Question

CASE #5

APR vs EAR

Mary Jo plans to invest some money so that she has $8,000 at the end of three years.

	Option A	Option B	Option C	Option D
APR	5.19%	5.11%	5.20%	5.40%
m/per year	365	12	4	1

How much should she invest today given the choices above?

What is her best choice?

What is the EAR for each account? Support your decision in question 2.

Physical therapy equipment purchase

Your firm needs to buy additional physical therapy equipment that costs $20,000. The equipment manufacturer will give you the equipment now if you will pay $6,000 per year for the next four years. If your firm can borrow money at a 9 percent interest rate, should you pay the manufacturer the $20,000 now or accept the four-year annuity offer of $6,000?

Answer 1

1.Formula for compound interest can be used to compute principal amount as:

A = P x (1 + r/m)^mt

P = A/(1+r/m)^mt

A = Future value of investment = $ 8,000

P = Principal

r = Rate of interest

m = No. of compounding in a year

t = No. of years = 3

Option A:

r = 5.19 % p.a.

m = 365

r /m=0.0519/365 = 0.00014219

P = $ 8,000/ (1+0.00014219)^365x3

= $ 8,000/ (1.00014219)¹⁰⁹⁵

= $ 8,000/ 1.16846267452562

= $ 6,846.602955 or $ 6,846.60

Option B:

r = 5.11 % p.a.

m = 12

r/m = 0.0511/12 = 0.00425833

P = $ 8,000/ (1+0.00425833)^12x3

= $ 8,000/ (1.00425833)³⁶

= $ 8,000/ 1.16529528965770

= $ 6,865.212681 or $ 6,865.21

Option C:

r = 5.20 % p.a.

m = 4

r/m = 0.052/4 = 0.01300000

P = $ 8,000/ (1+ 0.01300000)^4x3

= $ 8,000/ (1.01300000)¹²

= $ 8,000/ 1.16765177626913

= $ 6851.35771 or $ 6,851.36

Option D:

r = 5.40 % p.a.

m = 1

r /m = 0.054/1 = 0.054

P = $ 8,000/ (1+0.054)^1x3

= $ 8,000/ (1.054)³

= $ 8,000/ 1.170905464

= $ 6,832.3193 or $ 6,832.32

2.The best choice for Mary Jo is Option D as it require less principal than other options.

3.

Formula for effective interest rate is:

r = (1+i/n) ⁿ – 1        

r = Effective interest rate

i = Stated interest rate

n = No. of compounding periods in a year

Option A:

r = (1+ 5.19%/365) ³⁶⁵ – 1

= (1.00014219) ³⁶⁵ – 1 = 1.05326652 – 1 = 0.05326652 or 5.33 %

Option B:

r = (1+ 5.11%/12) ¹² – 1

= (1.00425833) ¹² – 1 = 1.052313956 – 1 = 0.052313956 or 5.23 %

Option C:

r = (1+ 5.2%/4) ⁴ – 1

= (1.013) ⁴ – 1 = 1.053022817 – 1 = 0.053022817 or 5.30 %

Option D:

r = (1+ 5.4%/1) ¹ – 1

= (1.054) ¹ – 1 = 1.054 – 1 = 0.054 or 5.4 %

As the EAR for option D is highest, option D should be chosen.

4.

PV of four year annuity is:

PV = C x PVIFA (i, n)

C = periodic cash flow = $ 6,000

i = Rate of interest = 9 %

n = No. of periods = 4

PV = $ 6,000 x 3.2397 = $ 19,438.20

As PV of annuity payment is less than $ 20,000; firm should accept the four-year annuity offer.