Chapter 6
Which of the following statements are true? (all else equal) A. For a given time period and future value- the higher the interest rate, the smaller the present value. B. For a given interest rate and future value- the longer the time period, the higher the present value. C. For a given future value and time period- the higher the present value, the lower the rate of return. D. The higher the future value, the higher the rate of return. E. As the number of periods increases, PV decreases. F. PV and r are positive related. G. FV and r are negative related.
a c d e
calculating PV of annuities due
2 STEP PROCESS 1: Calculate the PV in a normal way 2: multiple that answer by (1 + r)
You love your coffee! You purchase coffee 5 times per week at $4 per drink. You do this 52 weeks per year for 45 years but then quit, cold turkey. Instead of spending money on coffee, you consider investing this money each week in retirement portfolio that will return 6% annually. What would be the value of these weekly savings at your coffee retirement date?
6% = EAR! => Want weekly periodic rate => 0.06=(1+r)^52-1 r=? FV=1.06 PV=-1 N=52 => I/Y= 0.11211840% (weekly rate) PMT= 4*5 = 20 N= 45yrs*52 weeks/yr= 2340 I/Y= 0.11211840 => FV= 227,698.78
What's the EAR for a 9% APR with continuous compounding? What's the APR for a continuously compounded EAR of 15%?
APR=9% EAR = e^0.09 -1 = 9.42% EAR=15%: 15%=e^APR-1 -> APR=ln(1.15)=13.98%
APR red flags
"with" "compounded..." "per & with"
Your bank offers one-year CDs with an APR of 4%, compounded quarterly. What APR, compounded monthly, would give the same EAR?
EAR = (1+(.04/4))^4-1 = 4.060% (1+APR/12)^12-1= 4.060% => APR= 3.987%
You could buy an investment that will pay $500 semi-annually for the next year and a half. If the bank offers an APR of 6%, compounded quarterly, what is the most you would be willing to pay for this investment?
Effective Semi-annual Rate = (1+.06/4)^(4/2)-1 = 0.030225 PMT = 500 N = 3 I/Y = 3.0225% -> PV = -1,413.6941
annuity due
If the first payment occurs at the beginning of the period rent, leases when payment starts at time 0
ordinary annuity
If the first payment occurs at the end of the period fixed rate mortgage, car loan when payment starts at time 1
You must choose between the following investments: Investment V will pay $8,650 per year for 6 years or Investment VV will pay $6,920 for 9 years. If the discount rate is 6%, which investment has the higher present value? What if the discount rate is 17%? Rates are annual.
Investment VV at 6%; Investment V at 17% Investment V Investment VV CF0 = 0 CF0 = 0 C01 = 8,650 F01 = 6 C01 = 6,920 F01 = 9 I = 6 => NPV = 42,534.86 I = 6 => 47,067.71 I = 17 => NPV = 31,046.45 I = 17 => 30,797.91
Calculate the present value of an annuity that pays $100 annually with the first payment in 1 year for 8 years. The discount rate is 7% annually. Is this an ordinary annuity or annuity due?
N 8 I/Y 7 PMT 100 PV CPT = 597.13 Is this an ordinary annuity or annuity due? Ordinary annuity
You have received a loan to purchase your first home. The house sold for 350,000 and the bank gives you a mortgage for 75% of the purchase price. The mortgage is a standard 30-year fixed rate mortgage at 3.764% per year compounded monthly. What monthly payment will you face?What is the effective annual rate of interest?
N: 30*12 = 360 MONTHS I/Y: 3.764/12 = .31366667% PER MONTH PV: 350,000 *.75 = 262,500 CPT PMT = 1,217.76 PER MONTH EAR = (1+(.03764/12))^12-1 = 3.829619%
If you invested $10,000 and received back $17,000 fives year later, what annual rate of return would you have received?
PV = -10,000 FV = 17,000 N = 5 => 11.1962%
You are considering 2 securities, both will pay you $9000 ten years from today. Security A earns 6.5% annual interest for 7 years, then 4% annually until maturity. Security B can be purchased 3 years from today for $967.92 less than the future value of Security A at that time. What is the interest rate of Security B?
PVA3=FV/(1+R)^t = 9000/(1.04)^3/(1.065)^4 = 6219.34 PVB3= 6219.34 - 967.92 = 5251.41 PVB3= 5251.41 N = 7 FV = 9000 => I/Y = 8%
Continuous Compounding
Sometimes investments or loans are figured based on continuous compounding EAR = e^APR - 1 m = infinite periodic = epsilon, closest number to 0 without being 0 no rounding necessary on problems because you don't have to wait until the ned of the period to get interest
Multiple Cash Flows in calc
Using the CF keys, CF, 2nd [CLR WORK] CF0 = 0 C01 = 20 ENTER F01 = 1 C02 = 40 ENTER F02 = 1 C03 = 60 ENTER F03 = 1 C04 = 80 ENTER F04 = 1 NPV, I = 12 ENTER CPT is down arrow helpful to make yourself a timeline!
calculating annuities
annuity value it PMT key
EAR ceiling
as m increases, EAR increases with diminishing returns and will eventually reach an asymptote/ceiling
in annuity problems, what drives the equation
c, payment amount
beginning of year 2 vs at year 2
check which part of the year we are working with at is the end
Annuity
finite series of equal payments that occur at regular intervals. countable number of periods downpayment on house/car are not included in the annuity stream
with more than 1 period, the EAR will be strictly ______ than the APR
greater
frequency using cash flows
how many in a row not years, breaking down frequency into periods
perpetuity
infinite series of equal payments occurring at regular intervals PVo = C1 / r payment drives formula, r needs to match payment time going back a period aka r = dividend/price
What is the EAR if you borrow $1,000 for a year and pay back $100 at the end of the first, second and third quarters, and $1,100 at the end of the fourth quarter? And what is the APR?
