Question

Assume that Miss Bebe Nouveaune is born on July 1 of this year (July 1 is the first day of the KiwiSaver financial year). On this day her grandparents deposit $4,000 into a KiwiSaver account for her ($1,000 from each). So, that’s $4,000 deposited at t = 0. Assume that on every subsequent birthday up to and including her 18th birthday, Bebe’s parents deposit an additional $1,000 into her KiwiSaver account. That’s almost $20 a week. Assume that the deposit on her 18th birthday does not attract any slice of the member tax credit. So, that’s $1,000 deposited on each of t = 1 through t = 18. Once Bebe turns 18, she starts university and her parents never again contribute money to her KiwiSaver account. Bebe studies at university for three years during which she takes a “contributions holiday” and does not contribute money to her KiwiSaver account. (As an aside, if you are working, and you take a contributions holiday, then so too does your employer.) So, that’s $0 deposited from t = 19 through t = 21. Bebe graduates from university on her 21st birthday with a BCom in Finance. Hooray! Unfortunately, Bebe is unemployed for one year after graduation (because two recessions from now she has trouble getting a job). So, she does not start working until t = 22. Bebe’s banking salary is $100,000 in her first year of work, paid annually in arrears, but grows by a 4% combined COLA and promotion adjustment per year until she is 65. So, to be clear, she gets a lump sum gross salary of $100,000 at t = 23, and then $104,000 at t = 24, etc. Note that these cash flows can go on the timeline, but they are not being discounted, because we are modelling only the KiwiSaver account itself. I figure $100,000 at t = 23 is comparable to a good banking salary for a new graduate in today’s dollars. On Bebe’s 23rd birthday, and every birthday up to an including her 65th, Bebe contributes 8% of her gross annual salary to her KiwiSaver account as a lump sum; Bebe’s employer also contributes 3% (the minimum compulsory employer contribution) of Bebe’s gross annual salary to her KiwiSaver account at the same time, but only 67% of this, that is 2.01%, actually arrives in the KiwiSaver account.12 So, that’s 10.01% of her growing salary deposited on each of t = 23 through t = 65 (i.e., 43 deposits). Assume that the Government pays into Bebe’s KiwiSaver account a “member tax credit” (it’s actually just cash) of $521.43 on each of Bebe’s birthdays from 23 through 65 (anyone 18 or over gets this $521.43 if they contribute at least $1,042.86 each year; employer contributions and government contributions do not count towards this).13 That’s an extra $521.43 deposited on each of t = 23 through t = 65 (i.e., 43 more deposits). Assume that Bebe’s KiwiSaver account earns a constant 7.5% per annum after taxes and fees over her entire lifetime.

?a?Assume that at age 65 Bebe annuitizes her KiwiSaver account balance.14 She withdraws her money as a growing annuity due, growing at a COLA adjustment of g = 0.05 per annum. That is, I want you to assume she will withdraw a cash flow C on her 65th birthday, C×(1+g) on her 66th birthday, ..., up to C×(1+g)29 on her 94th birthday (a growing annuity due of 30 payments) and show me how you solved algebraically for C. Give me a ballpark figure for what this C number represents in today’s dollars?

Answer 1

Time periods (age)	0	1-18	19-22	23-65 (43 years)
Dollar contributions to KS Account	$4000	$1000	0	10.1%*100000+521.43
growth rate for salary	4%
Interest earned on KS account (discount rate)	7.50%
growth rate for annuity	5%
Annual annuity payments	30

Calculating forward value of KS account at the time of retirement		=FV(7.5%,18,-1000,-4000,1)
Annual contributions till age 18	$       53,056	=53056.41*(1+7.5%)^4
FV of accumulated contributions at age 23	$       70,855	=70,855.2*(1+7.5%)^(65-23)
FV of accumulated contributions at age 65	$ 1,477,499	First year contribution= 10.1%*100000+521.43--> growing at 4% salary growth rate for 43 years
FV Schedule of contributions in the emplyed life	$ 5,059,170	=FV of accumulated contributions at age 65+ FV Schedule of contributions in the emplyed life
Total funds accumulated in KS account at retirement	$ 6,536,669
Annuity value at age 65	$ 300,222.33
Annuity equivalent value at age 23	$ 14397.51	=300222.33/(1+7.5%)^(65-23)

