Shares Investment Analysis

8/18/2019 Shares Investment Analysis

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House price 300000 $

Loan term 30 years

House rent 280 $/week 1213.333333333

House maintainance 2500 $/year

Loan interest rate 1.06 annual

Deposit interest rate 1.06 annualHouse appreciation 1.015 annual

Ta 0.3 annual

!"# 1.03 annual

ent mana%ement &ee 0.0' ()rent

Loan min repayment 21'*5 $/year 1816

+tamp ,uty 11330

-ear House inest

sset Liaility et

0 $ 300000 $ 311330 $ 11330

1 $ 304500 $ 308215 $ 3'15

2 $ 30*068 $ 304*13 $ 4154

3 $ 313'04 $ 301413 $ 122*0

4 $ 31840* $ 2*''04 $ 20'05

5 $ 323185 $ 2*3''1 $ 2*414

6 $ 328033 $ 28*603 $ 38430

' $ 332*53 $ 285184 $ 4''6*

8 $ 33'*48 $ 280501 $ 5'44'

* $ 34301' $ 2'5536 $ 6'481

10 $ 348162 $ 2'02'3 $ ''88*

11 $ 353385 $ 2646*5 $ 8868*

12 $ 358685 $ 258'82 $ ***03

13 $ 364066 $ 252515 $ 111551

14 $ 36*52' $ 2458'1 $ 123656

15 $ 3'50'0 $ 238828 $ 136241

16 $ 3806*6 $ 231363 $ 14*332

1' $ 386406 $ 222864 $ 163542

18 $ 3*2202 $ 213184 $ 1'*018

1* $ 3*8085 $ 202231 $ 1*5854

20 $ 40405' $ 18**0* $ 214148

21 $ 41011' $ 1'6113 $ 234004

22 $ 41626* $ 160'34 $ 255535

23 $ 422513 $ 143655 $ 2'885*

24 $ 428851 $ 124'48 $ 304103

25 $ 435284 $ 103882 $ 331402

26 $ 441813 $ 80*13 $ 360*00


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2' $ 448440 $ 5568* $ 3*2'51

28 $ 45516' $ 28050 $ 42'116

2* $ 461**4 $ 21'6 $ 4641'1

30 $ 468*24 $ 351'4 $ 5040*8


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$/monts

$/mont

!as inest ent

sset Liaility et

$ $ $

$ 10423 $ $ 10423 $ 13*4'

$ 2112* $ $ 2112* $ 14365

$ 3212' $ $ 3212' $ 14'*6

$ 43423 $ $ 43423 $ 15240

$ 55024 $ $ 55024 $ 156*'

$ 66*3' $ $ 66*3' $ 16168

$ '*16* $ $ '*16* $ 16653

$ *1'2' $ $ *1'2' $ 1'153

$ 104620 $ $ 104620 $ 1'668

$ 11'854 $ $ 11'854 $ 181*8

$ 13143' $ $ 13143' $ 18'44

$ 1453'6 $ $ 1453'6 $ 1*306

$ 15*680 $ $ 15*680 $ 1*885

$ 1'4355 $ $ 1'4355 $ 20482

$ 18*410 $ $ 18*410 $ 210*6

$ 204852 $ $ 204852 $ 21'2*

$ 2212'5 $ $ 2212'5 $ 22381

$ 23880' $ $ 23880' $ 23052

$ 25'520 $ $ 25'520 $ 23'44

$ 2''486 $ $ 2''486 $ 24456

$ 2*8'86 $ $ 2*8'86 $ 251*0

$ 321503 $ $ 321503 $ 25*46

$ 345'28 $ $ 345'28 $ 26'24

$ 3'1553 $ $ 3'1553 $ 2'526

$ 3**081 $ $ 3**081 $ 28351

$ 42841' $ $ 42841' $ 2*202


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$ 45*6'6 $ $ 45*6'6 $ 300'8

$ 4*2*'6 $ $ 4*2*'6 $ 30*80

$ 528446 $ $ 528446 $ 31*10

$ 566221 $ $ 566221 $ 3286'

$ 663535


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Loan interest entrepayments Loan coer cost 7aintainance cost

