Present/Future Value of Annuity Question

**Jane Doe is 25 years old and starting next year wants to**

save and invest in order to ensure an annual income of $100kin each of the 20 years after she retires, till she is 85. She willretire at 65.

She expects to earn 10% interest till retirement and 5%interest in the years after retirement.

How much must she save each year from now till retirementstarting next year?

This part is easy; PVA with pmt=100k, t=20, r=5% --> ~$1,246,200

The second part: **As it turned out, Jane spent the next 5 years on the beach inBarcelona. She is 30 now and has $4,000 in the bank. Howmuch does she need to save each year, starting next year?All other data unchanged.**

What should be my thought process behind this? I know the future value when she's 65 should be $1,246,200, interest should be 10%, and there's 35 more years.

Re: Present/Future Value of Annuity Question

Hey youngb11.

What formulas do you have for the annuity?

What you can do is either re-arrange to solve for n or use a numerical scheme if you can't re-arrange n by itself on only one side of the equation.

Re: Present/Future Value of Annuity Question

Quote:

Originally Posted by

**youngb11** **Jane Doe is 25 years old and starting next year wants to**

save and invest in order to ensure an annual income of $100kin each of the 20 years after she retires, till she is 85. She willretire at 65.

She expects to earn 10% interest till retirement and 5%interest in the years after retirement.

How much must she save each year from now till retirementstarting next year?

This part is easy; PVA with pmt=100k, t=20, r=5% --> ~$1,246,200

The second part: **As it turned out, Jane spent the next 5 years on the beach inBarcelona. She is 30 now and has $4,000 in the bank. Howmuch does she need to save each year, starting next year?All other data unchanged.**

What should be my thought process behind this? I know the future value when she's 65 should be $1,246,200, interest should be 10%, and there's 35 more years.

I would give it a try, though I have had limited exposure to business math...I will do both the problems as I see that you have partly done the first problem and maybe it'll be more useful to everybody else...

**1**. Jane wants $1000 per year after retirement at 5% interest, so she would want to know the present value of annuity(pva) at the point of retirement,

$\displaystyle pva=p*\frac{1-\frac{1}{(1+r)^n}}{r}$ #p=>pension & r=>interest rate & n=> no. of years

putting the values,

$\displaystyle pva=100000*\frac{1-\frac{1}{(1+0.5)^{20}}}{0.5}=1246221$

Now that jane has to have $ 1246221 at retirement, it becomes a future value problem now:

$\displaystyle fv=c*\frac{(1+r)^n - 1}{r}$ # c=cash payments to bank

since jane wants to have 1246221 as future value at retirement,

$\displaystyle 1246221=c*\frac{(1+0.1)^{40} - 1}{0.1}$

therefore, $\displaystyle c=2815.72$

so jane would need to save at least $\displaystyle \$ 2815.72$ and invest it, to fulfill her plans...

**2**. I guess the $4000 accumulated by jane would get compounded annually and contribute in making the $ 1246221, which she requires at the point of retirement.

$\displaystyle A=p(1+r)^n=4000(1+0.1)^{35}=112409.74$

now we will again calculate c (cash payments to bank) like we did before but take into account the $ 1246221

$\displaystyle fv=1246221=112409.74 + c\frac{(1+0.1)^{35} -1}{0.1}$

$\displaystyle =1133811.26=c*\frac{(1+0.1)^{35} -1}{0.1}$

solving for c we get $\displaystyle c=\$ 4183.49$

note that in the second case jane's savings or cash payment to bank has increased from $2815.72 to $4183.49. This is because, Jane would have made more by saving what she had for 5 years than she had earned at barcelona.If she had saved $ 2815.72 per year for 5 years, her account would have gone to $\displaystyle 2815.72*\frac{(1+0.1)^5 -1}{0.1}$ or

$\displaystyle \$ 17190.25 $.

So jane made a terrible decision(Headbang) and I have tried to make approximations upto 2 decimal places

please correct me if i am wrong.