The Formula Guide

Every formula in words as well as symbols. Each worked example is generated by the same code as the calculator, so the numbers always agree.

Borrow

Chapters 3 and 11

Effective annual rate

EAR = (1 + i/m)^m − 1

Divide the quoted yearly rate by the number of compounding periods, add one, raise it to the number of periods, subtract one.

i: quoted (nominal) annual rate, as a decimal
m: times per year that interest is added
EAR: what a year of borrowing or saving really adds up to

Worked example. A card quoting 18% a year, charged monthly, is not 18%: (1 + 0.18/12)^12 − 1 = 19.56%. Over a year, $1,000 of debt grows to $1,196.

Use the Rate translator calculator →

Chapter 11 · The True Cost of Borrowing

Level payment (reducing balance)

Payment = P × r ÷ (1 − (1 + r)^−n)

Each month you owe interest on what is left, and the payment is set so the balance reaches exactly zero after n payments. A flat rate instead charges interest on the full amount for the whole term: Payment = (P + P × rate × years) ÷ n.

P: amount borrowed
r: monthly rate (annual rate ÷ 12)
n: number of monthly payments
Cost of credit: total repaid − amount borrowed + fees

Worked example. $10,000 at 6% flat over 24 months: payment = (10,000 + 10,000 × 6% × 2) ÷ 24 = $466.67. Interest is $1,200. But because you are repaying the balance down, this is equivalent to 11.1% on a reducing balance.

Use the True cost of a loan calculator →

Chapters 11 and 15

Compare on total cost of credit

Cost of credit = payment × n − P + fees

Calculate each offer’s total repayment and add its fees, then compare. The lowest payment is rarely the lowest cost.

P: amount borrowed (same for both)
n: months in the term
fees: every upfront charge

Worked example. Same 8% on $10,000: 24 months costs $854.55 in interest; 60 months costs $2,166. The longer loan has the lower payment ($202.76 vs $452.27) and the higher cost.

Use the Compare two loan offers calculator →

Chapter 12 · Credit Cards, Overdrafts and Revolving Debt

Month by month

New balance = balance × (1 + APR/12) − payment

Each month the balance gains a month of interest and falls by the payment. When the payment is a percentage of the balance, it shrinks as the balance shrinks, so the balance falls ever more slowly.

APR: yearly card rate
payment: minimum: a % of balance (with a floor), sometimes plus the month’s interest

Worked example. $3,000 at 20% with a 2% minimum (floor $25.00): first payment $60.00; clearing it takes 31 years and $10,210 in interest. Holding the first payment fixed clears it in 9 yr 1 mo for $3,504.

Use the The minimum-payment trap calculator →

Chapter 14 · Making Debt Decisions

Two orderings

Avalanche: highest rate first · Snowball: smallest balance first

Always pay every minimum, then send every spare pound, dollar or dirham to one debt. Avalanche picks the highest rate, which always minimises interest. Snowball picks the smallest balance, which clears accounts sooner and helps some people stay motivated.

Budget: total you can pay per month across all debts
Minimums: the smallest payment each lender requires

Worked example. With a card (24%), a car loan (7%) and a store card (29%) and $700.00 a month: avalanche clears everything in 1 yr 10 mo for $1,422 interest; minimums alone take 6 yr 10 mo and cost $5,566.

Use the Debt payoff planner calculator →

Chapters 10 and 14

Debt-to-income and utilisation

DTI = monthly debt payments ÷ gross monthly income

Lenders add up all your monthly debt payments and divide by income. Many keep this below 36–43%, but a lower figure is safer. Utilisation is balances ÷ credit limits and is often watched below 30%.

DTI: debt-to-income ratio
Utilisation: how much of your available card limit you use

Worked example. On $5,000 a month with $500.00 of debt payments, DTI is 10%. Adding a $400.00 loan payment takes it to 18%. A 36% limit leaves $1,300 of headroom: about $65,653 at 7% over five years. Card use: 30%.

