Financial mathematics formula sheet, with worked examples
The formula sheet I wish I had had at university: the financial mathematics formulas that appear in Business Administration exams, with a worked numerical example in each section. It follows the notation used at Spanish universities, which is what you will meet if you study here in English.
Notation: C0 is the initial capital, Cn the final amount, i the effective interest rate per period, n the number of periods and a each annuity payment.
1. Simple and compound interest
With simple interest, interest does not earn interest; with compound interest, it does. Unless the question says otherwise, operations longer than a year use compounding.
Simple interest: Cn = C0 · (1 + i · n)
Compound interest: Cn = C0 · (1 + i)n
Example. 10,000 euros at 5% a year compounded for 3 years: C3 = 10,000 · 1.053 = 11,576.25 euros.
2. Simple discounting: commercial and rational
Commercial (bank) discount charges interest on the face value; rational discount charges it on the present value. With the same data, commercial discount always gives a lower present value.
Commercial discount: E = N · (1 − d · t)
Rational discount: E = N / (1 + i · t)
Example. A 5,000 euro bill due in 90 days at 6%, using a 360-day year. Commercial: 4,925.00 euros. Rational: 4,926.11 euros.
3. Compound discounting
C0 = Cn · (1 + i)−n
4. Equivalent rates, nominal rates and APR
Equivalence: (1 + i) = (1 + im)m ⇒ im = (1 + i)1/m − 1
Nominal: jm = m · im Effective annual rate (no fees) = (1 + jm/m)m − 1
Example. 12% nominal compounded monthly is 1% a month, so the effective annual rate is 1.0112 − 1 = 12.68%. Conversely, 6% effective a year is 0.4868% a month.
The APR (TAE in Spain) is never below the nominal rate: the two only coincide when there are no fees and payments are annual.
5. Level annuities
Present value, ordinary annuity: V0 = a · [1 − (1 + i)−n] / i
Future value, ordinary annuity: Vn = a · [(1 + i)n − 1] / i
Annuity due: multiply the ordinary result by (1 + i)
Deferred d periods: V0 = (1 + i)−d · a · [1 − (1 + i)−n] / i
Perpetuity: V0 = a / i
Example. 1,000 euros at the end of each year for 5 years at 4%: present value 4,451.82 euros, future value 5,416.32 euros. As a perpetuity: 1,000 / 0.04 = 25,000.00 euros.
6. Growing annuities
Ordinary, n periods (q ≠ 1 + i): V0 = a · [1 − qn · (1 + i)−n] / (1 + i − q)
Growing perpetuity (q < 1 + i): V0 = a / (1 + i − q)
7. Loans
French method instalment: a = C0 · i / [1 − (1 + i)−n]
Interest in period t: It = Ct−1 · i Principal repaid: At = a − It At+1 = At · (1 + i)
Example. 10,000 euros at 5% over 4 annual instalments: a = 2,820.12 euros. Year one pays 500.00 euros of interest and repays 2,320.12 euros of principal.
Constant principal: A = C0 / n, with interest on the outstanding balance, so the total payment falls over time. American (bullet): interest C0 · i every year and the full principal at the end.
8. Investment appraisal: NPV and IRR
NPV = −A + Σ Qt / (1 + k)t
IRR: the rate r that makes NPV = 0
Example. An investment of 1,000 euros returning 400, 500 and 300 euros in years 1 to 3, with k = 8%: NPV = 37.19 euros. Being positive, the project earns more than the 8% required, so its IRR is above 8%.
Would you rather go through it with a tutor?
I give one-to-one online Business Administration lessons, at the hour you choose, in English or Spanish.
Common questions on this topic
What is the formula for a loan instalment under the French method?
a = C0 · i / [1 − (1 + i)^−n], where C0 is the amount borrowed, i the effective rate per period and n the number of instalments. 10,000 euros at 5% over 4 years gives an annual instalment of 2,820.12 euros.
How do I convert a nominal rate into an effective annual rate?
Without fees, effective annual rate = (1 + j/m)^m − 1, where j is the nominal rate and m the number of periods per year. 12% nominal compounded monthly is 12.68% effective.
Is financial mathematics in Spain different from UK or US courses?
The mathematics is the same, but Spanish universities use their own notation and emphasise topics such as commercial discount, the French loan method and APR (TAE). This sheet follows the Spanish approach in English.