$ $ $ $

$ 18680 $ '848 $ '84'.65 $ 25'5

$ 184*3 $ '42* $ '42*.24 $ 2652

$ 182*5 $ 6**8 $ 6**8.28 $ 2'32

$ 18085 $ 6554 $ 6554.38 $ 2814

$ 1'862 $ 60*' $ 60*'.18 $ 28*8

$ 1'626 $ 5626 $ 5626.25 $ 2*85

$ 1'3'6 $ 5141 $ 5141.20 $ 30'5

$ 1'111 $ 4642 $ 4641.5* $ 316'

$ 16830 $ 412' $ 412'.00 $ 3262

$ 16532 $ 35*' $ 35*6.*' $ 3360

$ 16216 $ 3051 $ 3051.04 $ 3461

$ 15882 $ 248* $ 2488.'3 $ 3564

$ 1552' $ 1*10 $ 1*0*.55 $ 36'1

$ 15151 $ 1313 $ 1313.00 $ 3'81

$ 14'52 $ 6** $ 6*8.55 $ 38*5

$ 14330 $ 66 $ 65.66 $ 4012

$ 13882 $ 586 $ $ 4132

$ 133'2 $ 1258 $ $ 4256

$ 12'*1 $ 1*4* $ $ 4384

$ 12134 $ 2662 $ $ 4515

$ 113*5 $ 33*5 $ $ 4651

$ 1056' $ 4151 $ $ 4'*0

$ *644 $ 4*2* $ $ 4*34

$ 861* $ 5'31 $ $ 5082

$ '485 $ 655' $ $ 5234

$ 6233 $ '40' $ $ 53*1


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$ 4855 $ 8283 $ $ 5553

$ 3341 $ *186 $ $ 5'20

$ 1683 $ 10115 $ $ 58*1

$ 131 $ 110'2 $ $ 6068

$ 384618 $ *6*5


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Total cas input

$

$ 10423

$ 10081

$ *'30

$ *368

$ 8**5

$ 8611

$ 8216

$ '80*

$ '38*

$ 6*5'

$ 6512

$ 6053

$ 5581

$ 50*4

$ 45*3

$ 40''

$ 4132

$ 4256

$ 4384

$ 4515

$ 4651

$ 4'*0

$ 4*34

$ 5082

$ 5234

$ 53*1


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$ 5553

$ 5'20

$ 58*1

$ 6068


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House price 330000 $

Loan amount 1*0000 $

Loan term 30 years


Deposit interest rate 1.035 annual

House appreciation 1.045 annual

Loan actual repayment 38400 $/year 3200

-ear House inest

House price 7oney owe, et

0 $ 330000 $ 1*0000 $ 140000

1 $ 344850 $ 160150 $ 184'00

2 $ 360368 $ 128*5' $ 231412

3 $ 3'6585 $ *6360 $ 2802254 $ 3*3531 $ 622*6 $ 331235

5 $ 411240 $ 266** $ 384541

6 $ 42*'46 $ 104** $ 440245

' $ 44*084 $ 4*3'2 $ 4*8456

8 $ 46*2*3 $ 8***3 $ 55*28'

* $ 4*0411 $ 132443 $ 622855

10 $ 512480 $ 1'6803 $ 68*283

11 $ 535542 $ 22315* $ '58'01

12 $ 55*641 $ 2'1601 $ 831242

13 $ 584825 $ 322223 $ *0'048

14 $ 611142 $ 3'5123 $ *86265

15 $ 638643 $ 430404 $ 106*04'

16 $ 66'382 $ 4881'2 $ 1155554

1' $ 6*'414 $ 548540 $ 1245*54

18 $ '28'*8 $ 611624 $ 1340422

1* $ '615*4 $ 6''54' $ 143*141

20 $ '*5866 $ '4643' $ 1542303

21 $ 831680 $ 81842' $ 1650106

22 $ 86*105 $ 8*3656 $ 1'62'61

23 $ *08215 $ *'22'0 $ 1880485

24 $ *4*085 $ 1054423 $ 200350'

25 $ **1'*3 $ 11402'2 $ 2132065

26 $ 1036424 $ 122**84 $ 2266408

2' $ 1083063 $ 1323'33 $ 2406'*6

28 $ 1131801 $ 1421'01 $ 2553502

2* $ 1182'32 $ 15240'8 $ 2'06810

30 $ 1235*55 $ 1631061 $ 286'016


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$/mont

0 2 4 6

$600000

$400000

$200000

$

$200000

$400000

$600000

$800000

$1000000

$1200000


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8 10 12 14 16

!olumn

!olumn !

