year's cashflow will be and still say that it exists and belongs to this asset.
Show solution
Check each candidate against the conditions on an entry — a real number, dated, from t forward:
a — real entries, all negative. Money leaves you at every date, so you would call it a liability, but the definition places no condition on the sign: it qualifies.
b — 0∈R at every date, so the zero sequence qualifies. The patent is an asset worth nothing.
c — unknown entries are still entries. The definition needs them to exist and to be dated, not to be known.
d — 4+3i∈/R. A cashflow is money changing hands, and money has no imaginary component, so this is not a sequence of cashflows.
Nothing else is required: the definition does not even forbid two different assets from sharing a cashflow.
Problem 3 · One Index Share, Entry by Entry
Given: the notes' 2008 figures for the firms in the S&P 500, adjusted to the index — expected earnings of $66 per index share this year, dividends of $24, with dividends and earnings having grown 6.6% a year since 1926 (about 3.2% after inflation). Year 0 is this year; carry the growth rate forward.
What is the year-0 entry of one index share's sequence?
What is the year-2 entry, in nominal dollars?
✅ Correct! The sequence opens at the dividend, $24.00, and each entry is the previous one times 1.066: $24.00, $25.58, $27.27, and so on.
❌ Not quite. $66 is what the firms earn. The entry that reaches the holder of an index share is the dividend, $24 — a little over a third of earnings.
❌ Not quite. $66 − $24 = $42 is the part that stays inside the firms. It never reaches you, so it is not an entry of your sequence.
❌ Close, but… $25.58 is the year-1 entry. The sequence opens at year 0 with this year's dividend, before any growth.
❌ Close, but… $24.00 × 1.066 = $25.58 is one year of growth. Year 2 grows the year-1 entry once more.
❌ Not quite. $25.56 grows at 3.2% a year: it is the year-2 entry of the other sequence, counted in constant 2008 dollars. Nominal dollars grow at 6.6%.
❌ Close, but… $27.17 adds 6.6% of $24 (that is, $1.584) each year. Each entry grows the previous entry: $25.58 × 1.066 = $27.27.
Show solution
Step 1 — which number is the entry? The holder of an index share receives the dividend, not the earnings: $24 of the $66 is paid out, about 36 percent, and $66 − $24 = $42 stays inside the firms. The sequence opens at $24.00 in year 0.
Step 2 — carry the growth rate forward (an assumption about the future borrowed from the past):
Year 1: $24.00 × 1.066 = $25.58
Year 2: $25.58 × 1.066 = $27.27
Step 3 — the other sequence. Strip inflation out and the growth rate is 3.2%: $24.00 × 1.032 = $24.77 in year 1, and $24.77 × 1.032 = $25.56 in year 2, counted in constant 2008 dollars. In dollars per index share:
One index, two sequences of numbers, depending on which dollars you choose to count in.
Problem 4 · A Different Jet Forecast
Given: the Boeing regional jet as in the video — development takes three years (years 0, 1 and 2) and costs about $850MM — but now the forecast is 18 planes a year from year 3, unit costs come down only to $35MM, and the average price is still $41MM a plane. All entries in $MM.
What is each entry from year 3 onward?
Which sequence is the programme?
✅ Correct! Three negative development entries whose phasing is not specified, then 18×(41−35)=108 a year for as long as the production line stays open.
❌ Close, but…30×6=180 keeps the old volume. The new forecast is 18 planes a year.
❌ Not quite.18×41=738 is revenue alone. Each plane also costs $35MM to build, so only the margin, 41−35=6 per plane, is left.
❌ Close, but… 6 is the margin on one plane. The entry is for the whole year: 18 planes at that margin.
❌ Not quite. Development runs across three years, so it occupies three dated entries — years 0, 1 and 2 — not one lump at year 0. Only their total, about −850, is given.
❌ Not quite. That adds the entries into one total. The asset is the list, ordered by date; nothing is summed.
❌ Not quite. The development outflows are dated today and the next two years, so they belong to the sequence from today forward. They come first, as negative entries.
Show solution
Step 1 — the margin per plane:41−35=6 ($MM).
Step 2 — the entry per year:18×6=108 ($MM), from year 3 onward, for as long as the production line stays open.
Step 3 — the front of the sequence: years 0, 1 and 2 carry the development spending — money going out, so negative entries. The description gives only their total, about −$850MM, not how it is phased, so write them as symbols:
{D0,D1,D2,108,108,108,…},D0,D1,D2<0
Every entry is a forecast, and the list is never added up: the list itself is the asset.