Question

Professor,

In trying to apply my knowledge in the real world, I am trying to create a realistic retirement schedule. However, I am running into difficulties using both a financial calculator as well as our equations from class in doing this.

I am trying to do the following: plan a retirement schedule between the ages of 22 and 68, in which I would deposit 25% of my income each year. The income starts at 80,000 with an annual growth rate of 5% and, to be realistic, assuming an interest rate of 2.5%. I will assume for simplicity that I receive my first salary ($80,000) when I turn 22, and my last salary when I turn 68. As soon as I receive a salary, I will save 25% of it.

However, this raises issues, as if I try to use the equation for the present value of a growing annuity with a 5% growth rate and 2.5% discount rate, r-g will yield a negative number. Also, I could not find online how to do this on my HP 10bII+ financial calculator and I don't want to manually enter 47 payments.

Do you know how I could overcome this obstacle?

1. Briefly comment on Jimmy’s concern that “if I try to use the equation for the present value of a growing annuity with a 5% growth rate and 2.5% discount rate, r-g will yield a negative number”. (3 sentences at most)
2. How much will Jimmy have in his retirement account when he turns 68, immediately after the last deposit?
3. What single deposit made on Jimmy’s 22^nd birthday would give the same account balance when he turns 68?

Answer 1

	a
		We need to calculate Future Value, not the present Value
		We can calculated Future Value even if g is greater than r
		Because (((1+r)^n)-((1+g)^n)) will also be negative
	b	Total number of deposits=68-21=47
		(First salary saved at age 22 and last salary at age 68)
		First year saving =80000*25%	$20,000
		g=Growth rate of savings=5%	0.05
		r=Discount Rate=2.5%=	0.025
		It can also be calculated by using the formula
		Future Value =(P/(r-g))*(((1+r)^n)-((1+g)^n))
		P=First Payment=$20000, r=discount rate=2.5%, g=growth rate=5%
		r-g=0.025-0.05=-0.025
		(((1+r)^n)-((1+g)^n))=(((1+0.025)^47)-((1+0.05)^47))	-6.71427396
		Future Value =20000*(-6.71427396/-0.025)=	$5,371,419.17
		Amount available at retirement	$5,371,419.17
	c	Assume , amount of single deposit =X
		Future Value of single deposit =X*(1+r)^n
		r=discount rate =2.5%=0.025
		n=68-22=46
		X*(1.025^46)=	$5,371,419.17
		X=5371419.17/(1.025^46)=	$1,725,008.49
		Amount of Single deposit at 22nd birthday	$1,725,008.49