Question

Assume that it is now January 1, 2001 and you will need $1,000 on January 1, 2005. Your bank compounds interest rate at an 8 per cent annual rate.

I. How much must you deposit of January 1, 2002, have a balance of $1,000 on January 1, 2005?

II. If you want to make equal payments on each January 1 from 2002 through 2005 to accumulate the $1,000, how large must each of the 4 payments be?

III. If your father were to offer either to make the payments calculated in (b) or to give you a lump sum of $750 on January 1, 2002, which would you choose?

IV. If you have only $750 on January 1, 2002, what interest rate, compounded annually would you have to earn to have the necessary $1,000 on January 1, 2005?

V. To help you reach you $1,000 goal, your father offers to give you $400 on January 1, 2002. You will get a part-time job and make 6 additional payments of equal amounts each 6 months thereafter. If all of this money is deposited in a bank which pays 8 per cent, compounded semiannually, how large must each of the 6 payments be?

VI. What is the effective annual rate being paid by the bank in part v.?

Answer 1

I)	Amount to be deposited on January 1, 2002 = 1000/1.08^3 =	$         793.83
II)	$1000, it is the FV of the annuity. The annuity would be 1000*0.08/(1.08^4-1) =	$         221.92
III)	FV of the lump sum would be 750*1.08^3 =	$         944.78
	Wheras, FV of the payments in [2] would be $1000.
	The annuity payments would be preferred.
IV)	1000 = 750*(1+r)^3
	r = (1000/750)^(1/3)-1 =	10.06%
V)	FV of 400 given by father = 400*1.04^6 =	$         506.13
	Balance to be made up by January 1, 2005 = 1000-506.13 =	$         493.87
	Halfyearly payments required = 493.87*0.04/(1.04^6-1) =	$           74.46
	[=74.46*1.04^5+74.46*1.04^4+74.46*1.04^3+74.46*1.04^2+74.46*1.04+74.46 = 493.89]
VI)	Effective interest rate = 1.04^2-1 =	8.16%