Actuarial Mathematics for Modelling · Financial instruments and insurance contracts as cashflow models

Cashflow Models: Basic Concepts and Notation Explained

Updated 11 October 2026 · Fact-checked

A cashflow model describes a contract as a list of payments, each with an amount, a time and a condition. Fixed payments are certain. Payments that depend on events, such as death or default, are contingent. To solve questions, list every flow, date it, give it a sign, then value it.

Understand Cashflow Models: Basic Concepts and Notation

A cashflow model reduces a contract to the money that moves in and out, and when. A bond, a loan, a life policy and a share can all be written this way. Once you have the cashflows, you can value them, price them or compare them.

Each cashflow has three parts: the amount, the time it is paid, and the condition under which it is paid. Time is usually measured in years from a chosen start point, called time 0. A payment of amount C_t at time t is written C_t. Take one viewpoint, such as the investor or the insurer, and keep it throughout. Money received is positive. Money paid out is negative.

A flow is certain (fixed) if its amount and timing are known in advance. A fixed-interest bond coupon is an example, ignoring default. A flow is uncertain if the amount, the timing, or whether it is paid at all is not known. An equity dividend has an uncertain amount. A death benefit has an uncertain timing and an uncertain payment, because it is paid only if the life dies within the term.

For continuous payment, you use a rate of payment ρ(t) per year at time t. The total paid between times t1 and t2 is the integral of ρ(t) from t1 to t2. Contingent flows are modelled with probabilities or random variables. The expected value of a contingent flow is the amount times the probability of payment.

This topic sets up the language for later work: present values, equation of value, annuities, bonds, and the present value random variables for life contingencies. Get the notation and the timeline right first. Most errors later start here.

Key rules to remember

Discrete cashflow: C_t = amount paid at time t (t in years from time 0); Use positive for receipts and negative for payments, from one stated viewpoint.
Continuous cashflow: Total paid from t1 to t2 = ∫ ρ(t) dt, integrated from t1 to t2; ρ(t) is the rate of payment per year at time t.
Expected value of a contingent payment: E[payment] = C × P(payment is made); For a single payment C at a fixed time, due only if an event occurs.
Net cashflow at time t: Net_t = Income_t − Outgo_t; Combine all flows at the same time before valuing.
Present value of a set of certain flows: PV = Σ C_t × v^t, where v = 1 ÷ (1 + i); Assumes a constant effective annual rate i. Covered fully in the equation of value topics.

How to solve Cashflow Models: Basic Concepts and Notation questions

Use this method for any question asking you to model a contract as cashflows.

1Read the contract and decide the viewpoint (investor, borrower, insurer, policyholder). State it.
2Choose time 0 and the time unit, usually years.
3List every payment: amount, time and condition. Include premiums, benefits, coupons, redemption, expenses and any charges.
4Mark each flow as certain or contingent. For contingent flows, state the event and the assumption about its probability.
5Give each flow a sign from your viewpoint. Be consistent.
6Draw a timeline or table with time, flow in, flow out and net flow.
7If asked, value the flows: use PV = Σ C_t v^t for certain flows, or multiply by the probability for expected values.
8Check the answer: sign, timing and whether the units are sensible.

Quickest way: Timeline table method

When to use it: Use it for MCQs and short written parts where you must list or value flows fast.

Write the times across the top: 0, 1, 2, and so on.
Under each time, write the flow with its sign.
Cross-check the count of flows against the contract terms, such as the number of coupons or premiums.
For contingent flows, write the amount × probability beside it.
Only then discount or sum.

