You just won the lottery, which will pay you $1,000 a year until you pay it off.

Question

You just won the lottery, which will pay you $1,000 a year until you pay it off. The current interest rate is 5%. The first payment happens in 1 year. a) As we know the lottery will pay 10 times, what is this lottery actually worth today? b) Forget about what you got in part a), if you know the prize is worth $10,000, and you will still receive $1000 a year until the prize is paid off. The interest rate is still 5%. How long does it take for you to be paid off? c) It turns out the prize would pay you $1000 a year forever. For the first 4 years, the interest rate is 5%, then the rate falls into 4% for the next 3 years, then goes up to 6% forever. What is your prize worth today? I know how to do this with excel but the instructor is requiring we show our work. Can not have any excel, all formulas.

Explanation / Answer

1.
=1000/0.05*(1-1/1.05^10)
=7721.734929
2.
=log(1/(1-10000*0.05/1000))/log(1.05)
=14.20669908
3.
=1000/1.05+1000/1.05^2+1000/1.05^3+1000/1.05^4+1000/(1.04^1*1.05^4)+1000/(1.04^2*1.05^4)+1000/(1.04^3*1.05^4)+1000/(1.04^2*1.05^4*1.06)+1000/(1.04^2*1.05^4*1.06^2)...
=1000/1.05+1000/1.05^2+1000/1.05^3+1000/1.05^4+1000/(1.04^1*1.05^4)+1000/(1.04^2*1.05^4)+1000/(1.04^3*1.05^4)+1000/(1.04^2*1.05^4*0.06)
=18506.26951

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