Financial Mathematics

Stage 10 of 23 Strand 8 of 9 4 lessons

4 illustrated lessons, each teaching the why before the how.

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The Time Value of Money

Any four of the five quantities fix the fifth.

Value now and value later differ by one growth factor, so either can be found from the other

A finance calculator holds five quantities, and any four of them fix the fifth.

With no payments the whole relationship is one growth factor, applied N times.

Turn it around and a future amount is dragged back to what it is worth today.

$1000 due in five years is worth $747.26 today at 6% — that is the time value.

Discounting and compounding are one rule read in two directions, never two rules.

Now you

PV = $2000, I = 4%, N = 2, PMT = 0. What is FV?

$2000 is due in 3 years. What is it worth today at 6% a year?

Real Value After Inflation

Growth divided by prices, never growth minus prices.

Real value divides the grown balance by the price rise, so only the excess growth counts

An account grows 6% a year while the prices it will be spent on rise 4% a year.

After a year the balance is $1060 and the basket that cost $1000 now costs $1040.

Divide by the price rise: the $1060 buys what $1019.23 would have bought before.

The real growth is 1.92% a year, close to the 2% a subtraction gives but not equal.

Over ten years $1000 becomes $1791, but only $1210 of today’s buying power.

Now you

$2000 grows at 8% a year for 4 years while prices rise 4% a year. What is it worth in today’s money?

$5000 grows at 10% a year for 4 years while prices rise 2% a year. What is it worth in today’s money?

Amortizing a Loan

Interest on what is left, and the split moves.

Each equal loan payment pays interest on the outstanding balance first, and the rest reduces it

Borrow $200,000 at 0.5% a month and repay $1200 at the end of every month.

Month one: 0.5% of $200,000 is $1000, so only $200 of the $1200 pays off the loan.

Month two charges 0.5% of $199,800, not of the original loan — $999, a dollar less.

The payment never changes, but the gold interest slice shrinks as the debt falls.

Three steps, repeated: interest on the balance, the rest off the debt, then start again.

Now you

The payments on a loan are all equal. What happens to the split inside them?

A $150000 loan at 0.5% a month is down to $70000. How much of the next $1200 payment is interest?

The Value of an Annuity

Equal payments, summed as a geometric series.

A stream of equal payments is a geometric series, and one formula sums it

$1000 paid at the end of each year for five years — that stream is an annuity.

Each payment is worth less the further off it is, so drag every one back to today.

The five discounted payments form a geometric series with ratio 1 ÷ 1.05.

Summing that series gives one formula: the five payments are worth $4329.48 now.

Left to grow instead, the same five payments are worth $5525.63 at the end.

Now you

$2000 saved at the end of each year for 4 years, at 10%. What has it grown to by the end?

Five yearly payments of $1000. Why is the stream worth less than $5000 today?

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