Question

You are currently 23. You can afford to deposit $850 per month into an account that earns 4.8% annual interest compounded monthly. Your goal is to retire with $1 million. How old will you be when you retire? Show what you typed into the TVM solver below. Put a ? for the value you solve for. N= I%= PV= PMT= FV= P/Y= C/Y= Round your answer to the nearest year. Age at retirement:

Answer 1

We will use the formula

`\begin{gathered} PMT=P(((r)/(n))/(1-(1+(r)/(n))^(-nt))) \\ where\text{ PMT = per month deposit = \$850} \\ P=1,000,000 \\ r=interest\text{ rate = 4.8\% = }(4.8)/(100)=0.048 \\ n=number\text{ of compounding = 12} \\ t=time\text{ in years = ?} \end{gathered}`

Applying we have

`\begin{gathered} 850=1,000,000(((0.048)/(12))/(1-(1+(0.048)/(12))^(-12t))) \\ 850=1,000,000((0.004)/(1-(1+0.004)^(-12t)) \\ 850=(4000)/(1-(1.004)^(-12t)) \\ 850(1-(1.004)^(-12t))=4000 \\ (1-(1.004)^(-12t))=(4000)/(850) \end{gathered}`

Continuing we have

`\begin{gathered} 1-(1.004)^(-12t)=4.7058824 \\ (1.004)^(-12t)=1-4.7058824 \\ (1.004)^(-12t)=-3.7058824 \\ taking\text{ log} \\ log(1.004)^(-12t)=-log(3.7058824) \\ -12tlog(1.004)=-log(3.7058824) \\ t=(-log(3.7058824))/(-12log(1.004)) \\ t=27.34457 \\ t\approx27\text{ years } \end{gathered}`

So when you retire you will be

`23+27=50\text{ years }`

So you will be approximately 50 years at retirement