Present value of a cash-flow schedule

What you will be able to do

After this lesson, you should be able to:

interpret present value from the same perspective as its cash flows;
discount each dated deterministic amount;
add discounted amounts without mixing payment dates.

Make cash flows comparable before adding

Use the local present-value-payment-index $\explain{present-value-payment-index}{k}$ to select one row and the local number-of-future-payments $\explain{number-of-future-payments}{n}$ to count the included future rows. At row $\explain{present-value-payment-index}{k}$ , write the signed-cash-flow as $\explain{signed-cash-flow}{CF_{\explain{present-value-payment-index}{k}}}$ , its payment-time as $\explain{payment-time}{t_{\explain{present-value-payment-index}{k}}}$ , and the matching discount-factor as $\explain{discount-factor}{D}(0,\explain{payment-time}{t_{\explain{present-value-payment-index}{k}}})$ . Write the schedule’s present-value at valuation-time $0$ as $\explain{present-value}{PV_0}$ .

The present value of the deterministic schedule is the sum of the discounted cash flows:

\explain{present-value}{PV_0}=\sum_{\explain{present-value-payment-index}{k}=1}^{\explain{number-of-future-payments}{n}}\explain{signed-cash-flow}{CF_{\explain{present-value-payment-index}{k}}}\explain{discount-factor}{D}(0,\explain{payment-time}{t_{\explain{present-value-payment-index}{k}}})

(1.5.1)

The index $\explain{present-value-payment-index}{k}$ and the count $\explain{number-of-future-payments}{n}$ are defined only in this lesson. The other symbols are shared across the course.

Two schedules are compatible when they use the same valuation time, perspective, currency, and discount factors. Let $\mathrm{A}\mathbin{\|}\mathrm{B}$ be the schedule that contains every row of both schedules $\mathrm{A}$ and $\mathrm{B}$ . For two compatible schedules, the present value is additive:

\explain{present-value}{PV_0}(\mathrm{A}\mathbin{\|}\mathrm{B})=\explain{present-value}{PV_0}(\mathrm{A})+\explain{present-value}{PV_0}(\mathrm{B})

(1.5.2)

Example 1.5.1 Present values of dated cash flows

One payment

A deterministic receipt of USD 1,000 is paid after two years, and $\explain{discount-factor}{D}(0,2)=0.9070294785$ . The present value of the receipt is:

\explain{present-value}{PV_0} =1000\times0.9070294785 =\text{USD }907.0294785 \approx\text{USD }907.03

The present value is in USD at valuation-time, from the holder’s perspective.

Mixed signed cash flows

Suppose the holder receives USD 100 at $\explain{payment-time}{t}_1$ , pays USD 40 at $\explain{payment-time}{t}_2$ , and receives USD 300 at $\explain{payment-time}{t}_3$ . The discount factors are 0.97, 0.93, and 0.88.

Payment index $\explain{present-value-payment-index}{k}$	Cash flow $\explain{signed-cash-flow}{CF_{\explain{present-value-payment-index}{k}}}$ , USD	Discount factor $\explain{discount-factor}{D}(0,\explain{payment-time}{t_{\explain{present-value-payment-index}{k}}})$	Discounted cash flow $\explain{signed-cash-flow}{CF_{\explain{present-value-payment-index}{k}}}\explain{discount-factor}{D}(0,\explain{payment-time}{t_{\explain{present-value-payment-index}{k}}})$ , USD
1	+100	0.97	+97.00
2	-40	0.93	-37.20
3	+300	0.88	+264.00

The present value is the sum of the discounted cash flows:

\explain{present-value}{PV_0}=97.00-37.20+264.00=\text{USD }323.80

Discounting does not change the sign of a cash flow. Reversing the perspective would change the sign of the present value.

Additivity

Combining two schedules keeps every row, including duplicate payments. For compatible schedules, a property test of the domain code checks (1.5.2). The compatibility conditions are necessary: present values in different currencies, or at different valuation dates, cannot be added without another model.

The lab below is a calculator for the one-payment case. It calculates the discount-factor from one discounting-lab-rate with annual compounding. It then multiplies one discounting-lab-cash-flow by the discount factor to give the present-value at valuation-time $0$ . The lab formulas use the same symbols as the text above.

