Question

Bilbo Baggins wants to save money to meet three objectives. First, he would like to be able to retire 30 years from now with retirement income of $25,000 per month for 25 years, with the first payment received 30 years and 1 month from now. Second, he would like to purchase a cabin in Rivendell in 15 years at an estimated cost of $574,000. Third, after he passes on at the end of the 25 years of withdrawals, he would like to leave an inheritance of $700,000 to his nephew Frodo. He can afford to save $1,800 per month for the next 15 years. If he can earn a 10 percent EAR before he retires and a 6 percent EAR after he retires, how much will he have to save each month in years 16 through 30?

Answer 1

PV of annuity for making pthly payment
P = PMT x (((1-(1 + r) ^- n)) / i)
Where:
P = the present value of an annuity stream
PMT = the dollar amount of each annuity payment
r = the effective interest rate (also known as the discount rate)
i=nominal Interest rate
n = the number of periods in which payments will be made
Nominal Interest rate	6%
Frequency	monthly
Effective Interest rate	(1+6%/12)^12)-1)
Effective Interest rate	6.168%
Monthly payment	25000	Monthly
Annual payment	300000
PV of annuity	P = PMT x (((1-(1 + r) ^- n)) / i)
PV of annuity	= 300000* (((1-(1 + 6.18%) ^- 25)) / 6%)
PV of annuity	3,883,388.80
PV of inheritance=	700000/(1+6%)^25
PV of inheritance=	163,099.04
Total Corpus required=	4,046,487.84
FV of First 15 years investment @ 10%
Nominal Interest rate	10%
Frequency	monthly
Effective Interest rate	(1+10%/12)^12)-1)
Effective Interest rate	10.471%
Monthly payment	1800
Annual payment	21600
FV of this annuity
FV of annuity
The formula for the future value of an ordinary annuity, as opposed to an annuity due, is as follows:
P = PMT x ((((1 + r) ^ n) - 1) / i)
FV=	21600*((((1 + 10.471%) ^ 15) - 1) / 10%)
FV=	746,006.55
Cabin Cost	540000
Corpus remaining	206,006.55
Value of this corpus at the time of retirement i.e. after 15 years
Value of this remaining corpus=	206006*(1+10%)^15
Value of this remaining corpus=	860,538.19
Corpus to be built with annuity from year 16 to year 30
Required corpus=	=4046487.85-860538.186
Required corpus=	3,185,949.66
3,185,949.66	=P*((((1 + 10.471%) ^ 15) - 1) / 10%)
3,185,949.66	=P*34.537
P=	92,247.44
Monthly depost from t 16-T30=	7,687.29