Schedule of contribution during working life-

		Age->	23	24	25	26	27	28	29	30	31	32	33	34	35	36	37	38	39	40	41	42	43	44	45	46	47	48	49	50	51	52	53	54	55	56	57	58	59	60	61	62	63	64	65
Gov. share	521.43	Salary->	100000	104000	108160	112486.4	116985.9	121665.3	126531.9	131593.2	136856.9	142331.2	148024.4	153945.4	160103.2	166507.4	173167.6	180094.4	187298.1	194790	202581.7	210684.9	219112.3	227876.8	236991.9	246471.6	256330.4	266583.6	277247	288336.9	299870.3	311865.1	324339.8	337313.3	350805.9	364838.1	379431.6	394608.9	410393.3	426809	443881.3	461636.6	480102.1	499306.1	519278.4
		10.1% contribution to KS acc	10100	10504	10924.16	11361.13	11815.57	12288.19	12779.72	13290.91	13822.55	14375.45	14950.47	15548.49	16170.43	16817.24	17489.93	18189.53	18917.11	19673.8	20460.75	21279.18	22130.34	23015.56	23936.18	24893.63	25889.37	26924.95	28001.94	29122.02	30286.9	31498.38	32758.31	34068.65	35431.39	36848.65	38322.6	39855.5	41449.72	43107.71	44832.02	46625.3	48490.31	50429.92	52447.12
		+GOC share	10621.43	11025.43	11445.59	11882.56	12337	12809.62	13301.15	13812.34	14343.98	14896.88	15471.9	16069.92	16691.86	17338.67	18011.36	18710.96	19438.54	20195.23	20982.18	21800.61	22651.77	23536.99	24457.61	25415.06	26410.8	27446.38	28523.37	29643.45	30808.33	32019.81	33279.74	34590.08	35952.82	37370.08	38844.03	40376.93	41971.15	43629.14	45353.45	47146.73	49011.74	50951.35	52968.55
	FV=	(Annual compounding after interest received)	221482.027	213866.4	206527	199452.7	192633.3	186058.5	179719	173605.5	167709.4	162022.2	156536.1	151243.2	146136.4	141208.7	136453.1	131863.5	127433.5	123157.3	119029.2	115043.8	111195.8	107480.2	103892.3	100427.3	97080.93	93848.84	90726.96	87711.34	84798.21	81983.94	79265.02	76638.08	74099.9	71647.36	69277.46	66987.31	64774.14	62635.27	60568.13	58570.23	56639.19	54772.7	52968.55
	sum of FV=		5059170.26

Solve for C by solver or by manually adjusting C in the below schedule-

Age->	65	66	67	68	69	70	71	72	73	74	75	76	77	78	79	80	81	82	83	84	85	86	87	88	89	90	91	92	93	94
Annuity received	300222.33	315233.4	330995.1	347544.9	364922.1	383168.2	402326.6	422443	443565.1	465743.4	489030.5	513482.1	539156.2	566114	594419.7	624140.7	655347.7	688115.1	722520.8	758646.9	796579.2	836408.2	878228.6	922140	968247	1016659	1067492	1120867	1176910	1235756
Funds remaining after annuity	$ 6,236,447	6388947	6537123	6679862	6815930	6943956	7062426	7169665	7263825	7342869	7404553	7446413	7465738	7459554	7424601	7357305	7253755	7109672	6920377	6680758	6385236	6027720	5601570	5099548	4513767	3835641	3055821	2164141	1149541	0.860935
Annual compounding after interest received	6704180.48	6868118	7027407	7180852	7327125	7464753	7592108	7707390	7808612	7893584	7959895	8004894	8025668	8019021	7981446	7909103	7797787	7642897	7439405	7181815	6864128	6479799	6021688	5482014	4852300	4123314	3285008	2326451	1235757	0.925505
Remaining value of funds at age 94	0.92550543