Determine the Present Value of an Annuity

Question

Dave, a recent retiree, receives his $600 pension at the END of each month. He will receive this

pension for 20 years. If Dave can invest his funds at an interest rate of 10 percent compounded

annually, he should be just as satisfied receiving this pension as receiving a lump sum payment

today of what amount?

What if he receive it at the Beginning of the month?

How to do this without financial calculator? with formula?

I know how to do it yearly, but for monthly I am not sure

also, I have just started learning about finance. please write the details. Thank you

Answer 1

We need to look at the value for a monthly annuity which pays 600 per month for 20 years.

20 years = 20*12 = 240 months. He gets the first payment in 1 month Or Period.

First we have to Calculate the periodic interest rate. We need the monthly rate which compounded for 12 months yiled a 10% annual return. or (1+i^Monthly)¹²= 1+i^annual i^annual = 10% so we have to take the 12th root of (1+10%) or 1+i^Monthly = (1+i^annual )^(1/12) =0.797% to check this. Put 1 dollar in an account and let it compound at 0.797% per month and you will have (1*(1+.00797)¹² = 1.1 =1 + 10%

Now, what is an Annuity paying 600 per period for 240 periods worth today.

PV = PMT* β(i,n) where beta is the multiple of PV/PMT and β= ((1/i)(1-(1/(1+i))^N) this looks a little scary . You should really memorize this formula. β is a factor you can pick up from standard loan tables However, this is not a typical monthly rate. You can also do these calculations in a financial calculator or in Excel(recommended) Here, β(.00797,240) = (1/.00797)(1-(1/(1.00797))²⁴⁰) =(1/.00797)(1- .14864)=.85136/.00797 =106.82 so PV= 600*106.82=64,092.07

The second part of the question asks "what would it be worth if we got all of the payments 1 month earlier. We don't have to discount the first payment as we are getting it today. we then get 239 more payments of 600 so the early payment is 600 today plus the PV of an annuity for 239 more periods.

If you need help with this course please reach out. Not only did I teach this stuff at Tufts But used variations throughout my professional career.

Best,

Lenny

So we calculate β(.00797,239) = (1/.00797)(1- (1/(1.00797))²³⁹= (1/.00797)(1-.14998)=.85002/.00797 =106.65 so the last 239 payments are worth 106.6529*600 =63991.76. Now we add the 600 we get today and get $64,591.76. the difference is $499.69 that is how much better it is to get the payments now instead of starting in a month.