Essential Mathematics of Money by Tim Hill
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Essential Mathematics of Money by Tim Hill

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Essential	Mathematics	of	Money:	A	Self-Teaching	Guide
by	Tim	Hill

Copyright	©	2018	by	Questing	Vole	Press.	All	rights	reserved.

Editor:	Kevin	Debenjak
Proofreader:	Diane	Yee
Compositor:	Kim	Frees
Cover:	Questing	Vole	Press

Self-Teaching	Guides	by	Tim	Hill
Essential	Trigonometry

Essential	Algebra

Essential	Geometry

Essential	Precalculus

Essential	Advanced	Precalculus

Essential	Permutations	&	Combinations

Essential	Mathematics	of	Money

Essential	Excel	Loan	&	Mortgage	Formulas

For	a	complete	list	of	titles,	go	to


1.	Interest

2.	Present	Value

3.	Annual	Percentage	Rate

4.	Mortgages

5.	Annuities

6.	Stocks

7.	Personal	Savings

8.	Student	Loans

9.	Kelly	Strategy

10.	Mathematics	Review

1.	Interest
Mathematical	prerequisites.	The	material	in	this	book	requires	a	command
of	 high-school	 mathematics,	 especially	 exponents,	 logarithms,	 geometric
series,	 sigma	 notation,	 and	 some	 statistics.	 In	 a	 few	 places,	 the	 tools	 of
elementary	 calculus—derivatives,	 Taylor	 series,	 L’Hôpital’s	 rule,	 and
iteration—are	 used	 to	 prove	 a	 result,	 solve	 an	 equation,	 or	 derive	 an
optimal	 value.	 If	 you	 like,	 you	 can	 skip	 or	 skim	 the	 calculus-based	 parts
without	 loss	 of	 continuity.	 Chapter	 10	 provides	 a	 mathematical	 refresher
and	a	short	list	of	key	mathematical	facts.

Keep	in	mind	that	mathematical	concepts	aren’t	confined	to	 the	context	 in
which	they’re	introduced.	You	can	apply	the	methods	in	this	book	to	a	wide
variety	 of	 problems.	 As	 illustrated	 by	 various	 examples,	 the	 same
calculations	 that	 determine	mortgage	 payments	 can	 also	 help	 you	 decide
how	to	pay	down	credit-card	debt,	and	the	optimal	strategy	for	investing	in
speculative	business	ventures	also	applies	to	casino	betting.

When	you	borrow	money	for	a	period	of	 time,	 interest	will	be	charged	by	 the
lender	(think	of	interest	as	the	cost	of	using	someone	else’s	money).	Conversely,
when	 you	 lend	 money	 you	 will	 receive	 interest.	 Interest	 is	 payment	 above
repayment	 of	 the	 amount	 borrowed,	 at	 a	 particular	 rate.	 This	 interest	 is	 either
simple	 or	 compound,	 and	 the	 dramatic	 difference	 between	 the	 two	 is	 best
illustrated	with	 an	 example.	Suppose	 that	 you	 invest	 $100	 for	 100	years	 at	 an
annual	interest	rate	of	5%.

With	simple	interest,	$5	is	added	each	year,	or	$5	×	100	=	$500	in	total,	so
the	end	total	is	$100	+	$500	=	$600.

With	compound	interest,	let	S	and	E	denote	the	amounts	at	the	start	and	end
of	 a	 given	 year,	 respectively.	During	 the	 year,	 5%	 of	S	 is	 added,	 so	E	 =
S(1	+	5/100).	This	addition	happens	every	year,	so	after	100	years	the	total
amount	that	you	have	is	$100(1	+	5/100)100	=	$13,150.13.

Note	that	interest	is	distinct	from	any	fees	that	a	lender	might	charge	a	borrower
in	 addition	 to	 interest.	 Unless	 otherwise	 specified,	 interest	 calculations	 in	 this

Use	compound	interest

Ignore	fees	and	other	transaction	costs

Use	natural	(base	e)	logarithms,	denoted	by	log(x),	loge(x),	or	ln(x)

Example	1.1		Suppose	that	you	borrow	an	amount	C	on	which	you’re	charged
an	annual	rate	of	interest	of	r%.	If	you	make	no	repayments,	then	how	much	will
you	owe	after	n	years?

Solution		After	n	years,	you’ll	owe

C(1	+	r/100)n

Example	1.2		If	the	annual	interest	rate	is	12%,	and	you	borrow	$1000	for	six
months,	then	how	much	should	you	repay?

Solution		 It’s	 tempting	 to	 think	 that	 by	 borrowing	 for	 half	 a	 year	 the	 interest
charge	is	6%	(half	of	12%),	so	the	repayment	should	be	$1060.	This	answer	is
incorrect,	however.	Let	x%	be	the	interest	rate	for	six	months.	After	six	months
you	owe	$1000(1	+	x/100),	 so	after	another	 six	months,	one	year	 in	 total,	you
would	owe	$1000(1	+	x/100)2.	Because	the	annual	rate	is	12%,	we	have

leading	to

The	interest	rate	for	six	months	is	actually	5.83%,	so	you	should	repay	$1058.30.

Example	1.3		If	the	annual	rate	is	12%,	and	you	borrow	$1000	for	one	month,
then	how	much	should	you	repay?

