Question

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? $

Answer 1

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