Actuarial Mathematics for Modelling · Project appraisal

Project Appraisal with Borrowing and Different Interest Rates

Updated 11 October 2026 · Fact-checked

When you borrow at one rate and invest at another, appraise the project with an accumulated profit calculation. Roll the running balance forward year by year. Charge the borrowing rate while the balance is negative and the investment rate while it is positive. A positive final balance means the project adds value; compare options on that basis.

Understand Project Appraisal with Borrowing and Different Interest Rates

A basic net present value (NPV) calculation uses one interest rate. It treats money you owe and money you hold as earning or costing the same rate. In practice they differ. A company usually pays more to borrow than it earns on spare cash.

The fix is to track the project's cash balance through time. At each date, add the interest on the previous balance, then add the new cash flow. If the balance is negative, you are in debt, so interest is charged at the borrowing rate. If the balance is positive, you hold surplus cash, so it earns the investment rate (also called the lending or reinvestment rate).

The balance at the end of the project is the accumulated profit. A positive value means that, after paying all funding costs, the project leaves you better off. A negative value means it loses money on these funding terms. You can convert it to a present value by discounting at a chosen rate, if the question asks for that.

The key idea is that the rate applied can change from period to period, depending on the sign of the balance. This is why a single NPV or a single IRR can mislead here. The IRR assumes one rate for both borrowing and reinvesting, and that is not the situation in the question.

When the project is financed entirely by borrowing and the balance never turns positive, the accumulated profit equals the NPV at the borrowing rate, multiplied by (1 + i)ⁿ. The two methods give the same decision. They differ only when the balance changes sign.

Key rules to remember

Balance roll-forward: Bₜ = Bₜ₋₁ × (1 + i) + Cₜ, where i = borrowing rate if Bₜ₋₁ < 0 and i = investment rate if Bₜ₋₁ > 0; Cₜ is the net cash flow at time t (inflow positive, outflow negative). Start with B₀ = C₀.
Accumulated profit: Accumulated profit = Bₙ, the balance at the end of the project; Accept the project on this basis if Bₙ > 0. Say clearly which date the figure applies to.
Present value of accumulated profit: PV = Bₙ ÷ (1 + i)ⁿ; Use the discount rate the question specifies. Do not assume it equals the borrowing rate.
Single-rate check: Bₙ = NPV(i) × (1 + i)ⁿ; Holds only when one rate i is used for every period. Use it as a sense check, not when the rates differ.
Interest on a balance: Interest for one year = Bₜ₋₁ × i; For other periods, use (1 + i)^(fraction) − 1 or the stated convention. Stay consistent with the rate's compounding basis.

How to solve Project Appraisal with Borrowing and Different Interest Rates questions

Use this method for any question where the borrowing rate and the investment rate differ.

1List all net cash flows by date. Write outflows as negative and inflows as positive. Net any flows that fall at the same time.
2Note the borrowing rate, the investment rate, and the project end date. Convert any nominal rates to the effective rate for the period you are using.
3Set B₀ equal to the time 0 cash flow. This is normally negative.
4Move to the next date. Check the sign of the previous balance. Multiply it by (1 + borrowing rate) if negative, or (1 + investment rate) if positive. Then add the new cash flow.
5Repeat until the end date. Keep every balance, because the sign of each one decides the next rate.
6Read off the accumulated profit Bₙ. If asked for a present value, discount Bₙ at the rate given.
7State the decision: accept if Bₙ > 0, reject if it is negative, or choose the option with the larger accumulated profit if you are comparing projects over the same horizon. Add one line on what drives the result.

Quickest way: Running balance on a single line

When to use it: Use this under time pressure when there are four or more cash flows and the question asks only for accumulated profit or a yes/no decision.

Write the balance after each date in a column, one line per date. Do not recompute from time 0 each time.
Mark the sign of each balance with a small plus or minus next to it, so you pick the right rate on the next line.
Round only at the end. Keep at least four decimal places of rupees in working.
As a check, recompute the final line using the single borrowing rate. The two answers should differ only by the extra interest from the positive balances, so any gap should have a clear explanation.

Common mistakes in Project Appraisal with Borrowing and Different Interest Rates

Using one rate for every period
Students are used to NPV with a single rate and apply it automatically.
Fix: Check the sign of the balance at the start of every period before choosing the rate.
Choosing the rate by the sign of the cash flow instead of the balance
It is easy to confuse the new cash flow with the accumulated position.
Fix: The rate depends on the opening balance Bₜ₋₁, not on Cₜ. Add Cₜ after you have applied interest.
Accumulating each cash flow separately at one rate
Students accumulate every payment to the end at the borrowing rate, which is quick but ignores the investment rate.
Fix: Use this shortcut only when the balance stays negative throughout. Otherwise roll the balance forward.
Quoting the accumulated profit without its date
The final number looks like a normal NPV.
Fix: Say it is the value at time n. Discount it only if the question asks for a present value.
Using an IRR to decide when the rates differ
IRR is a familiar decision rule.
Fix: The IRR assumes one rate for borrowing and reinvesting. When the rates differ, use the accumulated profit to decide, and note the IRR assumption if you quote it.
Mixing nominal and effective rates
Questions may give a rate convertible half-yearly while cash flows are annual.
Fix: Convert to an effective annual rate first, or move the time step to match the compounding period.