Solution		The	appropriate	interest	rate	is	that	value	x	such	that

giving	 1	 +	 x/100	 =	 1.121/12	 ≈	 1.0094888.	 The	 interest	 rate	 for	 one	 month	 is
0.94888%,	so	you	should	repay	$1009.49.

Example	 1.4	 	What	 is	 the	 annual	 interest	 rate	 if	 a	 bank	 advertises	 (a)	 a
quarterly	interest	rate	of	2%,	(b)	1%	per	month?

Solution		(a)	After	a	 full	year	of	 four	quarters,	with	2%	interest	charged	each
time,	a	loan	of	size	C	becomes	C(1.02)4	≈	1.0824C.	The	true	annual	interest	rate
is	8.24%,	not	the	8%	that	the	bank	suggests.

(b)	A	 rate	 of	 1%	per	month	means	 that	 the	 debt	 is	C(1.01)12	 ≈	 1.1268C	 at	 the
year’s	end—a	true	annual	rate	of	12.68%,	not	12%.

No	matter	how	an	interest	rate	is	presented	or	quoted,	you	can	use	a	simple	two-
step	process	to	calculate	how	much	is	owed	at	any	specified	future	time.

1.	 If	 necessary,	 convert	 the	 rate	 to	 the	 true	 annual	 rate	 by	using	 the	method
shown	in	the	preceding	example.

2.	 With	r%	as	the	true	annual	rate,	calculate	the	amount	owed	when	borrowing
C	for	t	years	(t	can	be	a	whole	number	or	a	fraction)	as

We	say	that	interest	is	compounded	continuously.

Example	1.5		If	the	true	annual	interest	rate	is	10%,	then	how	much	should	you
repay	if	you	borrow	$1000	for	(a)	six	months,	(b)	40	months?

Solution		 (a)	 Six	months	 is	 half	 a	 year,	 so	 you	 should	 repay	 $1000(1.10)1/2	 =


(b)	40	months	is	three	and	one-third	years,	so	you	should	repay	$1000(1.10)10/3	=

Example	1.6		A	 loan	 is	offered	at	15%	 interest	 compounded	quarterly.	Show
that	 the	 true	 annual	 rate	 is	 about	 15.865%.	 Find	 the	 true	 annual	 rate,	 to	 three
decimal	places,	if	the	nominal	rate	is	15%	compounded	(a)	monthly,	(b)	weekly,
(c)	daily,	(d)	hourly,	(e)	more	frequently	(continuously).

Solution		If	you	borrow	an	amount	C	on	these	terms,	then	after	one	year	of	four
quarters	 you’ll	 owe	C(1	+	0.15/4)4	 =	C	 ∙	 1.158650...,	 so	 the	 true	 rate	 is	 about

(a)	Monthly,	(1	+	0.15/12)12	=	1.160754...,	so	the	rate	is	16.075%.

(b)	Weekly,	(1	+	0.15/52)52	=	1.161583...,	so	the	rate	is	16.158%.

(c)	Daily,	(1	+	0.15/365)365	=	1.161798...,	so	the	rate	is	16.180%.

(d)	Hourly,	(1	+	0.15/8760)8760	=	1.161832...,	so	the	rate	is	16.183%.

(e)	 Continuously,	 exp(0.15)	 =	 e0.15	 =	 1.161834...;	 the	 limit	 is	 also	 a	 rate	 of

Example	1.7		You	invest	$10,000	in	property,	which	 increases	 in	value	at	 the
compound	rate	of	5%	annually,	but	you	are	charged	2%	of	the	total	value	at	the
end	of	each	year.	What	will	your	investment	be	worth	at	the	end	of	five	years?

Solution		If,	at	 the	beginning	of	any	year,	your	 investment	 is	worth	C,	 then	 it
increases	to	1.05C,	but	the	2%	charge	then	turns	this	amount	into	1.05C	×	0.98	=
1.029C,	so	at	the	end	of	5	years,	you’ll	have	1.0295C,	or	about	$11,536.57.	Note
that	a	5%	increase,	with	a	2%	decrease,	is	not	the	same	as	a	3%	increase.

Example	1.8		You	borrow	an	amount	C	at	r%	annual	interest.	If	you	make	no
repayments,	then	at	what	time	will	your	debt	double?

Solution		Your	debt	will	double	at	that	time	n	when

C(1	+	r/100)n	=	2C;

that	is,	when	n	log(1	+	r/100)	=	log(2)	=	0.69314718....	Thus

where	x	=	r/100.	When	|x|	<	1,	we	have	the	Taylor	series	log(1	+	x)	=	x	−	x2/2	+
x3/3	−	⋯,	so	when	x	 (>	0)	 is	small,	 the	right	side	of	 the	equation	above	 is	 just
under	x	and	consequently	n	is	just	over	0.693/x	=	69.3/r.

The	 preceding	 example	 justifies	 a	 handy	 method	 for	 estimating	 the	 doubling
time	of	a	debt.

The	 Rule	 of	 72	 	 At	 an	 annual	 interest	 rate	 of	 r%,	 it	 will	 take
approximately	72/r	years	for	a	debt	to	double,	if	no	repayments	are	made.

Naturally,	the	same	argument	shows	that	if	you	invest	a	sum	of	money	that	earns
annual	 interest	of	r%,	then	 it	will	 take