Actuarial Mathematics for Modelling · Present value and accumulated value of cashflow streams

Valuing Cashflows and the Equation of Value

Updated 11 October 2026 · Fact-checked

An equation of value sets the present value of money paid in equal to the present value of money paid out, at the same date and the same interest rate. To solve it, choose a focal date, discount or accumulate every cashflow to it, write the equality and solve for the unknown.

Understand Valuing Cashflows and Equation of Value

Money has a time value. ₹1,000 today is not worth the same as ₹1,000 in five years, because today's money can earn interest. So you can only compare cashflows after moving them to the same date.

The equation of value does this. Pick a focal date. Move every inflow and every outflow to that date using the interest rate. Then state that the value of inflows equals the value of outflows. This is the idea of equivalence: two sets of cashflows are equivalent if their values at one date are equal.

The focal date is your choice. With a constant effective rate, any date gives the same answer, so choose the one that makes the algebra easiest. Usually that is the date of the unknown payment, or time 0 if the unknown is a time or a rate.

The unknown can be a payment amount, a payment time, or an interest rate. Unknown payment: the equation is linear and easy. Unknown time: you often need logarithms. Unknown rate: you usually need trial and error, or interpolation, or a quadratic if there are only two or three terms.

The method of equated time finds a single time t at which one payment equal to the sum of several payments is equivalent to them. An exact answer comes from the equation of value. An approximation is the weighted average of the payment times, with the payment amounts as weights. This is only approximate, since it ignores compounding.

Key rules to remember

Discount factor: v = 1 ÷ (1 + i), and v^t = (1 + i)^(−t); Present value at time 0 of ₹1 paid at time t, for effective rate i per period.
Accumulation factor: (1 + i)^t; Value at time t of ₹1 invested at time 0.
Equation of value at time 0: Σ (inflows at time t) × v^t = Σ (outflows at time t) × v^t; Use any focal date. Rates and times must be in the same unit.
Equation of value at focal date T: Σ C_t × (1 + i)^(T − t) = 0, with inflows positive and outflows negative; Works for payments before and after T. Use (1 + i)^(T − t) for every payment.
Exact equated time: Σ C_t × v^t = (Σ C_t) × v^t̄, so t̄ = −ln[ Σ C_t v^t ÷ Σ C_t ] ÷ δ, where δ = ln(1 + i); Exact for a single replacement payment equal to the total of the C_t.
Approximate equated time: t̄ ≈ Σ (t × C_t) ÷ Σ C_t; Weighted average of times. Only an approximation, and it is not exact for compound interest.
Net present value: NPV(i) = Σ C_t × v^t; Inflows positive and outflows negative. Equation of value is NPV(i) = 0 for the rate that makes it so.

How to solve Valuing Cashflows and Equation of Value questions

This method works for unknown payments, unknown times and unknown rates. Take your time on the setup, because most marks go there.

1Draw a timeline. Mark each cashflow with its amount, its time and its direction (in or out).
2Convert the interest rate to an effective rate per the time unit you use. For example, convert a nominal rate convertible monthly to an effective annual rate if times are in years.
3Choose a focal date. If the unknown is a payment, pick its date. If the unknown is a time or rate, time 0 is usually simplest.
4Move every cashflow to the focal date using v^t or (1 + i)^t. Check the sign of each exponent.
5Write: value of inflows = value of outflows. Keep the unknown as a symbol.
6Solve. For a payment, rearrange. For a time, take logarithms. For a rate, use a quadratic, trial and error, or linear interpolation.
7Check the answer: it should be sensible in size and time. Substitute it back if there is time, and state units.

Quickest way: Pick the focal date at the unknown

When to use it: Use when you need one unknown payment and the rate is constant. It removes the need to discount the unknown itself.

Put the focal date at the unknown payment so it has factor 1.
Write every other payment with a factor (1 + i)^(T − t). Payments before the focal date are accumulated and payments after it are discounted.
Collect the known terms on one side and the unknown on the other.
Divide to find the unknown. Do one sanity check. A replacement payment made between the payment dates should be close to the total of the payments it replaces. It lies slightly below the total when the larger payment is the one being discounted, as in the example below (about ₹98,756 against ₹1,00,000). A replacement payment made later than all the payments it replaces should exceed their total when interest is positive.

