1. APRs, EARs, APYs, and practice problems
FINC311
September 14, 2015
Jay Coughenour
APRs, EARs, APYs, and practice problems

2. APR = “Annual Percentage Rate”
APR = periodic rate * comp periods per year
Examples:
1% compounded per month = 12% APR Or
6% APR compounded monthly = 0.5% compounded per month
Typically, we are given the APR and compounding periods
“9.99% APR compounded daily”
“3.75% APR compounded monthly”

3. Effective Annual Returns (EARs)
Suppose we are interested in FV of $100 in 1 year at the following rates:
10% APR compounded annually:
FV = 100(1.1)1=
10% APR compounded daily:
.1/365 = , FV = 100( )365=
Therefore, 10% APR compounded daily has an effective annual rate of %
That is, FV = 100( )1 =

4. APYs APYs = EARs APY = “Annual Percentage Yield”
Some banks state rates as APRs while others use APYs or EARs

5. Santander CD advertisement
I snapped this on May 4, 2015 in Wilmington
Dirty details
$10K minimum + checking acct

6. As of Sept 7, 2015

7. By the way…
What is the difference between 0.90% and 1.29% APY for a 2 year 10,000 CD?
10,000(1.0090)2= 10,180.81
10,000(1.0129)2= 10,259.66
diff =
Note: 10,000(1.0039)2 = 78.15
Can’t just use r2 – r1 = to get the difference
Difference = 1o,000(r22 +2(r2-r1)-r12) = 78.85

8. Examples

9. Our one equation so far…
FV = PV(1 + r/m)m*t
This equation relates the value of a SINGLE cash flow at different points in time.
FV increases with PV, r, t and m
PV increases with FV, decreases with r, t, and m

10. Example 1
How much money do I need to deposit today in an account earning 12% APR, compounded annually, to have $4, in 4 years?
PV = 4250/(1.12)4 = 4250/ =
*BTW: PV = 4250*(1.12)-4 =
BAII Functions: 4250,+/-,FV,12,I/Y,4,N,CPT,PV
Conceptual extension: Would I need to deposit more or less than $ if the rate was 12% APR compounded monthly?
Less! Because I’m earning interest at a higher EAR.

11. Example 2
Today the average cost of a box of cereal is $ If the inflation rate is expected to be 2.9% compounded annually, how much will an average box of cereal cost in 20 years?
FV = 3.59(1.029)20 = $6.36
BAII Functions: 3.59, +/-, PV, 2.9, I/Y, 20, N, CPT, FV

12. Example 3
Your friend saved $9500 for a new car. However, she wants a used BMW that should cost $17500 in 2 years according to Kelly Blue Book. If she saves no more, what rate must she earn on her $9500 to obtain the car in 2 years?
r= (FV/PV)1/t – 1 = (17500/9500)1/2 – 1 = = 35.72%
BAII Functions: 17.5, FV, 9.5, +/-, PV, 2, N, CPT, I/Y

13. Example 4
My PNC savings deposits currently earn interest at 1.1% APR compounded monthly. How long will it take for my current balance to double if I make no more deposits?
t = ln(FV/PV)/ln(1+r) = ln(2)/ln(1+.011/12) = ln(2)/ln( )
t = / = months OR years
BAII Functions: 2, FV, 1, +/-, PV, , I/Y, CPT, N
Gives you the answer in months
Also, note we can use annual r if we use appropriate EAR:
EAR = ( )12 – 1 =
T = ln(2)/ln( ) = /.0110 = years
BAII Functions: 2, FV, 1, +/-, PV, 1.106, I/Y, CPT, N
Gives you the answer in years

14. Example 5
You work for Bank CNP. A competitor, Bank AB, is offering 2-year small business loans at 6.9% APR, compounded quarterly. If your bank always quotes monthly compounded rates, what monthly compounded APR must your bank offer to equate with Bank AB?
Step 1: What is Bank AB EAR?
Bank AB EAR = (1+.069/4)4 – 1 = ( )4 – 1 = = 7.08%
Step 2: Now, what monthly compounded APR has same EAR?
= (1+APR/12)12 – 1
[ / ]*12 = APR = 6.86%

