You can afford monthly deposits of $90 into an account that pays 3.9% compounded monthly. How long will it be until you have $7,700 to buy a​ boat?

Question

You can afford monthly deposits of $90 into an account that pays 3.9% compounded monthly. How long will it be until you have $7,700 to buy a​ boat?

Type the number of​ months: _______

​(Round to the​ next-higher month if not​ exact.)

Answer 1

Aff... this is a horrible bit of maths if you really wanted an equation, and then be able to work it out for yourself!

Why? Because you need:

Total Savings = [ Compound interest for principal, the starting amount ] + [ Future value of a series of monthly payments ]

first part is compound interest on principal: P ( 1 + ^r/_n ) ^nt

^{P = principal, r = interest n = number of times interest applied per period t = number of periods}

second part is future value of annuity: C * [ ( (1 + i )ⁿ − 1 ) / i ​]

_{C = cash flow (deposit) per period i = interest n = number of periods}

I recommend simply plugging your numbers into a compound interest calculator!

Moreover... there is some ambiguity in your current question:

Is there any deposit (the principal) in your account to start? I assume $90 initial principal, then monthly deposits.

The interest rate of 3.9% paid monthly is far too high (because 12 x 3.9%. I think it's meant to be 3.9% annual interest, paid monthly (i.e. 3.9%/12 or approx ¹/₃ % each month), so I will assume this. However, 3.9% nominal, is 4% or so APY, so if the interest rate was specified as APY we would need to use the APY formula to get the nominal monthly interest.

For a long period of savings for a specific goal (e.g. a boat costing $7,700 today) we should account for inflation where your savings are losing value, and the boat price is rising. In other words, the adjustment for inflation changes the calculation so that we aim not for $7,700 but the inflation-adjusted value of $7,700. I will ignore this as you don't mention it.

We would also typically need to factor in income tax on the interest too. But I will ignore this as it's not mentioned. I think the easiest way to deal with inflation and income tax, if required, is to simply change the interest rate to a net interest rate i.e. (interest paid per period) x (1 - tax rate on interest income) - (inflation rate per period). Inflation in US runs about 2% so this has a significant effect on net interest (3.9-2 = 1.9), and the amount of time to reach the savings goal.

Based on above assumptions: 75 months (6 years 3 months) [from calculator]

Savings Projection:

$6,840.00 deposits

$929.41 interest

$7,769.41 total

Or, using annuity equation only (slightly different assumption to above, with no initial principal, so amounts will differ slightly):

FV = C * [ ( (1 + i )ⁿ − 1 ) / i ​] and interest per month as 0.039/12 and deposit is 90$ per month

FV = 90 * ( (1 + 0.039/12)^75 - 1) / (0.039 / 12 ) = $7,630

or for 76 periods = $7,745

Of course, to be fair, I was plugging in possible number of months by "trial and error" (which is sometimes easier/quicker).

To use proper maths, I need to rearrange the FV equation to make n (number of periods) the subject instead, and substitute 7,700 (the boat) for FV:

FV = C * [ ( (1 + i )ⁿ − 1 ) / i ​]

=> FV i / C = (1 + i )ⁿ − 1

=> FV i / C + 1 = (1 + i )ⁿ

=> n = log ( FV i / C + 1 ) / log (1 + i )

=> n = log ( ( 7700 * (0.039/12) / 90 ) + 1 ) / log (1 + (0.039/12) ) = 75.6 periods rounded to 76

Answer 2

The compound interest equation is:

A=P(1+r/n)^nt

Solving for t = (1/n)ln(A/P)/ln(1+r/n) , which is time in years

where ln() is the natural log

Here n = 12, A= 7700, P = 90, r = 3.9% = .039

t = 12(1/12)(ln(7700/90)/ln(1+.039/12) to get the time in months

t = ln(85.56)/ln(1.00325) ≈4.45 /.003245 ≈1371 months