!olumn D


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#nest amount 10000 $


+ares %rowt rate 1.05 annual

+ares ,ii,en, 1.05 annual

Ta 0.3 annual

!"# 1.03 annual

-ear House pay o9 +are

Total sain%s Dii,en,s :#T Dii,en, Ta

0 $ 10000 $ $

1 $ 10600 $ 500 $ 150

2 $ 11236 $ 543 $ 163

3 $ 11*10 $ 58* $ 1''

4 $ 12625 $ 63* $ 1*2

5 $ 13382 $ 6*3 $ 2086 $ 14185 $ '52 $ 226

' $ 15036 $ 816 $ 245

8 $ 15*38 $ 885 $ 266

* $ 168*5 $ *60 $ 288

10 $ 1'*08 $ 1042 $ 313

11 $ 18*83 $ 1130 $ 33*

12 $ 20122 $ 122' $ 368

13 $ 2132* $ 1331 $ 3**

14 $ 2260* $ 1444 $ 433

15 $ 23*66 $ 156' $ 4'0

16 $ 25404 $ 1'00 $ 510

1' $ 26*28 $ 1844 $ 553

18 $ 28543 $ 2001 $ 600

1* $ 30256 $ 21'1 $ 651

20 $ 320'1 $ 2356 $ '0'

21 $ 33**6 $ 2556 $ '6'

22 $ 36035 $ 2''3 $ 832

23 $ 381*' $ 300* $ *03

24 $ 4048* $ 3265 $ *'*

25 $ 42*1* $ 3542 $ 1063

26 $ 454*4 $ 3843 $ 1153

2' $ 48223 $ 41'0 $ 1251

28 $ 5111' $ 4525 $ 135'

2* $ 54184 $ 4*0* $ 14'3

30 $ 5'435 $ 5326 $ 15*8


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inest #nest montly

+ares price et

$ 10000 $ 10000

$ 10500 $ 10850

$ 113*3 $ 11''2

$ 12361 $ 12''3

$ 13412 $ 1385*

$ 14552 $ 1503'$ 15'88 $ 16315

$ 1'130 $ 1''01

$ 18586 $ 1*206

$ 20166 $ 2083*

$ 21880 $ 22610

$ 23'40 $ 24532

$ 25'58 $ 2661'

$ 2'*48 $ 288'*

$ 30323 $ 31334

$ 32*01 $ 33**'

$ 356*' $ 3688'

$ 38'32 $ 40023

$ 42024 $ 43425

$ 455*6 $ 4'116

$ 4*4'1 $ 51120

$ 536'6 $ 55466

$ 5823* $ 60180

$ 6318* $ 652*6

$ 68560 $ '0846

$ '4388 $ '6868

$ 80'11 $ 83401

$ 8'5'1 $ *04*0

$ *5015 $ *8182

$ 1030*1 $ 106528

$ 111854 $ 115583


14/69

#nest amount 10000 $

einest 100 $/mont


+ares %rowt rate 1.05 annual

+ares ,ii,en, 1.05 annual

Ta 0.3 annual!"# 1.03 annual

-ear House pay o9 +are

Total sain%s Dii,en,s :#T Dii,en, Ta

0 $ 10000 $ $

1 $ 10600 $ 500 $ 150

2 $ 11236 $ 543 $ 163

3 $ 11*10 $ 58* $ 1''

4 $ 12625 $ 63* $ 1*25 $ 13382 $ 6*3 $ 208

6 $ 14185 $ '52 $ 226

' $ 15036 $ 816 $ 245

8 $ 15*38 $ 885 $ 266

* $ 168*5 $ *60 $ 288

10 $ 1'*08 $ 1042 $ 313

11 $ 18*83 $ 1130 $ 33*

12 $ 20122 $ 122' $ 368

13 $ 2132* $ 1331 $ 3**

14 $ 2260* $ 1444 $ 433

15 $ 23*66 $ 156' $ 4'0

16 $ 25404 $ 1'00 $ 510

1' $ 26*28 $ 1844 $ 553

18 $ 28543 $ 2001 $ 600

1* $ 30256 $ 21'1 $ 651

20 $ 320'1 $ 2356 $ '0'

