Question

John makes regular $2000 deposits in his savings account every year for the next 10 years, starting today(total 11 deposits). The bank will pay annual intrest rate of 5% for the next 10 years and will increase the intrest rate to 10%, thereafter. When will John have $100,000 in his account?

Please show all working with formulas. Don't solve it through excel.

Answer 1

We can use Future value (FV) of an Annuity due formula to find out the Future value from the annual savings of $2,000 in 10 years (as the payments are made at the starting of the year)

FV = PMT*(1+i) *{(1+i) ^n−1} / i

Where FV =?

PMT = Annual saving = $2,000

Annual interest rate = 5% per year

n = N = number of payments =11

Therefore,

FV = $2,000 * (1+5%)* [(1+5%) ^11 -1]/5%

FV = $29,834.25

Now we can use Future value (FV) formula to find out the time period of getting desired Future value of $100,000 from accumulated amount of $29,834.25 in 10 years

FV = PV * (1+r %) ^n

Where,

Future value of investment FV =$100,000

Present value of investment PV = $29,834.25

Interest rate = 10% per year (after 10 years)

Time period n =?

Therefore

$100,000 = $29,834.25 * (1 + 10%) ^n

Or $100,000/ $29,834.25 = (1 + 10%) ^n

Taking natural log (ln) from both sides

ln ($100,000/$29,834.25)= n * ln (1.10)

Or n = ln (3.35) / ln (1.10)  

Or n = 1.2095 /0.0953

Or n = 12.69 years

Therefore it will take 10 + 12.69 = 22.69 years; when John will have $100,000 in his account.