Try discounting a future payment

Future payment $1,000

Annual rate 5%

Time to payment 2 years

Present value: $907.03
Discount factor: 0.907029

\explain{discount-factor}{D(0,t)} = (1 + \explain{discounting-lab-rate}{y})^{-\explain{payment-time}{t}} = (1 + \htmlData{lab-slot=rate}{0.0500} )^{- \htmlData{lab-slot=years}{2} } \;=\; \htmlData{lab-slot=factor}{0.907029}

\explain{present-value}{PV_0} = \explain{discounting-lab-cash-flow}{C} \cdot \explain{discount-factor}{D(0,t)} = \htmlData{lab-slot=payment}{1{,}000.00} \times \htmlData{lab-slot=factor}{0.907029} \;=\; \htmlData{lab-slot=present}{907.03}

Value today across the whole 0–30 year range at the current rate. The dot is your selected payment; the dashed line is its undiscounted amount.

The future payment is unchanged. Increasing the rate or waiting longer lowers its value today because the discount factor becomes smaller.

Assumptions and a text alternative

This toy model uses a deterministic payment, a whole number of years, and annual compounding. It has no curve, calendar, uncertainty, credit, tax, or funding.

Value today at the current rate (5%)
Time to payment	Value today
0 years	$1,000.00
5 years	$783.53
10 years	$613.91
15 years	$481.02
20 years	$376.89
25 years	$295.30
30 years	$231.38

The tested domain code does the lab’s calculation. The lab converts its percentage input to a decimal before the calculation.

Check your understanding

These items separate interpretation from calculation and include an unfamiliar multi-payment case. Each question stays collapsed until you open it; answers and explanations appear once you check.

Knowledge check 1.5.1 Present value

Model boundary and review note

This lesson covers deterministic present value only. It omits curves, calendars, uncertainty, credit, recovery, liquidity, tax, and funding.

The rule (1.5.1) follows Tuckman & Serrat[1]; Hull states the same principle[2]. The lesson stays draft pending human confirmation of the printed locators.

References

Tuckman & Serrat, Fixed Income Securities: Tools for Today's Markets (4th ed., 2022). §1.2, price as the sum of each cash flow times its discount factor (eqs. 1.1-1.3); full price equals present value (eq. 1.5). draft ↩
Hull, Options, Futures, and Other Derivatives (8th ed., 2012). Ch. 4, “Bond Pricing”. draft ↩

Notation used on this page (13)

C

Discounting lab cash flowdraft

The one dated amount the discounting lab discounts. It is page-local to the lab's one-payment model; the lesson's schedule notation writes an indexed amount instead.

PV_0 = \explain{discounting-lab-cash-flow}{C} \cdot D(0,t)

Units: stated currency at payment time

y

Discounting lab ratedraft

The one annual rate the discounting lab compounds to build each discount factor. It is page-local to the lab's toy model; elsewhere this lesson takes every discount factor as given rather than deriving it from a rate.

D(0,t) = (1 + \explain{discounting-lab-rate}{y})^{-t}

Units: decimal rate per year

n

Number of future paymentsdraft

The final index in the finite present-value sum. It counts included future rows and does not itself specify maturity in years.

\explain{number-of-future-payments}{n} \in \mathbb{N}

k

Present value payment indexdraft

Selects one dated cash flow in the finite schedule. This lesson-local index keeps sequence bookkeeping separate from the associated time measured in years.

\explain{present-value-payment-index}{k} \in \{1,\ldots,n\}

A(0,t)

Accumulation factordraft

Grows one current unit over a stated future horizon under the selected rate model.

Units: dimensionless currency-units per current currency-unit

m

Compounding frequencydraft

Number of equal compounding periods per year under the stated rate convention, a positive integer fixed by the model convention.

Units: compounding periods per year

D(0,t)

Discount factordraft

Converts one deterministic future unit into value at valuation time.

Units: current currency-units per future currency-unit

j^{(m)}

Nominal annual ratedraft

Annualized rate quote that must be paired with its compounding frequency.

Units: decimal rate per year

t_k

Payment timedraft

Time from the valuation date to one scheduled cash-flow event; a time coordinate, not a calendar date or payment index.

Units: years from the valuation date

r_m

Periodic ratedraft

Rate applied once in each compounding period under the stated convention, derived from the stated nominal annual quote in this model.

Units: decimal per compounding period

PV_0

Present valuedraft

Combines dated signed cash flows into one value at valuation time, using the same holder perspective as the signed cash flows.

Units: stated currency at the valuation time

CF_k

Signed cash flowdraft

Amount received or paid at one event from the stated holder perspective; positive means received and negative means paid by that holder.

Units: stated currency units

0

Valuation timedraft

Common origin from which later model times and present values are measured.