Question

Suppose that you have $1,000 at hand that you can place in a bank gaining 5% annually in interests; assume that this interest rate is your personal discount rate. Your friend asks you to lend him $1,000 so he can buy a new gadget that he does not really need. He promises to pay back the full amount, plus interest, in a certain amount time. Assume that you really do not think that your friend should buy this new gadget, except, of course, if you were to make a profit out of it.

1. (a) Your friend offers to pay you back $1,006 in one year. What should you do? Demonstrate how munch money you would make if you were to place your $1,000 in the bank and compare it with the amount of money you would make if you were to accept your friend’s offer.

2. (b) Your friend offers to pay you back $1,060 in one year. What should you do? Demonstrate.

3. (c) Your friend offers to pay you back $500 in one year, another $400 in two years and $200 in three years. What should you do? Demonstrate.

4. (d) Your friend offers to pay you back $200 in one year, another $400 in two years and $500 in three years. What should you do? Demonstrate.

5. (e) Suppose instead that your friend just saw a publicity for this gadget he really does not need but really wants, and that this gadget will come into the market in x years and will cost $1,000. Your friend is worried that he will spend his money on other useless gadgets before this one comes out, so he offers to pay you $y each year if you agree to give him $1,000 in xyears when the gadget gets on the market. Assume you can still earn 5% annual interest by placing the money your friend gives you in a bank, and that in x years you have to withdraw all of the money from the bank; assume again that this interest rate is your personal discount rate.

   1. (i) If x = 1, that is, your friend pays you $y today, and you pay him back $1,000 in a year. For what amount of money y will you be indifferent between accepting or refusing his offer?

   2. (ii) If x = 5, for which amount of money y will you be indifferent between accepting or refusing his offer?

   3. (iii) Suppose now that instead of giving you the money each year, your friend gives youonly $y today. If x = 10, for which amount of money y will you be indifferent between accepting or refusing your friend’s offer?

Answer 1

a) If i invest 1000 in bank at 5% for 1 year i would get 1050 after 1 yr. but if i give the same amount to friend so i would recieve 1006 in 1 yr so i am better off with investing that amount in the Bank.

Bank :

Amount	1000
Rate	5%
Time (years)	1
Compounding factor	105%
Final Amount	1050

Note: Compounding factor = (1+rate)^time

With Friend:

Final Amount = 1006

b) similarly in this case

with Bank:

Amount	1000
Rate	5%
Time (years)	1
Compounding factor factor	105%
Final Amount	1050

Note: Compounding factor = (1+rate)^time

With Friend:

Final Amount = 1060 ( return would be 6%)

So i am better off paying the money to My Friend where i would get 6% return compared with Bank which is giving 5%

c)

with Friend:

cash flow	500	400	200
time	1	2	3
rate	5%
discounting factor	0.952380952	0.90702948	0.8638376
PV	476.1904762	362.811791	172.76752
Total	1011.769787

Note: discounting factor = 1/ [(1+rate)^time]

with Bank:

Amount	1000
Rate	5%
Time (years)	1
Compounding factor	105%
Final Amount	1050

I am better off investing that money in the Bank.

d)

cash flow	200	400	500
time	1	2	3
rate	5%
discounting factor	0.952380952	0.90702948	0.8638376
PV	190.4761905	362.811791	431.918799
Total	985.2067811

with Bank:

Amount	1000
Rate	5%
Time (years)	1
Compounding factor factor	105%
Final Amount	1050

I am better off investing that money in the Bank.

e) i)

time	1
rate	5%
compounding factor	1.05
PV	952.3809524
amount paid in 1 year	1000

At 952.38 i am indifferent.

ii)

cash flow	230.9747981	230.974798	230.974798	230.974798	230.974798
time	1	2	3	4	5
rate	5%	5%	5%	5%	5%
discounting factor	0.952380952	0.90702948	0.8638376	0.82270247	0.78352617
PV	219.9759982	209.500951	199.524715	190.023538	180.974798
Total	1000

PV = 230.9747981 * 0.952380952 (similarly for others)

At 230.97 i am indifferent.

iii)

time	10
rate	5%
compounding factor	1.628894627
PV	613.9132535
amount paid in 1 year	1000

At 613.91 i am indifferent.

Note: Using goal seek you can find the solution from excel.

Note: discounting factor = 1/ [(1+rate)^time]

Note: Compounding factor = (1+rate)^time