21 $ 33**6 $ 2556 $ '6'

22 $ 36035 $ 2''3 $ 832

23 $ 381*' $ 300* $ *03

24 $ 4048* $ 3265 $ *'*

25 $ 42*1* $ 3542 $ 1063

26 $ 454*4 $ 3843 $ 1153

2' $ 48223 $ 41'0 $ 1251

28 $ 5111' $ 4525 $ 135'

2* $ 54184 $ 4*0* $ 14'3

30 $ 5'435 $ 5326 $ 15*8


15/69

inest #nest montly

+ares price et

$ 10000 $ 10000

$ 10500 $ 10850

$ 113*3 $ 11''2

$ 12361 $ 12''3

$ 13412 $ 1385*$ 14552 $ 1503'

$ 15'88 $ 16315

$ 1'130 $ 1''01

$ 18586 $ 1*206

$ 20166 $ 2083*

$ 21880 $ 22610

$ 23'40 $ 24532

$ 25'58 $ 2661'

$ 2'*48 $ 288'*

$ 30323 $ 31334

$ 32*01 $ 33**'

$ 356*' $ 3688'

$ 38'32 $ 40023

$ 42024 $ 43425

$ 455*6 $ 4'116

$ 4*4'1 $ 51120

$ 536'6 $ 55466

$ 5823* $ 60180

$ 6318* $ 652*6

$ 68560 $ '0846

$ '4388 $ '6868

$ 80'11 $ 83401

$ 8'5'1 $ *04*0

$ *5015 $ *8182

$ 1030*1 $ 106528

$ 111854 $ 115583


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s 250 7oney to inest per p

, 0.01*230'6*2 "erio,s per year

r 0.05 nnual interest on ca

% 0.0* nnual interest on s 14.*5 rokera%e &ee

. .

1.000*600013

; 1.00165863'4

alpa 0.41433*6324

eta 0.1'44306464

%amma 0.23*885200*

. .

c0 1.386146580:00'

c1 2.6110152*8:00'

c2 2.36628'802:005

. .

n ?


17/69

Maximising the future value:

To simplify the equation, we can define some extra constants:

Solutions:From here there are two options:

Approximating the solution with a Taylor series:

Truncating the series to the first three terms, we get a quadratic polynomial (with constant coefficients

So if money is ready to invest everyn+1 periods, there will bet /(n+1) separate investments, and the fu

This formula is exact in the case of integert andn, and a good approximation whent andn are not in

Derivative of the future value with respect ton:Fortunately, only a couple of terms inFV depend onn, so we can find the derivative after some effort:

Equating the derivative to zero, we can remove the denominator, and assumingt is greater than zero,

Then, we can define a functionf(n) and write the equation as:

Note thatα, β,γ,G, andR are all constant.

1. Cse ewtons meto, or anoter numerical meto, &or En,in% te positie root o& &


18/69

Closed-form approximate solution:

Then,

Example:Using the example from the question:

For example, I might put aside $100 every week to invest into a stock with an expected growth of 9%

Spreadsheet:Here's one way to implement the above solution with a spreadsheet. StackExchange doesn't allow ta

Formulae:

Copy and paste column A:

s

,

r

%

.

UsingR,G,α, β andγ defined above, letc0,c1 andc2 be the coefficients of the truncated Taylor seri

n should be rounded to the nearest whole number. To be certain, check the values above and below

Using Newton's method to find roots off(n) above, we getn = 14.004.

Using the closed-form approximate solution, we getn = 14.106.

