1. Find Time n = ?
Regular Deposit
(PMT Amount) C = 1000, Interest Rate i = 10%, Future value FV = 6715.61,
Deposit Frequency = at the beginning (Annuity Due) of every Year (1/year)
for Future value of Annuity Due method
Solution:
`C=1000` (Cash flow per year)
`i=10%=0.1` per year (Interest rate)
`FV=6715.61` (Future value)
Now, Future value (Annuity Due) formula is
`FV_("Annuity Due")=C*[((1+i)^n-1)/(i)]*(1+i)`
`:.6715.61=1000*[((1+0.1)^n-1)/(0.1)]*(1+0.1)`
`:.6715.61=1100*[((1.1)^n-1)/(0.1)]`
`:.(6715.61)/(1100)*0.1=(1.1)^n-1`
`:.0.61=(1.1)^n-1`
`:.1.1^n=1+0.61`
`:.1.1^n=1.61`
taking natural log on both the sides
`:.ln(1.1^n)=ln(1.61)`
`:.n*ln(1.1)=ln(1.61)`
`:.n=ln(1.61)/ln(1.1)`
`:.n=(0.48)/(0.1)`
`:.n=5` year
2. Find Time n = ?
Regular Deposit
(PMT Amount) C = 5000, Interest Rate i = 10%, Future value FV = 18205,
Deposit Frequency = at the beginning (Annuity Due) of every Year (1/year)
for Future value of Annuity Due method
Solution:
`C=5000` (Cash flow per year)
`i=10%=0.1` per year (Interest rate)
`FV=18205` (Future value)
Now, Future value (Annuity Due) formula is
`FV_("Annuity Due")=C*[((1+i)^n-1)/(i)]*(1+i)`
`:.18205=5000*[((1+0.1)^n-1)/(0.1)]*(1+0.1)`
`:.18205=5500*[((1.1)^n-1)/(0.1)]`
`:.(18205)/(5500)*0.1=(1.1)^n-1`
`:.0.33=(1.1)^n-1`
`:.1.1^n=1+0.33`
`:.1.1^n=1.33`
taking natural log on both the sides
`:.ln(1.1^n)=ln(1.33)`
`:.n*ln(1.1)=ln(1.33)`
`:.n=ln(1.33)/ln(1.1)`
`:.n=(0.29)/(0.1)`
`:.n=3` year
This material is intended as a summary. Use your textbook for detail explanation.
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