Question

In: Economics

After shopping for a car Amelia ended up borrowing $11000 from her grandparents at 9% per...

After shopping for a car Amelia ended up borrowing $11000 from her grandparents at 9% per year compounded annually with repayment at the end of 5 years. Her Grandparents asked her to develop some alternative repayment options.
If Amelia's TVOM is 11%, what is the present worth for Amelia of each of the following 3 alternatives?
1) Interest only at the end of each year and principal at the end of the fifth year. $  
2) Equal annual payments. $  
3) Pay the principal and interest in one lump sum after 5 years? $


If Amelia's TVOM is 6%, what is the present worth for Amelia of each of the following 3 alternatives?
1) Interest only at the end of each year and principal at the end of the fifth year. $  
2) Equal annual payments. $  .
3) Pay the principal and interest in one lump sum after 5 years? $

Solutions

Expert Solution

IF TVOM=11%

Rate of intrest=9%

Intrest amount to be paid annually=0.09*11000=990

Present value factor=

where n= no. of years

End of year Intrest paid(I) Present value factor(PVF) Present value(PV=I*PVF)
1 990 0.9009 891.891
2 990 0.8116 803.484
3 990 0.7312 723.888
4 990 0.6587 652.113
5 990 0.5934 587.466

Total Intrest present value=3658.842

Now present value for the principal amount= principal amount*(PVF for 5th year)=11000*0.5934=6527.4

Total value=present value intrest+present value principal=6527.4+3658.842=10186.242

2) Equal annual installment

Installment=(11000/PVIFA(5yrs,11%))=11000/3.6959=2976.271

PVIFA(5yrs,11%) can be found from the present value of annuity table at 11% and 5 yrs. The image will be attached below.

End of year Installment paid(I) Present value factor(PVF) Present value(PV=I*PVF)
1 2976.271 0.9009 2681.322
2 2976.271 0.8116 2415.542
3 2976.271 0.7312 2176.250
4 2976.271 0.6587 1960.470
5 2976.271 0.5934 1766.120

Total value=sum of present value=10999.704

3)Intrest after 5 years=principal*((1+i)n-1)=11000*((1+0.09)5-1)=5924.863

Total amount to be paid=Intrest+principal=5924.863+11000=16924.8635

Present value=total amount*(PVF for 5th year)=16924.8635*0.5934=10043.214

When TVOM is changed to 6% the problem can be solved similar to above.

IF TVOM=6%

Rate of intrest=9%

Intrest amount to be paid annually=0.09*11000=990

Present value factor=

where n= no. of years

End of year Intrest paid(I) Present value factor(PVF) Present value(PV=I*PVF)
1 990 0.9434 933.962
2 990 0.88999 881.090
3 990 0.8396 831.204
4 990 0.7921 784.179
5 990 0.7472 739.728

Total Intrest present value=4170.163

Now present value for the principal amount= principal amount*(PVF for 5th year)=11000*0.7472=8219.2

Total value=present value intrest+present value principal=8219.2+4170.163=12389.363

2) Equal annual installment

Installment=(11000/PVIFA(5yrs,6%))=11000/4.21236=2611.363

PVIFA(5yrs,6%) can be found from the present value of annuity table at 6% and 5 yrs. The image will be attached below.

End of year Installment paid(I) Present value factor(PVF) Present value(PV=I*PVF)
1 2611.363 0.9434 2463.56
2 2611.363 0.88999 2324.087
3 2611.363 0.8396 2192.500
4 2611.363 0.7921 2068.461
5 2611.363 0.7472 1951.210

Total value=sum of present value=10999.818

3)Intrest after 5 years=principal*((1+i)n-1)=11000*((1+0.09)5-1)=5924.863

Total amount to be paid=Intrest+principal=5924.863+11000=16924.8635

Present value=total amount*(PVF for 5th year)=16924.8635*0.5934=12646.258


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