15. Example 5 (again) Now solve using BAII financial functions
Step 1: Find FV of $1 using Bank AB quarterly rate
BAII Functions: 1, +/-, PV, 1.725, I/Y, 4, N, CPT, FV
Your calculator now shows:
(this lets you know the Bank AB EAR = 7.08%
Step 2: Find monthly compounded rate that yields the same FV of in 1 year
BAII Functions: 1, +/-, PV, , FV, 12, N, CPT, I/Y
Your calculator now shows: which is monthly rate.
Now find *12 = or 6.86%

16. Now solutions not visible

17. Example 6
A firm just issued a ‘zero coupon bond’ that only pays the holder $1,000 in 15 years. What is the value of the bond if the market requires a 15% annually compounded rate of return?
PV = $1000/(1.15)15 = $122.89
BAII Functions: 1000, +/-, FV, 15, I/Y, 15, N, CPT, PV

18. Example 7
Google stock increased 46.5% during Suppose you invest $10,000 today in Google. How much will you have in 5 years if it compounds annually at that rate each year?
FV = $10,000(1.465)5 = $67,482.03
BAII Functions: 10000, +/-, PV, 46.5, I/Y, 5, N, CPT, FV
Note that some students prefer to solve for $1, then multiply answer by 10,000

19. Example 8
Your 3-year old child just received a $3,500 gift for her college education. She will start college in 15 years. Based on inflation estimates, experts believe 4 years of tuition, room, and board will cost $83,000 in 15 years. If you deposit the $3,500 in an ING account earning 4% APR compounded annually, what is the present value of your shortfall? What rate would you need to earn to eliminate the shortfall without further deposits?
FV of $3500 = 3500(1.04)15 = 6,303.30
Shortfall in 15 years = 83,000-6,303.30= 76,696.70
PV of shortfall = 76,696.70/(1.04)15 = 42,586.95
This means that you would need to add $42, today to the $3,500 gift to “fully fund” the college expense that is due in 15 years.
There is zero shortfall when FV=$83,000
r = (83000/3500)1/15 – 1 = 23.5%
You need something much more risky than an ING account!

20. Example 8 (again) What is PV of shortfall?
BAII Functions: 3500, +/-, PV, 4, I/Y, 15, N, CPT, FV
You now have on your calculator.
While it is there, hit: +/-, +, 83000, =
You should now have the FV of your shortfall “76,696.70” showing on your calculator.
While that is there, hit 2nd, FV, and is still showing but TVM is cleared
+/-, FV, 4, I/Y, 15, N, CPT, PV and now calc shows
What rate eliminates shortfall?
BAII Functions: 3500, +/-, PV, 83000, FV, 15, N, CPT, I/Y
Now you should have “23.5” on calc which is 23.5%

21. Example 9
Bank A is offering a loan at 12% APR compounded daily, and Bank B is offering a loan of 12% APR compounded monthly. Who offers best borrowing rate? Bank A or B? (Calculations are not necessary)
Bank B. The effective rate on Bank A > Bank B effective rate since they differ only in their compounding periods.
EARs increase with m!

22. Example 10
Each month, starting 1 month from today, I can make a monthly deposit $350 into an account earning 3% APR compounded monthly. How much will I have saved at the end of the year (12 months from today)?
Given what we have learned in Ch 4, this involves 12 FV calculations! Need to find the sum of FV11, FV10, FV9, …, FV0

23. Example 10 in Excel PV rate periods FV 350 0.0025 11 $359.75 10
$358.85 9
$357.95 8
$357.06 7
$356.17 6
$355.28 5
$354.40 4
$353.51 3
$352.63 2
$351.75 1
$350.88
$350.00
$4,258.23

24. Example 10
There has got to be an easier way to handle cash flows that are constant through time!
That is what we cover (in part) in Ch 5 next week!

25. END.