Checking this against theFV witht = 1680 (evenly divisible by eachn + 1 tested):

Whenn = 13,FV = $903,861.85

Whenn = 14,FV = $903,891.13

Whenn = 15,FV = $903,865.89

Therefore, you should wait forn = 14 periods, keeping that money in the bank, investing it together wi


19/69

;

alpa

eta

%amma

.

c0

c1c2

.

n


20/69


21/69

erio,

res

a new variable:

returnsan,minimise&ees/30''430''4

ng the truncated Taylor series.See below for a spreadsheet you can use.

interest forn periods. That is,

http://money.stackexchange.com/questions/30734/frequency-of-investments-to-maximise-returns-and-minimise-fees/30774#30774http://money.stackexchange.com/questions/30734/frequency-of-investments-to-maximise-returns-and-minimise-fees/30774#30774


22/69

):

ture value of the investments will be:

egers. SubstitutingS, we get the version of the formula which explicitly depends onn:

we can divide by the constant ( 1-G t ):

. Tis can e ,one in a numer o& so&tware packa%es like 7TL Fctae etc or y u

ere.

http://en.wikipedia.org/wiki/Newton%27s_methodhttp://en.wikipedia.org/wiki/Taylor_serieshttp://en.wikipedia.org/wiki/Taylor_serieshttp://en.wikipedia.org/wiki/Newton%27s_method


23/69

.a., but brokerage fees are $10/trade. For how many weeks should I accumulate the $100 before in

les in their syntax at this time, so I'll show a screenshot of the formulae and columns you can copy

es forf(n):

using the formula forFV.

h the money in the next period (so you will make an investment every 14 + 1 = 15 weeks.)


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in% a %rapics calculator.

http://en.wikipedia.org/wiki/Newton%27s_methodhttp://en.wikipedia.org/wiki/Newton%27s_method


28/69

vesting, if I can put it in my high-interest bank account at 4% p.a. until then?

and paste:


29/69


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step 0.64 18'*.28

era%e annual can%e 4.'8( 18'*.*2

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36/69

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37/69

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39/69

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40/69

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41/69

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42/69

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43/69

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44/69

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45/69

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46/69

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47/69

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48/69

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49/69

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50/69

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51/69

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52/69

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53/69


54/69


55/69

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56/69

1860 1880 1*00 1*20 1*40 1*60 1*80 2000 2020

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57/69


58/69


59/69


60/69


61/69

!olumn >

:ponential =

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62/69

!olumn >

:ponential =

!olumn L


63/69


64/69


65/69


66/69


67/69

7in H:L" repay 4000

H:L" interest 5.00(

H:L" early repay onus 5.00(

+tocks interest 4.50(

H:L" D:T


68/69

s+tock cost cumulatie et H:L" ,et alance

2*165.00 30'00.00 0.00

2*165.00 30*23.*1 0.00

2*165.00 30*65.63 0.00

2*165.00 30816.35 0.00

2*165.00 30466.83 0.00

2*165.00 2**0'.38 0.00

2*165.00 2*12'.85 0.00

2*165.00 2811'.58 0.00

2*165.00 26865.41 0.002*165.00 2535*.63 0.00

2*165.00 2358'.*' 0.00

2*165.00 2153'.56 0.00

2*165.00 1*1*4.8* 0.00

2*165.00 16545.82 0.00

2*165.00 135'5.50 0.00

2*165.00 10268.3* 0.00

2*165.00 6608.1' 0.00

2*165.00 25''.'3 0.00

0 2 4 6 8 10 12 14 1


69/69

+cenario 2A >ull payo9 H:L" cost cumulatie +tocks amount +tock cost cumulatie et

2*165.00 0.00 0.00 2*165.00

2*165.00 4000.00 4000.00 2*165.00

2*165.00 81'*.82 8000.00 28*85.18

2*165.00 1254'.53 12000.00 2861'.4'

2*165.00 1'111.60 16000.00 28053.40

2*165.00 21880.84 20000.00 2'284.16

2*165.00 26864.4' 24000.00 26300.53

2*165.00 320'2.15 28000.00 250*2.85

2*165.00 3'513.*3 32000.00 23651.0'2*165.00 43200.34 36000.00 21*64.66

2*165.00 4*142.3' 40000.00 20022.63

2*165.00 55351.53 44000.00 1'813.4'

2*165.00 6183*.82 48000.00 15325.18

2*165.00 6861*.'8 52000.00 12545.22

2*165.00 '5'04.53 56000.00 *460.4'

2*165.00 8310'.'' 60000.00 605'.23

2*165.00 *0843.81 64000.00 2321.1*

2*165.00 *8*2'.63 68000.00 1'62.63

6 18

+cenario 1A 7in repayments

+cenario 2A >ull payo9

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