Question

I am currently 25 years old. I have $20,000 invested in a brokerage account that I expect to give me a return of 8% compounded semi-annually. I am planning to work until I am 60 and put some money away monthly in a saving account that gives me a return of 2.05% compounded daily (First payment will be one month from now and last payment will be the month I retire. You can assume I turned 25 today. Also, assume a year has 365 days).

As soon as I retire, I plan to move somewhere warm and buy a house for $60,000 and invest all my remaining money into risk free treasury bonds that will give me a return of 2% compounded quarterly. My expected monthly expense for the remainder of my life, after retirement, is $1000 a month that I will be needing at the beginning of each month. A fortune teller told me I’ll live until I am 80 years old and I believe them.

a.) (10 points): How much should I put away each month in my savings account until I retire, so I can meet all my retirement needs. (Try not to round any intermediate results)

b.) (5 points): Recalculate the same above question but assume my saving account that gives me 2.05% compounds continuously instead of daily.

Answer 1

There are (60-25) = 35 years ie. 35*12 = 420 months until he retires.

The amount needed after 35 years = $60,000 to buy a house + PV of all future cash-flows of $1000 for 20 years or 240 months compounded 2% quarterly

2% compounded quarterly is equivalent to ((1+0.02/4)^4) -1 annually = 2.015% annually

We calculate the PV of all future monthly cash-flows of $1000 for 20 years using excel function 'PV'

Rate = 0.02015/12

N = 240

pmt = -1000

fv = 0

PV = $197,396.72

The amount needed after 35 years = $60000+ $197,396.72 = $257,396.72

The value of 20,000 today, compounded semi-annually at 8%, after 35 years = 20000*(1+(0.08/2))^35

The value of 20,000 after 35 years = $ 78921.78

There are (60-25) = 35 years ie. 35*12 = 420 months until he retires.

FV of monthly deposits after 35 years = $257,396.72 - $78921.78 =$178,474.94

a)

We calculate the PMT needed for FV to be $178,474.94 after 35 years or 420 months at 2.05% compounded daily

Rate = 2.05%/12

N = 420

PV = 0

fv = 178474.94

PMT(2.05%/12,420,0,178474.94 )

We get, monthly deposits required = $290.92

b)

We calculate the PMT needed for FV to be $178,474.94 after 35 years or 420 months at 2.05% compounded continuously

Rate = (e^(0.0205)-1)/12 = 0.00172596402

N = 420

PV = 0

fv = 178474.94

PMT(0.00172596402,420,0,178474.94 )

We get, monthly deposits required = $289.72