Question

[Please show all work - thanks]

Suppose an agent has $100,000 today that he wants to save for 10 years. Compare the following two savings plans.

Bank A offers the following alternative:

For the first $50,000 the agent obtains 8% p.a. (per annum) for 10 years. For the other amount he obtains 4% p.a. for the first four years. Then he obtains 1% p.a.

Bank B offers the following alternative:

The interest in year 1 is 2%, in year 2 is 4%, in year 3 is 8%, in year 4 is 20%, then for years 5 to 10 the agent obtains 3% p.a.

For both plans, interest payments are reinvested.

(a) The agent maximizes the amount at t=10. Which plan is better? How much more can he spend at t=10, if he chooses the better one?

(b) Suppose bank B wants to match the offer of bank A. Interest rates for years 2 to 10 are as above. What interest rate for the first year must bank B offer the agent so that he gets the same amount as from bank A? [4p]

Answer 1

A

A
Calculations For Bank A
	P	i (Interest)	t (Years)	amount at end of period	Formula
First	50000	8%	10	107946.2499	P(1+i)^t                           50000*(1+8/100)^10
Second(first 4 years)	50000	i1=4%	4
Next 6 years	58492.93	i2=1%	6	62091.42174	(P(1+i1)^t)((1+i2)^t)                       (50000*(1+4/100)^4)*((1+1/100)^6)
	Value of 100000$ after 10 years			170037.6716
Calculations For Bank b
Year	P	i (Interest)	t (Years)	amount at end of period
1	100000	2%	1	102000	P(1+i)^t                          100000*(1+2/100)^1
2	102000	4%	1	106080	P(1+i)^t                          102000*(1+4/100)^1
3	106080	8%	1	114566.4	P(1+i)^t                          106080*(1+8/100)^1
4	114566.4	20%	1	137479.68	P(1+i)^t                         114566.4*(1+20/100)^1
5 to 10	137479.7	3%	6	164157.9276	P(1+i)^t                         137479.68*(1+3/100)^6
	Value of 100000$ after 10 years			164157.9276

Direct Formula

(((((100000*(1+2/100)^1)*((1+4/100)^1))*((1+8/100)^1))*((1+20/100)^1))*((1+3/100)^6))

164157.9276

Value if we go with bank A			170037.67
Value if we go with bank B			164157.93
Value of bank A is Higher so Bank A alternative is better .
He can spend 5879.74 $(170037.67-164157.93) more if he chooses Bank A over Bank B

B

Calculations For Bank b

Direct Formula

(((((100000*(1+2/100)^1)*((1+4/100)^1))*((1+8/100)^1))*((1+20/100)^1))*((1+3/100)^6))

164157.9

This was the original situation. Now we need to change the first years's 2% interest rate so that the final amount becomes 170037.67 instead of 164157.9. That means we have to increase value of 2 to get a higher amount. We have to replace 2 With X in the above equation and we have to find the value of X

(((((100000*(1+X/100)^1)*((1+4/100)^1))*((1+8/100)^1))*((1+20/100)^1))*((1+3/100)^6))=164157.9

(((((100000*(1+5.654/100)^1)*((1+4/100)^1))*((1+8/100)^1))*((1+20/100)^1))*((1+3/100)^6))=170037.67

Solving For X we get value of x = 5.654 %

So if bank B keeps interest rate for first year as 5.654% then it can give same results as bank A