Question

1.Ms. Miller is 25 years of age as of today. She wants to retire at...

1.     Ms. Miller is 25 years of age as of today. She wants to retire at age 65 and wants to save equal amounts annually over next forty years so that she can have annual income of $80,000 from the accumulated funds from her annual savings over her retirement age from age 65 to age 95. Her savings will start when she will be of age 26 and her first retirement income will be made from the accumulated funds when she will be age 66.

 

Suppose that she will earn on her savings annual interest rate of 6%, compounded annually, throughout the time horizon, find her equal annual saving during her working life


2.     Consider the same data as in Question above. Assume that, with prices of goods and services at date zero as the base, the expected annual rate of inflation of 1.0% throughout the time horizon. Suppose that she will like to have annual real income (or income in real terms) of $80,000 during her retirement age of 65-95 years. Answer the following questions:

 

(i)                Find the equal annual real saving during her working life.

(ii)              Calculate the present value of the equal annual real savings over forty years of working life?

Answer & Explanation
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A

 Her annual savings during her working life would be $7115.36.

 

2.

i.

Annual real savings would be $10030.05 during  her working life

 

ii.

Present value of real savings would be $150915.12

Step-by-step explanation

1

Annual income per year in retirement (P)= 80000

number of years from age 65 to 95 (n) =30

interest rate is 6% compounded annually. So annual interest rate (i)=6%

 

Amount required at time of age 65 would be present value of annual income required that will be calculated by present value of annuity formula.

 

Present value of annuity formula = P*(1-(1/(1+i)^n))/i

=80000*(1-(1/(1+6%)^30))/6%

=1101186.492

 

This would be amount required at age 65 which will be accumulated by annual savings made upto  age 65 in 40 years

 

So future value required to be saved (FV)=1101186.492

number of yearly deposit (n) =40

interest rate (i) =6%

 

Amount required to save annually = FV*i/(((1+i)^n)-1)

=1101186.492*6%/(((1+6%)^40)-1)

=7115.356073

 

So Her annual savings during her working life would be $7115.36.

 

2.

i.

Real income today (present value) =80000

Inflation rate or Growth rate Required in income (g)= 1%
first withdrawal is age 66 that will be 41 years from now
Number of years (n)= 41

 

First income in  nominal dollars (future  value)= Present value*(1+g)^n

=80000*(1+1%)^41

=120300.1897

 

First payment of $120300.1897 needs on 66th year. so first annuity ( P) =120300.1897

growth rate in income (g) =1%

interest rate (i) =6%

number of withdrawal (n) = 95-65 = 30

 


We have to calculate present value of retirement needs with inflation rate by present value of growing annuity formula.


Present value of growing annuity formula =(Annuity *(1-((1+g)^n/(1+i)^n))/(i-g)
(120300.1897*(1-((1+1%)^40/(1+6%)^30)))/(6% -1%)
=1782304.43

 

This is Value at age 65. So it is also Future Value of Annuity deposits made.


Future value = 1782304.43
Increase or Growth rate in Savings (g)= 1%

interest rate (i) =6%
Number of years (n)=  40

 

 It is growing Annuity.

 

Future value of growing annuity =first Annuity/(i-g)*(((1+i)^n)-((1+g)^n))
1782304.43=P/(6%-1%)*(((1+6%)^40)-((1+1%)^40))
1782304.43= P*175.9370841
P =1782304.43/175.9370841
P or first deposit in nominal terms (n)= 10130.35108

 

 

First Annuity Deposit for savings is $10130.35108

it is in year 1

Present  value today in real terms = 10130.35108/(1+1%)

=10030.05057

 

So Annual real savings would be $10030.05 during  her working life

 

ii.

Real savings per year (P)=10030.05057

number of years (n) =40

interest rate (i) =6%

 

Present value of real savings = P*(1-(1/(1+i)^n))/i

=10030.05057*(1-(1/(1+6%)^40))/6%

=150915.1185

 

So present value of real savings would be $150915.12