Use the Borrowing capacity & debt-to-income calculator →

Chapter 13 · Mortgages and the Rent-or-Buy Question

Mortgage payment

Payment = P × r ÷ (1 − (1 + r)^−n)

The same level-payment formula as any amortising loan. In the early years most of each payment is interest; the balance falls slowly at first and faster later.

P: amount borrowed (price − deposit)
r: monthly rate (annual ÷ 12)
n: number of monthly payments (years × 12)

Worked example. $200,000 borrowed at 6% over 30 years: payment $1,199.10, total interest $231,676. Overpaying $200.00 a month ends the mortgage 9 years early and saves $79,801 in interest.

Use the Mortgage explorer calculator →

Chapter 13 · Mortgages and the Rent-or-Buy Question

Compare net wealth at the horizon

Buy: value × (1 − selling costs) − mortgage balance vs Rent: deposit and extra cash invested

If you buy, your wealth is the home’s value after selling costs, minus the mortgage still owed. If you rent, you invest the deposit and purchase costs, and each month you also invest whatever buying would have cost you above your rent (or draw it down if renting costs more).

Deposit + costs: cash a buyer ties up on day one
Running costs: tax, insurance, maintenance as % of value
Investment return: what the renter’s money earns

Worked example. At these assumptions, after 3 years the buyer is $13,401 behind; after 25 years $327,808 ahead. Time in the home changes the answer.

Use the Rent or buy? calculator →

Save & grow

Chapter 3 · The Price of Time

Future value

FV = P × (1 + r)^t + C × ((1 + i)^(12t) − 1) ÷ i

A lump sum grows by (1 + r) each year. Monthly contributions each grow for the time they remain invested; the second term adds them up. To see buying power, divide by (1 + inflation)^t.

P: starting amount
C: amount added each month
r: yearly return
i: equivalent monthly return
t: years

Worked example. $5,000 plus $300.00 a month for 25 years at 6% becomes $224,346. You paid in $95,000; growth supplied $129,346. After 2.5% inflation the buying power is $121,010.

Use the Compounding & time calculator →

Chapter 2 · The Quiet Tax

Purchasing power and real return

Buying power = M ÷ (1 + π)^t Real return = (1 + r) ÷ (1 + π) − 1

Prices rise by π each year, so money buys 1/(1 + π) as much a year later. The real return is not return minus inflation: it is the ratio of growth to price rises.

M: amount today
π: inflation per year
r: nominal return per year
t: years

Worked example. $10,000 left idle for 20 years at 3% inflation buys what $5,537 buys today: 45% of its buying power gone. At a 5% return, the real return is 1.94% and it ends at $14,691 in today’s money.

Use the The quiet tax calculator →

Chapter 7 · Resilience

Buffer target

Target = essential monthly costs × months of cover

Months of cover rises with income uncertainty and with the number of people who rely on you. The gap is the target minus accessible savings; dividing by what you can add each month gives the time to close it.

Essential costs: housing, food, utilities, transport, insurance, minimum debt payments
Months: 3 for steady pay, up to 6 or more for irregular income, plus 1 with dependants (a rule of thumb)

Worked example. Essentials $2,400 a month, mixed income, one dependant: target 5.5 months = $13,200. With $3,000 saved the gap is $10,200, closed in 3 yr 5 mo saving $250.00 a month.

Use the Buffer calculator calculator →

Chapters 8 and 25

Required monthly saving

C = (Target − P × (1 + i)^n) × i ÷ ((1 + i)^n − 1)

First grow what you already have. Whatever is still missing must come from monthly contributions, each growing for the time left. The formula solves for the contribution.

Target: amount you want
P: amount you already have
i: monthly return
n: months until the goal

Worked example. To reach $60,000 in 8 years from $5,000 at 4%: $472.16 a month. At 0% growth it would take $572.92; time and return carry the difference.

