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Estimates, Compound Interest, Payment Options

by Alphie Vills
(Port Moresby, Papua New Guinea)











































Please,I need help on this problem.

A friend is planning on buying a major appliance.A store which has the item being purchased is offering 'No deposit,No interst,No repayments for 24 months for purchases over $1000 or a 10% discount for cash.

He asks you if he should pay for the appliance immediately to obtain the 10% discount or to accept delayed payment option at a higher price.

Estimates and Assumptions

Make the following estimates however,you may choose any that seem realistic.
1) Purchase price of major appliance, $C (must be at least $1000).
2) Interest rate earned by a Savings account, S% pa compounded monthly.
3) Credit card interest rate, I% pa compounded daily.

Assume any fees and charges involed with the delayed payment offer are minimal.

Task
A) Calculate the discounted cash price.
B) Calculate the effective discount rate being used.

Assume the delayed payment option is chosen.
C) If the cash price is avaiable assume it is invested in a savings account earning interest at S% pa
compounded mounthly (estimate 2).
D) How much interest has been earned?
E) How much needs to be invested in savings account,so the full price of the appliance is avaliable in
two years?

Comments for Estimates, Compound Interest, Payment Options

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Apr 30, 2012
Estimates, Compound Interest, Payment Options
by: Staff



Part I

Question:

by Alphie Vills
(Port Moresby, Papua New Guinea)


Please, I need help on this problem.

A friend is planning on buying a major appliance. A store which has the item being purchased is offering 'No deposit, No intestine repayments for 24 months for purchases over $1000 or a 10% discount for cash.

He asks you if he should pay for the appliance immediately to obtain the 10% discount or to accept delayed payment option at a higher price.

Estimates and Assumptions

Make the following estimates however, you may choose any that seem realistic.
1) Purchase price of major appliance, $C (must be at least $1000).
2) Interest rate earned by a Savings account, S% pa compounded monthly.
3) Credit card interest rate, I% pa compounded daily.

Assume any fees and charges involved with the delayed payment offer are minimal.

Task
A) Calculate the discounted cash price.
B) Calculate the effective discount rate being used.

Assume the delayed payment option is chosen.
C) If the cash price is available assume it is invested in a savings account earning interest at S% pa
compounded monthly (estimate 2).
D) How much interest has been earned?
E) How much needs to be invested in savings account, so the full price of the appliance is available in
two years?



Answer:


Calculate the discounted cash price.

You are probably familiar with the formula for evaluating the amount accumulated in a savings account, given a certain deposit and a fixed interest rate.

It is:

A = P*(1 + r)^t

A = final balance in the savings account
P = principle (the initial deposit in the bank, or the Present Value of the Investment)
r = decimal form of interest rate for each period of time (monthly)
t = number of monthly time periods

The Present Value “P” of the Future Balance “A” is computed by solving for P. (Just divide each side of the equation by (1 + r)^t .)

P = A / (1 + r)^t

The Present Value “P” is the discounted cash price.

Discounted Cash Price = P = A / (1 + r)^t



Option A: 'No deposit, No interest, No repayments for 24 months for purchases over $1000 or a 10% discount for cash

Your friend purchases an appliance for $1200 and accepts the No deposit, No interest, and No repayments for two years. At the end of two years your friend pays $1200 in cash.

P = A / (1 + r)^t

A = 1200
P = present value in today’s dollars
r = .005 (6% annual interest rate ÷ 12 months)
t = 24 months

P = 1200 / (1 + .005)²⁴

P = 1200 / (1.005)²⁴

P = 1064.6228026694018

P = $1064.62


A future payment of $1200 from a savings account in two years is worth $1064.62 in today’s dollars.


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Apr 30, 2012
Estimates, Compound Interest, Payment Options
by: Staff

-----------------------------------------------------------------

Part II

If you deposited $1064.62in a savings account for two years at 6% interest compounded monthly, the final balance at the end of two years would be exactly $1200 (the amount you owe for the appliance).


Option B: Pay cash now


If you took the 10% discount and paid cash today, you would pay

Discount Price = $1200 - 10% of $1200

Discount Price = $1200 - .1* $1200

Discount Price = $1200 - $120

Discount Price = $1080

The present value P can be computed for different interest rates.


Comparing these two values in today’s dollars:

No Deposit
No Interest
No Payments
(for 2 years) 10% cash discount

$1064.62 (6% int) $1080


If these assumptions are correct, you are slightly better off to delay payment for two years.

This comparison can be computed for a variety of interest rates.


No Deposit
No Interest
No Payments
(for 2 years)   10% cash discount


$1152.98 (2% int)      $1080 10% cash discount is less expensive

$1130.20 (3% int)      $1080 10% cash discount is less expensive

$1107.89 (4% int)      $1080 10% cash discount is less expensive

$1086.03 (5% int)      $1080 10% cash discount is less expensive

$1064.62 (6% int)      $1080 2 year delay is less expensive

$1043.65 (7% int)      $1080 2 year delay is less expensive


As a general rule of thumb, the 10% cash discount is less expensive if your only option is to deposit your money in a savings account which pays an annual rate of interest below 5.3%, compounded monthly.

If you savings account interest is greater than 5.3% per annum, compounded monthly, the two year delay is less expensive.

The 5.3% is the result of solving the following equation for r:

1080 = 1200 / (1 + r/12)²⁴









The annual effective discount rate:

From your point of view, the annual effective discount rate is the annual interest earned in your savings account divided by the balance in your savings account (including interest) at the end of the year.

Because the discount rate is computed using the year end balance in your savings account (interest rate is computed using the beginning amount), it will always be lower than the interest rate.

Discount Rate - The rate at which future cash flows are discounted

interest rate = rₑ = (interest paid during year)/(balance in account at beginning of year)

discount rate = dₑ = (interest paid during year)/(balance in account at end of year)

dₑ = rₑ / (1 + rₑ)

for a 6% interest rate, the annual effective interest rate for monthly compounding is 6.17%:

rₑ = (1+0.06/12)^12 -1 = 1.0616778118645 - 1 = .0616778118645 = 6.17%

dₑ = .0616778118645 / (1 + .0616778118645)

dₑ = 0.0580946603341

dₑ = 5.81%




Thanks for writing.

Staff
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