Cash-flow timelines and perspective

What you will be able to do

After this lesson, you should be able to:

• read an ordered schedule of payment amounts and times;
• state the time unit and the time origin of a timeline;
• apply a consistent sign convention from a named perspective.

Start with events, not formulas

A financial calculation begins with a list of events. The valuation-time, written $0$, is the origin of every time in this lesson. The local payment-index $\explain{payment-index}{k}$ identifies one row. The index is not a time or a date. At row $\explain{payment-index}{k}$, write the signed-cash-flow as $\explain{signed-cash-flow}{CF_{\explain{payment-index}{k}}}$ and its payment-time as $\explain{payment-time}{t_{\explain{payment-index}{k}}}$. Row $\explain{payment-index}{k}$ pairs one amount with one time:

$$\explain{signed-cash-flow}{CF_{\explain{payment-index}{k}}} \quad\text{occurs at}\quad \explain{payment-time}{t_{\explain{payment-index}{k}}}$$

(1.1.1)

The shared subscript $\explain{payment-index}{k}$ pairs each amount with its time. The amount, the time, and the row index still have different meanings.

Example 1.1.1 Reading cash-flow timelines

Read a timeline

Suppose, in a simplified primary-issuance example, a bondholder pays an issuer USD 980 now, receives USD 30 after one year, and receives USD 1,030 after two years.

Payment index $\explain{payment-index}{k}$	Payment time $\explain{payment-time}{t_{\explain{payment-index}{k}}}$, years from time 0	Signed cash flow $\explain{signed-cash-flow}{CF_{\explain{payment-index}{k}}}$, USD	Event
0	0	-980	Purchase price paid
1	1	+30	Coupon received
2	2	+1,030	Coupon and principal received

This table does not say whether USD 980 is a fair price. It only describes who exchanges what, and when.

Reverse the perspective

From that issuer’s perspective, every amount above changes sign:

Time	Bondholder	Issuer
0 years	USD -980	USD +980
1 year	USD +30	USD -30
2 years	USD +1,030	USD -1,030

Changing the perspective does not change the dates or the absolute values of the amounts. Cash flows from opposite perspectives must be converted to one perspective before they are added. Otherwise the sum mixes two different models.

Same amount, different time

USD +100 at $\explain{payment-time}{t}_1=0.5$ years and USD +100 at $\explain{payment-time}{t}_2=2$ years are two distinct events. The amounts match; the times do not. A later lesson introduces the discount factors needed to compare them at time zero.

Check your understanding

These direct and transfer items cover reading a schedule and reversing perspective. Each question stays collapsed until you open it; answers and explanations appear once you check.

Knowledge check 1.1.1 Cash-flow timelines

1. DirectQuestion 1Not answered

2. TransferQuestion 2Not answered

3. DirectQuestion 3Not answered

4. TransferQuestion 4Not answered

Model boundary and review note

The times in this lesson are inputs, measured in years. Real schedules require calendar dates, business-day adjustments, and year-fraction conventions; none is inferred in this lesson.

The dated-schedule and timeline treatment here follows Tuckman & Serrat[1]. Reversing the holder’s perspective changes the sign of every cash flow. This is a bookkeeping convention, not a claim from one source page. The lesson stays draft pending human confirmation of the printed locators.

References

1. Tuckman & Serrat, Fixed Income Securities: Tools for Today's Markets (4th ed., 2022). §1.1, the dated coupon schedule in Table 1.1; §1.6 and Figure 1.3, a timeline of dated events and day-count intervals. draft ↩