Worked examples

Example 1

A project needs an outlay of ₹2,00,000 now and a further ₹1,00,000 at the end of year 1. It then returns ₹1,50,000 at the end of each of years 2, 3 and 4. Money can be borrowed at 9% pa effective and surplus cash earns 5% pa effective. Find the accumulated profit at the end of year 4 and state whether the project should go ahead.

Show the solution

Cash flows: time 0: −₹2,00,000; time 1: −₹1,00,000; times 2, 3, 4: +₹1,50,000 each.
B₀ = −2,00,000.
B₁ = −2,00,000 × 1.09 − 1,00,000 = −2,18,000 − 1,00,000 = −3,18,000. The balance is negative, so the borrowing rate applies next.
B₂ = −3,18,000 × 1.09 + 1,50,000 = −3,46,620 + 1,50,000 = −1,96,620.
B₃ = −1,96,620 × 1.09 + 1,50,000 = −2,14,315.80 + 1,50,000 = −64,315.80.
B₄ = −64,315.80 × 1.09 + 1,50,000 = −70,104.22 + 1,50,000 = +79,895.78.
The balance was negative until the last date, so the 5% investment rate was never used. The final balance is positive.

Answer: The accumulated profit at the end of year 4 is about ₹79,896. It is positive, so the project should go ahead on these funding terms.

Example 2

A project costs ₹5,00,000 now. It returns ₹3,00,000 at the end of year 1 and ₹3,00,000 at the end of year 2. It then needs a closing outflow of ₹2,00,000 at the end of year 3. Borrowing costs 10% pa effective and surplus cash earns 6% pa effective. Find the accumulated profit at the end of year 3 and compare it with the figure from using 10% for every period.

Show the solution

Cash flows: time 0: −5,00,000; time 1: +3,00,000; time 2: +3,00,000; time 3: −2,00,000.
B₀ = −5,00,000.
B₁ = −5,00,000 × 1.10 + 3,00,000 = −5,50,000 + 3,00,000 = −2,50,000. Still negative, so the borrowing rate applies next.
B₂ = −2,50,000 × 1.10 + 3,00,000 = −2,75,000 + 3,00,000 = +25,000. The balance is now positive, so the investment rate applies next.
B₃ = 25,000 × 1.06 − 2,00,000 = 26,500 − 2,00,000 = −1,73,500.
Single-rate check at 10%: B₃ = 25,000 × 1.10 − 2,00,000 = 27,500 − 2,00,000 = −1,72,500.
The difference of ₹1,000 is the interest lost on the ₹25,000 surplus because it earned 6% instead of 10%.

Answer: The accumulated profit at the end of year 3 is −₹1,73,500, so the project should be rejected. Using 10% throughout would give −₹1,72,500, which is ₹1,000 too favourable.

Exam tips

Show the balance for each date in a clear list. Examiners award method marks for the rate you chose and the sign of each balance.
Write down the borrowing and investment rates at the top, with their compounding basis. State any assumption, such as 'surplus cash is invested at the stated rate'.
If a question asks you to compare projects, check that the end dates match. If not, bring the accumulated profits to a common date or compare present values.
Do a quick sense check. If the balance never turns positive, the answer must equal the single-rate result at the borrowing rate.
In a computer-based paper, build the balance in one row per period with an IF formula for the rate. Then check the final cell against a hand calculation of the first two periods.

Practice questions from Project appraisal

Project Appraisal with Borrowing and Different Interest Rates in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

ACCA Strategic ProfessionalMultinational Investment Appraisal and Foreign Projects

Project Appraisal with Borrowing and Different Interest Rates: frequently asked questions

What is the accumulated profit method in project appraisal?

It rolls a running cash balance forward through the life of the project. Interest is charged at the borrowing rate when the balance is negative and earned at the investment rate when it is positive. The balance at the end is the accumulated profit.

Why can't I just use NPV when the borrowing rate differs from the investment rate?

A standard NPV uses one rate for every period. If the balance changes sign during the project, the effective rate also changes. The accumulated profit method handles this directly, and a single NPV does not.

Does the accumulated profit decide accept or reject?

On these funding terms, yes. A positive accumulated profit means the project leaves you better off after funding costs, and a negative one means it does not. Always state the date at which the profit is measured.

How do I convert accumulated profit to a present value?

Divide the final balance by (1 + i)ⁿ, where i is the discount rate the question gives and n is the number of years. Do not assume the discount rate equals the borrowing rate unless the question says so.