1. Find Time n = ?
Regular Deposit
(PMT Amount) C = 1000, Interest Rate i = 10%, Present value PV = 4169.87,
Deposit Frequency = at the beginning (Annuity Due) of every Year (1/year)
for Present value of Annuity Due method
Solution:
`C=1000` (Cash flow per year)
`i=10%=0.1` per year (Interest rate)
`PV=4169.87` (Present value)
Now, Present value (Annuity Due) formula is
`PV_("Annuity Due")=C*[(1-(1+i)^(-n))/(i)]*(1+i)`
`:.4169.87=1000*[(1-(1+0.1)^-n)/(0.1)]*(1+0.1)`
`:.4169.87=1100*[(1-(1.1)^-n)/(0.1)]`
`:.(4169.87)/(1100)*0.1=1-(1.1)^-n`
`:.0.38=1-(1.1)^-n`
`:.(1.1)^-n=1-0.38`
`:.(1.1)^-n=0.62`
taking natural log on both the sides
`:.ln(1.1)^-n=ln(0.62)`
`:.-n*ln(1.1)=ln(0.62)`
`:.-n=ln(0.62)/ln(1.1)`
`:.-n=(-0.48)/(0.1)`
`:.-n=-5`
`:.n=5` year
2. Find Time n = ?
Regular Deposit
(PMT Amount) C = 5000, Interest Rate i = 10%, Present value PV = 13677.69,
Deposit Frequency = at the beginning (Annuity Due) of every Year (1/year)
for Present value of Annuity Due method
Solution:
`C=5000` (Cash flow per year)
`i=10%=0.1` per year (Interest rate)
`PV=13677.69` (Present value)
Now, Present value (Annuity Due) formula is
`PV_("Annuity Due")=C*[(1-(1+i)^(-n))/(i)]*(1+i)`
`:.13677.69=5000*[(1-(1+0.1)^-n)/(0.1)]*(1+0.1)`
`:.13677.69=5500*[(1-(1.1)^-n)/(0.1)]`
`:.(13677.69)/(5500)*0.1=1-(1.1)^-n`
`:.0.25=1-(1.1)^-n`
`:.(1.1)^-n=1-0.25`
`:.(1.1)^-n=0.75`
taking natural log on both the sides
`:.ln(1.1)^-n=ln(0.75)`
`:.-n*ln(1.1)=ln(0.75)`
`:.-n=ln(0.75)/ln(1.1)`
`:.-n=(-0.29)/(0.1)`
`:.-n=-3`
`:.n=3` year
This material is intended as a summary. Use your textbook for detail explanation.
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