Practical applications of the equation of value (loans and APR): formula sheet

Key formulas

Loan from level instalments (arrears)

L = X × a_n = X × (1 − v^n) ÷ i

i is the effective rate per payment period. v = 1 ÷ (1 + i). First payment one period after the loan.

Level instalment

X = L ÷ a_n

Rearrangement of the equation of value at the loan date.

Interest in an instalment

I_t = i × B_(t−1)

B_(t−1) is the balance just after the previous payment.

Capital in an instalment

C_t = X − I_t

Capital repaid equals the fall in the balance.

Closing balance

B_t = B_(t−1) − C_t = B_(t−1)(1 + i) − X

Use this to roll the schedule forward.

Prospective balance

B_t = X × a_(n−t)

For level instalments in arrears after t payments, with the same i throughout.

Retrospective balance

B_t = L(1 + i)^t − X × s_t

s_t = ((1 + i)^t − 1) ÷ i. Gives the same value as the prospective method.

Capital in the first instalment and growth

C_(t+1) = C_t × (1 + i)

For level instalments in arrears, capital repayments grow geometrically at rate i.

Total interest paid

Total interest = n × X − L

For level instalments with no other charges.

Equation of value for APR

Loan − upfront charges = Σ (payment at time t) × v^t, where v = 1 ÷ (1 + j)

j is the effective rate per payment period. Charges paid later go on the right-hand side as extra payments.

Level instalments

Net amount received = X × a_n at rate j

Use a_n = (1 − vⁿ) ÷ j for payments at the end of each period. Use the annuity-due form if payments start at once.

Periodic rate to APR

APR = (1 + j)^m − 1

m is the number of periods per year. For monthly payments, m = 12.

Nominal to effective

1 + i = (1 + i⁽ᵐ⁾ ÷ m)^m

i⁽ᵐ⁾ is the nominal annual rate convertible m times a year. Use it to compare with the APR.

Flat rate instalment

Instalment = Loan × (1 + flat rate × term in years) ÷ number of instalments

This is only the way the payment is set. The flat rate itself is not the true cost.

Flat rate total interest

I = L × f × n

L is the amount borrowed, f the flat rate per year, n the term in years. Interest is on the original loan throughout.

Level instalment under a flat rate

X = L(1 + f n) ÷ N

N is the total number of payments, for example N = 12n for monthly payments. Payments are equal.

Equation of value for the effective rate

L = X × a(N at j)

Here a(N at j) = (1 − (1 + j)^−N) ÷ j, with payments at the end of each period. Solve for j, the rate per payment period.

Annual effective rate from monthly rate

i = (1 + j)^12 − 1

The nominal rate convertible monthly is i(12) = 12j.

Rough estimate of the nominal rate

r ≈ 2 m I ÷ (P (N + 1))

m is payments per year, I total interest, P the amount borrowed, N total payments. Approximate only, and it tends to overstate. Use it as a starting guess.

Equation of value for a loan

L = Σ Pt × v^t, where v = 1 ÷ (1 + i)

L is the amount borrowed and Pt is the payment at time t. Use the effective rate per payment period.

Level annuity-immediate

aₙ = (1 − vⁿ) ÷ i

Use for n level payments of 1 at the end of each period, valued one period before the first payment.

Increasing annuity-immediate (payments 1, 2, ..., n)

(Ia)ₙ = (äₙ − n × vⁿ) ÷ i

Payment at time t is t. For a different start and step, split into a level part plus an increasing part.

Payments rising in geometric progression

PV = P × Σ for t = 1 to n of (1 + g)^(t−1) × v^t = P × [1 − ((1 + g) ÷ (1 + i))ⁿ] ÷ (i − g)

Valid when i ≠ g. If i = g, PV = n × P ÷ (1 + i).

Balloon payment

B = (L − PV of regular payments) × (1 + i)ⁿ

n is the time of the balloon. The balloon is paid on top of any regular payment due at that time.

Deferred loan

PV at time 0 = v^d × (PV of payments valued at time d)

d is the number of periods the payment stream is shifted back to reach time 0.

Outstanding balance (prospective)

Bt = PV at time t of all future payments, at the loan rate

Right after the payment at time t, exclude that payment.

Outstanding balance (retrospective)

Bt = L × (1 + i)^t − accumulated value of payments made up to t

Gives the same result as the prospective method at the same rate.

Quick revision

• Equation of value: PV of outflows = PV of inflows at the same rate and date.
• Any valuation date works. Choose the one that makes the equation simplest.
• Interest in a payment = rate × opening balance; capital repaid = payment − interest.
• Outstanding balance prospectively = PV of the remaining payments at the loan rate.
• Outstanding balance retrospectively = accumulated loan − accumulated payments made.
• APR is the effective annual rate that equates the amount received with the PV of all payments and charges.
• Compute APR on the actual cash flows, including fees, not on the quoted rate alone.
• Flat rate interest is charged on the original loan for the whole term, so the effective rate is higher than the flat rate.
• Convert nominal rates to effective rates before comparing loans.
• For early repayment, settle the outstanding balance at that date, plus any stated penalty.
• Check every schedule: closing balance after the last payment should be zero.
• Show the equation, the rate, the time point and the result, with units, in every written answer.

Common mistakes

• Using the annual rate for monthly or half-yearly instalments. Fix: Convert first. For a nominal rate compounded monthly, use i^(12) ÷ 12 per month and count n in months.
• Calculating interest on the original loan every period. Fix: Interest is always the period rate times the current outstanding balance.
• Using the full loan as the amount received when a fee is deducted upfront. Fix: Subtract the fee at time 0. The borrower receives less than the loan, but repays on the full loan.
• Multiplying the monthly rate by 12 and quoting it as the APR. Fix: APR is an effective annual rate. Compute (1 + j)¹² − 1. The product 12j is only the nominal rate convertible monthly.
• Treating the flat rate as the effective rate Fix: A flat rate is only a way of working out total interest. Always solve the equation of value to get the true rate.
• Using the wrong number of payments Fix: Set N = payments per year × years. Match the rate j to the same period as the payment.
• Valuing an annuity at the wrong date, for example using aₙ for a deferred loan without discounting further Fix: State the valuation date of the annuity factor on your timeline. Then discount or accumulate it to the date you need.
• Forgetting that the balloon is paid in addition to the last regular instalment Fix: Read carefully. If the question says 'in addition', include both at that time. If it says 'the final payment is', it replaces the regular one.

Exam tips

• Always state the valuation date and write the equation of value in words before the numbers. Examiners give method marks for it.
• Use the prospective method as your main route. It needs fewer steps. Use the retrospective method only as a check or when asked.
• Show a short schedule table in written answers and confirm the final balance is zero. This catches errors fast.
• In computer-based Paper B questions, set the rate per period in a cell, use a formula for X, and fill the schedule down with formulas rather than typed numbers.
• Read the timing carefully. Arrears versus advance, and a deferred first payment, are the usual traps in MCQs.
• Write the equation of value in words first, then in symbols. Examiners give marks for the correct setup even if the solving goes wrong.
• State your time unit and payment timing as assumptions at the start of the written answer.
• For Paper B, show the formula, the rate you solved for and the conversion step. Do not give only the final APR.