Use the Goal calculator calculator →

Chapter 7 · Resilience

Buffer as a shock absorber

Buffer(month) = buffer(month − 1) + income − steady salary

Every month, add what you earned and take out your steady salary. The buffer rises in good months and falls in lean ones. The largest steady salary that never takes the buffer below zero is the lowest running average of your income, including the starting buffer.

income: what actually arrives each month
steady salary: what you pay yourself every month
buffer: accessible cash that absorbs the difference

Worked example. A year of lumpy income averaging $2,958 a month. Paying yourself $2,500 steadily from a $3,000 buffer: the buffer never drops below $1,700 and ends at $8,500. Spending what arrives, 5 months would have fallen below $2,500 of essentials.

Use the Irregular income smoother calculator →

Protect

Chapter 17 · Insurance: What to Protect, What to Absorb

Expected loss and loading

Expected payout = probability × (loss − excess) Loading = premium ÷ expected payout − 1

Multiply the chance of a claim by what the policy would actually pay: that is its average value to you. Loading shows how much more you pay in premium than the policy returns on average, which is how insurers cover costs and profit. The ruin test asks: if this happened, could my savings absorb it?

probability: chance of the loss in a year
loss: size of the loss
excess: the part you pay on any claim (deductible)
premium: yearly price of the cover

Worked example. Phone cover: $300.00 loss, 10% chance, $60.00 premium: average payout $25.00, so you pay 140% more than it returns, and the loss is affordable: absorb. Liability or income cover on a $200,000 loss you cannot fund: the loading barely matters, protect.

Use the Insure or absorb? calculator →

Invest

Chapter 21 · The Cost of Investing

Net return and end value

Net return = gross return − annual cost FV = P × (1 + net)^t + monthly contributions grown at net

Charges reduce the return you keep. Because the lost return would itself have compounded, the gap between two products widens year by year.

gross: return before costs
cost: all yearly charges: fund, platform, advice, trading
t: years

Worked example. $10,000 plus $400.00 a month at a 7% gross return for 30 years: at 0.25% a year it ends at $517,738; at 1.75% it ends at $387,295. The 1.5-point difference costs $130,443, or 25% of the cheaper outcome.

Use the The cost of investing calculator →

Chapter 20 · Risk and Return in Practice

Gain needed to recover

Gain needed = 1 ÷ (1 − loss) − 1 Years to recover = ln(1 ÷ (1 − loss)) ÷ ln(1 + r)

If you lose a fraction of your money, what remains is (1 − loss). To get back to 1 it must be multiplied by 1 ÷ (1 − loss). At a steady return r, the time to recover is how long that multiplication takes.

loss: fall in value, as a decimal
r: annual return after the fall

Worked example. A 50% fall needs a 100% gain. A 20% fall needs 25%. Gaining 50% then losing 50% does not return you to start: 75% is left.

Use the Losses and recovery calculator →

Chapter 22 · Building and Keeping a Portfolio

Trade needed

Trade = target weight × (total + new money) − current value

Multiply the total portfolio (plus any new money) by each target weight to get the value that holding should have. The trade is the difference from what you hold now: positive means buy, negative means sell.

target weight: the share of the portfolio you decided on
total: current value of everything
new money: any amount you are adding

Worked example. A 60/40 plan that has drifted to 70/30: sell $10,000 of the first holding and buy $10,000 of the second. Adding $10,000 of new money instead needs only $14,000 into the second and nothing sold.

Use the Rebalancing calculator →

Plan ahead

Chapters 1 and 6

Margin and net worth

Margin = take-home − committed − everyday Net worth = assets − liabilities

Margin is what is left of a month after the commitments you cannot easily change and the spending you can. Net worth is a snapshot: everything you own at today’s value minus everything you owe.

Committed: rent or mortgage, loan payments, insurance, contracts
Everyday: food, transport, leisure, anything flexible
Assets: cash, investments, pensions, property, vehicles at realistic value

Worked example. Take-home $4,000, committed $2,300, everyday $1,300: margin $400.00 (10%). Assets $25,000 minus debts $14,500: net worth $10,500.