Common mistakes in Cashflow Models: Basic Concepts and Notation

Mixing viewpoints, so the signs of some flows are reversed.
You switch between the insurer's and the policyholder's side while listing flows.
Fix: Write the viewpoint at the top and check each sign against it.
Putting a flow at the wrong time, for example a payment in advance placed at the end of the period.
You misread wording such as 'at the start of each year' or 'annually in arrear'.
Fix: Underline the timing words and convert them to exact time points on the timeline.
Treating a contingent flow as certain.
You list the death benefit or maturity benefit without its condition.
Fix: Attach the condition to every benefit and say whether you are using the amount or its expected value.
Leaving out flows such as expenses, redemption amount or final coupon.
You focus on the main payment and forget secondary ones.
Fix: Go through the contract clause by clause and tick each cashflow off.
Mixing a rate of payment ρ(t) with a single payment amount.
Continuous and discrete notation look similar.
Fix: A continuous flow must be integrated over a period to give an amount. Only then compare it with discrete payments.

Worked examples

Example 1

A 3-year bond with face value ₹1,00,000 pays an annual coupon of 6% in arrear and is redeemed at par at the end of year 3. Write the cashflows from the investor's viewpoint, with a purchase price of ₹98,000 at time 0. Then find the net cashflow at each time.

Show the solution

Viewpoint: investor. Time 0 is the purchase date.
Time 0: pay ₹98,000, so the flow is −₹98,000.
Coupon = 6% × ₹1,00,000 = ₹6,000.
Times 1 and 2: receive ₹6,000 each.
Time 3: receive the coupon ₹6,000 plus redemption ₹1,00,000 = ₹1,06,000.
All flows are certain, ignoring default.

Answer: Net flows: time 0: −₹98,000; time 1: +₹6,000; time 2: +₹6,000; time 3: +₹1,06,000.

Example 2

An insurer issues a 2-year term policy with sum assured ₹10,00,000, payable at the end of the year of death. The premium is ₹5,000, paid at the start of each year while the life survives. The probability of death in year 1 is 0.01 and in year 2 is 0.02, and the probability of surviving 1 year is 0.99. Ignore expenses. Find the expected cashflow at times 0, 1 and 2 from the insurer's viewpoint.

Show the solution

Viewpoint: insurer. Premiums are positive and benefits are negative.
Time 0: premium certain, as the life is alive at the start. Expected flow = +₹5,000.
Time 1: premium paid only if the life survives year 1, probability 0.99. Expected premium = 5,000 × 0.99 = ₹4,950.
Time 1: death benefit paid with probability 0.01. Expected benefit = 10,00,000 × 0.01 = ₹10,000.
Time 1 expected net = 4,950 − 10,000 = −₹5,050.
Time 2: no premium is due, since premiums are paid at times 0 and 1 only. Death benefit with probability 0.02: 10,00,000 × 0.02 = ₹20,000.
Time 2 expected net = −₹20,000.

Answer: Expected net cashflows: time 0: +₹5,000; time 1: −₹5,050; time 2: −₹20,000.

Exam tips

Always state the viewpoint and sign convention in the first line of a written answer. Examiners award marks for clear assumptions.
Draw a timeline even for short questions. It helps catch missing flows and wrong timing.
In life contingency questions, check that the premium is paid only while the life survives and the benefit only on the stated event.
In computer-based papers, lay out the cashflows in a column by time before you apply any discount formula.
In MCQs, check the timing words: in arrear, in advance, continuously. They change the answer.

Practice questions from Financial instruments and insurance contracts as cashflow models

Cashflow Models: Basic Concepts and Notation: frequently asked questions

What is a cashflow model in actuarial mathematics?

It is a representation of a contract as a set of payments, each with an amount, a time and a condition. You can then value, price or project the contract. It works for bonds, loans and insurance policies alike.

What is the difference between certain and contingent cashflows?

A certain cashflow has a known amount and time, such as a fixed coupon. A contingent cashflow depends on an event, such as death or survival. Its value is usually found using expected values or probabilities.

How do I decide the sign of a cashflow?

Pick one viewpoint and treat money received as positive and money paid as negative. Keep it fixed through the whole question. The insurer's and policyholder's signs are opposite.

What is the difference between a payment at time t and a rate of payment ρ(t)?

A payment C_t is a lump sum at one time. A rate ρ(t) is the amount per year paid continuously. You integrate ρ(t) over an interval to get the total amount paid in it.