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

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”

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=110.0000 10% APR compounded daily: .1/365 = .000274, FV = 100(1.000274)365= 110.5167 Therefore, 10% APR compounded daily has an effective annual rate of 10.5167% That is, FV = 100(1.105167)1 = 110.5167

APYs APYs = EARs APY = “Annual Percentage Yield” http://www.ally.com/cds/apy-vs-apr-what-is-apr-what-is-apy/ Some banks state rates as APRs while others use APYs or EARs

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

As of Sept 7, 2015 www.bankrate.com

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 = 78.85 Note: 10,000(1.0039)2 = 78.15 Can’t just use r2 – r1 =0.0039 to get the difference Difference = 1o,000(r22 +2(r2-r1)-r12) = 78.85

Examples

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

Example 1 How much money do I need to deposit today in an account earning 12% APR, compounded annually, to have $4,250.00 in 4 years? PV = 4250/(1.12)4 = 4250/1.5735 = 2700.95 *BTW: PV = 4250*(1.12)-4 = 2700.95 BAII Functions: 4250,+/-,FV,12,I/Y,4,N,CPT,PV Conceptual extension: Would I need to deposit more or less than $2700.95 if the rate was 12% APR compounded monthly? Less! Because I’m earning interest at a higher EAR.

Example 2 Today the average cost of a box of cereal is $3.59. 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

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 = .3572 = 35.72% BAII Functions: 17.5, FV, 9.5, +/-, PV, 2, N, CPT, I/Y

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(1.000917) t = .69315/.00091658 = 756.23 months OR 63.02 years BAII Functions: 2, FV, 1, +/-, PV, .091658, I/Y, CPT, N Gives you the answer in months Also, note we can use annual r if we use appropriate EAR: EAR = (1.000917)12 – 1 = .01106 T = ln(2)/ln(1.01106) = .69315/.0110 = 63.0136 years BAII Functions: 2, FV, 1, +/-, PV, 1.106, I/Y, CPT, N Gives you the answer in years

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 = (1.01725)4 – 1 = 1.0708-1 = 7.08% Step 2: Now, what monthly compounded APR has same EAR? 0.0708 = (1+APR/12)12 – 1 [1.07081/12 - 1]*12 = APR = 6.86%

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: 1.070805995 (this lets you know the Bank AB EAR = 7.08% Step 2: Find monthly compounded rate that yields the same FV of 1.070806 in 1 year BAII Functions: 1, +/-, PV, 1.070806, FV, 12, N, CPT, I/Y Your calculator now shows: 0.571725 which is monthly rate. Now find .00571725*12 = .0686 or 6.86%

Now solutions not visible

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

Example 7 Google stock increased 46.5% during 2007. 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

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 $42,586.95 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!

Example 8 (again) What is PV of shortfall? BAII Functions: 3500, +/-, PV, 4, I/Y, 15, N, CPT, FV You now have 6303.3 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 76696.7 is still showing but TVM is cleared +/-, FV, 4, I/Y, 15, N, CPT, PV and now calc shows 42586.95 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%

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!

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

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

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!

END.