Linus is 18 years old now, and is thinking about taking a 5-year university degr

Question

Linus is 18 years old now, and is thinking about taking a 5-year university degree. The degree will cost him $25,000 each year. After he's finished, he expects to make $50,000 per year for 10 years, $75,000 per year for another 10 years, and $100,000 per year for the final 10 years of his working career. If Linus lives to be 100, and if real interest rates stay at 5% per year throughout his life, what is the equal annual consumption he could enjoy until that date? Linus is also considering another option. If he takes a job at the local grocery store, his starting wage will be $40,000 per year, and he will get a 3% raise, in real terms, each year until he retires at the age of 53. If Linus lives to be 100, what is the equal annual consumption he could enjoy until that date? From strictly a financial point of view, is Linus better off choosing option A or B?

Explanation / Answer

Option 1 : Steps to find the Future value at age 18+5+10+10+10 =53.
1. FV of (-25000) for 5 Yrs at 5%. Age 23
2. FInd PV of annuity of $50000 for 10yrs at 5% (Y23). Then reduce FV in 1 above. Balance is amt at hand at Y10. Find its FV at Y53 ie after 30 yrs at 5%. Age 23+10 = 33
3. Find FV of annuity of 75000 for 10 yrs at 5%. Age 33+10=43. Then find Fv of this money at Y53.
4. Find FV of Annuity of 100,000 for 10 Yrs at 5%. Age 43+10 = 53.
5. Add the values of 2+3+4. This is the nest egg at age 53.
6. Now nest egg is PV of Annuity. We need to find annual PMT for 100-53=47 yrs at 5%

Now do maths :-
1. Age 23. FV of annuity FV = FV(Rate,nper,PMT) = FV(5%,5,-25000) = $138,141
2. Age 23. PV of annuity = PV=PV(Rate,nper,PMT) = PV(5%,10,-50000) = $386,087
So Net amount availble at Y23 = 2-1 = $247,946
FV of this Lump sum $247,946 after 30 yrs = PV*(1+i)^n = $247,946*(1+5%)^30
= $1,071,608
3. Age 33. FV of annuity of 75000 = FV(5%,10,-75000) = $943,342
FV of this Lump sum $943,342 after 20 yrs = PV*(1+i)^n = $943,342*(1+5%)^20
= $2,502,967
4. Age 43. FV of Annuity of 100,000 = FV(5%,10,-100000) = $1,257,789
FV of this Lump sum $1,257,789 after 10 yrs = PV*(1+i)^n =$1,257,7892*(1+5%)^10
= $20,488,061
5. Age 53. Add 2+3+4 = 24,062,636
6. Now we need to find annual PMT which can be withdrawn till age 100 ie 47 yrs
SO PMT=PMT(Rate,nper,PV) = PMT(5%,47,-24062636) = $1,338,225 .......Ans (A)

Option 2: We need to find FV of growing annuity of A=40000 with g=3% & Rate i=5% for (53-18) = n=35yrs.
FV of growing annuity = FVA = A*[(1+i)^n - (1+g)^n]/(i-g)
ie FVA = 40000*[(1+5%)^35 -(1+3%)^35]/(5%-3%) = $5,404,306
Now we need to find annual PMT which can be withdrawn till age 100 ie 47 yrs
SO PMT=PMT(Rate,nper,PV) = PMT(5%,47,-5404306) = $300,556 .....Ans (B)


So Linus is betteroff with Option A above

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