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.
Rearranged, it gives what a future amount is worth today: its present value.
$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?
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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?
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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?
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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 later it comes, so find the present value of each one.
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 is saved at the end of each year for 4 years, at 10%. What has it grown to by the end?
An annuity pays $1000 a year for five years. Why is it worth less than $5000 today?
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