Use the Margin & net worth calculator →

Chapter 4 · Every Choice Has a Price

Cash cost and opportunity cost

Cash cost = price − resale + running costs Opportunity cost = what that money would have grown to − resale − cash cost

First add up the cash that actually leaves you. Then ask what the same money (the price now, and the running costs as they are spent) would have become if invested, compared with what you get back.

resale: what you can sell it for at the end
running costs: fuel, insurance, servicing, subscriptions, per month
opportunity return: return you assume on the alternative use of money

Worked example. A $20,000 car kept 5 years, worth 40% at the end, costing $300.00 a month to run: cash cost $30,000 ($6,000 a year). Adding the growth the money could have earned at 5% brings the true cost to about $7,574 a year.

Use the True cost of owning calculator →

Chapter 24 · Retirement Across Systems

First-year income and sustainability

Income = pot × withdrawal rate Next balance = (balance − withdrawal) × (1 + r)

The first withdrawal is a share of the pot. Each following year it rises with inflation, while the remaining balance earns the return. The pot lasts if it stays above zero for the whole horizon.

withdrawal rate: share of the starting pot taken in year one
r: return while drawing
inflation: yearly increase in each withdrawal

Worked example. A $500,000 pot at 4% gives $20,000 in year one and still holds $292,496 after 30 years. At 7% it gives $35,000 but runs out in year 18.

Use the Withdrawal rates calculator →

Chapter 24 · Retirement Across Systems

Pot needed and projected

Pot needed = (income wanted − other income) × 12 ÷ withdrawal rate

Subtract income that does not come from your pot (state pension, workplace pension, rent) from the income you want. What remains must come from the pot, which at a chosen withdrawal rate tells you the pot required. Project the pot you will actually have, in today’s money, to compare.

income wanted: monthly spending you aim for, in today’s money
other income: monthly pension or other income expected, today’s money
withdrawal rate: share of the pot taken yearly

Worked example. From 35, retiring at 65, with $20,000 saved and $400.00 a month at 5% (2% inflation): the pot reaches about $227,778 in today’s money. Wanting $3,000 a month with $1,000 from elsewhere needs $600,000; the monthly shortfall is $1,241, closed by about $826.89 more a month.

Use the Retirement readiness calculator →

Chapter 23 · Your Largest Asset: Earning Power

Net present value of the uplift

NPV = −outlay + Σ uplift ÷ (1 + d)^t

Add the money you spend and the income you give up now. Then add the extra income each year, shrunk by a discount rate because money later is worth less than money now. A positive NPV means the investment pays more than your chosen return.

outlay: fees plus income forgone while studying
uplift: extra income per year, after tax
d: your discount rate: what the money could earn elsewhere

Worked example. A $3,000 course plus $1,000 of income forgone, raising earnings by $2,000 a year for 10 years: pays back in 2 years; total gain $16,000; NPV at 5% $11,443.

Use the Return on skills calculator →

Chapter 25 · Wealth, Time and Your Financial Strategy

Target pot and time to reach it

Target = yearly spending ÷ withdrawal rate then grow the pot with monthly saving until it reaches the target

Spending sets the target (what the pot must pay), and saving feeds it. A higher savings rate raises one and lowers the other.

savings rate: share of take-home pay saved
withdrawal rate: share of the pot you would take each year
return: assumed yearly growth

Worked example. On $4,000 a month take-home, from zero, at 5%: saving 10% reaches the target in 51 years; saving 50% in 16. Same pay, same return.

Use the Savings rate & independence calculator →

Chapter 24 · Retirement Across Systems

Income from each source

Pot income = pot × withdrawal rate ÷ 12 Entitlement = yearly amount ÷ 12

A pot you own converts to income through a withdrawal rate. A guaranteed pension already states a yearly amount. Add up the sources that have started by your retirement age; the rest arrive later.

pot: value you could draw on
entitlement: yearly pension a scheme states it will pay
start age: the age that source can first be drawn

Worked example. A $240,000 pot (at 4%) gives $800.00 a month from age 60. A guaranteed $12,000 a year from 67 adds $1,000. Retiring at 65 you have $800.00 a month; at 67 it rises to $1,800.

Use the Pension sources across systems calculator →