Common mistakes in Valuing Cashflows and Equation of Value

Discounting a payment that lies before the focal date, or accumulating one that lies after it
Students use v^t for everything without thinking about the direction of movement.
Fix: Use (1 + i)^(T − t) for every payment. If t is before T the exponent is positive and the payment grows. If t is after T the exponent is negative and it is discounted.
Using a nominal rate directly as the effective rate
The rate quoted, such as 6% convertible monthly, is read as 6% per year.
Fix: Convert first. The effective monthly rate is 0.06 ÷ 12 = 0.005. The effective annual rate is (1.005)^12 − 1.
Mixing time units, for example months for payments but an annual rate
The question states times in months and the rate per annum.
Fix: Choose one unit for both. Either write all times in years (6 months = 0.5) or use the effective monthly rate.
Using the weighted average time as the exact equated time
The approximation is quick and looks like a formula, so it is applied even when the question asks for the exact answer.
Fix: If the question says an interest rate, solve Σ C_t v^t = (Σ C_t) v^t̄ and use logarithms. Use the weighted average only if asked for an approximation.
Putting inflows and outflows on the same side with the wrong signs
Students forget which cashflow is paid and which is received.
Fix: Decide your perspective (lender or borrower) first. Write inflows to that party as positive and outflows as negative, then set the total to zero.
Rounding the discount factors too early
Four-figure factors seem enough, but the error builds up over several terms.
Fix: Keep full calculator precision until the final answer. Round only at the end.

Worked examples

Example 1

You must pay ₹40,000 in 1 year and ₹60,000 in 3 years. You want to replace both with a single payment of ₹X in 2 years. The effective annual interest rate is 8%. Find X.

Show the solution

Choose the focal date as time 2, the date of the unknown payment.
The payment at time 1 is accumulated by 1 year: 40,000 × 1.08 = 43,200.
The payment at time 3 is discounted by 1 year: 60,000 ÷ 1.08 = 55,555.56.
Equation of value at time 2: X = 43,200 + 55,555.56.
X = 98,755.56.

Answer: X ≈ ₹98,756 (to the nearest rupee)

Example 2

A payment of ₹50,000 is due now and another of ₹50,000 is due at the end of 4 years. Find the exact time t at which a single payment of ₹1,00,000 would be equivalent, at an effective annual rate of 10%. Compare it with the weighted average time.

Show the solution

Equation of value at time 0: 50,000 + 50,000 × v^4 = 1,00,000 × v^t.
Divide by 1,00,000: 0.5 + 0.5 × v^4 = v^t.
v^4 = 1.1^(−4) = 1 ÷ 1.4641 = 0.683013.
Left side: 0.5 + 0.5 × 0.683013 = 0.841507.
So v^t = 0.841507, which means 1.1^t = 1 ÷ 0.841507 = 1.188347.
t = ln(1.188347) ÷ ln(1.1) = 0.172700 ÷ 0.095310 = 1.812.
Weighted average time = (0 × 50,000 + 4 × 50,000) ÷ 1,00,000 = 2.
The exact time is shorter than 2 years because discounting gives more weight to the earlier payment.

Answer: Exact equated time ≈ 1.81 years. The weighted average approximation gives 2 years.

Exam tips

Draw the timeline before you write any algebra. Examiners reward a clear setup even when the arithmetic slips.
State your focal date and your effective rate per period explicitly. This earns method marks and helps if you make an error later.
For unknown rate questions with two or three payments, look for a quadratic in v before you try trial and error.
Show the full equation of value on one line, then solve. In the computer-based paper, build the cashflows in a column and use a goal seek or an IRR function, then check it against a hand calculation.
Check the answer for reasonableness. A replacement payment made between the payment dates should be close to the total, so a large gap signals an error. One made later than all the payments it replaces should exceed their total when interest is positive.

Practice questions from Present value and accumulated value of cashflow streams

Valuing Cashflows and Equation of Value: frequently asked questions

Does the choice of focal date change the answer?

Not if the interest rate is constant and compounding is consistent. All focal dates give the same unknown. Choose the date that makes the algebra simplest, usually the date of the unknown payment or time 0.

How do I solve an equation of value for an unknown payment?

Write each known cashflow at the focal date using the accumulation or discount factor. Set inflows equal to outflows, keep the unknown as X, and rearrange. Placing the focal date at the unknown payment saves a step.

What is the method of equated time in CM1?

It finds the single time at which one payment, equal to the total of several payments, is equivalent to them. The exact answer comes from the equation of value with logarithms. The weighted average of times is a quick approximation.

How do I find an unknown interest rate from an equation of value?

Set the net present value to zero. If there are two or three distinct times, treat v as the unknown and solve the quadratic. Otherwise use trial and error followed by linear interpolation between two rates that bracket zero.