n= 4 PMT= -100 FV= -1000 PV= 1000 CPT I/Y= 10% (Per quarter) EAR= (1+0.1)^4-1=46.41% APR=0.1*4=40%
I in the cash flows calc
needs to match the periods you use! divide if not taken annual periods
NPV
net present value difference bw the PV of the inflows and the PV of the outflows
growing annuity formulas
on formula sheet
PV lump sum amounts in ordinary vs annuity due
ordinary annuity - the lump sum value is 1 period prior to the cash flow in the stream occurring annuity due - the lump sum is at time 0 - same time as the first cash flow
Annuity due *payment*
ordinary payment / (1+r) ** when calculating payments you need to divide by (1+r), not multiple! bc making the payment sooner, less interest accrues
Effective Annual Rate (EAR)
the actual rate paid (or received) after accounting for compounding that occurs during the year aka APY (annual percentage yield) if you want to compare two alternative investments with different compounding periods you need to compute the EAR (APY) and use that for comparison EAR is the low frequency rate that is equivalent to a higher frequency interest rate -For example, 12.68% per year (low frequency) is equivalent to 1% per month (high frequency). -The Effective Annual Rate is very commonly used, but there can be effective weekly, monthly, quarterly, and semi-annual rates—and more. -A daily interest rate has an effective weekly rate. -A weekly interest rate has an effective monthly rate.
Annual Percentage Rate (APR)
the annual rate quoted by law periodic rate times the number of compounding periods per year represents the simple interest on a loan APR = r x m
C01, C02, etc in cash flows
these are just entries, not a timeline value, because the frequency may change
Congrats you won the lottery! However, to match government budgets, the payoff structure is unique, as follows. Years 1-5: $40,000 at the end of each year Years 6-10: $50,000 at the end of each year Years 11-20: $75,000 at the end of each year If your annual opportunity cost of capital is 5% what is the present value of your winnings? Assuming you invest the proceeds of each payment, how much will you have in 20 years?
CF Function CF0: 0 C01: 40,000 F01: 5 C02: 50,000 F02: 5 C03: 75,000 F03: 10 I: 5 NPV = 698,327.6358 PV: 698,327.6358 I/Y: 5 N: 20 =>CPT FV = 1,852,871.114
Suppose you have the opportunity to invest in a financial vehicle that guarantees the following returns: $10,000 in 2 years, $20,000 in 5 years, and $15,000 in 8 years. Assuming an annual interest rate of 4%, what is the present value of these cash flows?
CF0 = 0 C01 = 0 F01 = 1 C02 = 10,000 F02 = 1 C03 = 0 F03 = 2 C04 = 20,000 F04 = 1 C05 = 0 F05 = 2 C06 = 15,000 F06 = 1 => NPV = 36,644.46
A client has $202,971.39 in an account that earns 8% per year, compounded monthly. The client's 35th birthday was yesterday and she will retire when the account value is $1 million. 1. At what age can she retire if she puts no more money in the account? 2. At what age can she retire if she puts $250 per month into the account every month, beginning one month from today?
1. N (CPT = 240) I/Y .6667 PV -202,971.39 PMT 0 FV 1,000,000 55 years old 2. N (CPT = 220) I/Y .6667 PV -202,971.39 PMT -250 FV 1,000,000 53 years old
EAR formulas to memorize!!
1. [ 1 + (APR/m) ]^m - 1 2. [1+r]^m - 1 both are identical, just matters what you are given
Suppose you are evaluating a project that will cost $20,000 today but will return $5,000 per year for 3 years starting 4 years from the initial investment and $10,000 per year for 2 years after the final $5,000 payment. Assuming a discount rate of 6.5% per annum, what is the value of the project 2 years after the project ends?
CF0 = -20,000 C01 = 0 F01 = 3 C02 = 5000 F02 = 3 C03 = 10,000 F03=2 PV = 3440.06 I/Y = 6.5 NPV, I = 6.5 => NPV = 3,440.06 Then N = 10 => FV = 6457.47
You are valuing an investment that will pay you $12,000 the first year, $14,000 the second year, $17,000 the third year, $19,000 the fourth year, $23,000 the fifth year, and $29,000 the sixth year (all payments are at the end of each year). What is the value of the investment to you now if the appropriate annual discount rate is 11.00%?
CF0=0 C01=12000 C02=14000 C03=17000 C04=19000 C05=23000 C06=29000 I=11 NPV=76273.63
What is the present value of $50,000 to be received 10 years from now if the opportunity cost of capital (discount rate) is 7% for the first 5 years of the investment and 5.0% for the second 5 years? (assume annual compound interest) What is the value of the investment after 2 years? Rates are annual.
FV = 50000 N = 5 I/Y = 5 => PV = 39176.31 FV = 39176.31 N = 5 I/Y = 7 => PV = 27932.17 PV = 27932.17 N = 2 I/Y = 7 => 31979.54
What is the present value of $50,000 to be received 8 years from now if the opportunity cost of capital (discount rate) is 5.0% (annual compounded interest)?
FV = 50000 N = 8 I/Y = 5 => PV = 33841.97
You a make a monthly deposit of $1000, starting today and for the next 100 months, into an account earning an interest rate of 12% APR with monthly compounding.How much will you have in your account right after the final deposit?
PMT= 1000, n=101, r=1, FV? FV = 173,186.20
You deposit $6,000 into a saving account that earns 7% annual interest. How long (in years) will it take to double the money in the account? How many years to double by the Rule of 72?
PV = -6000 FV = 12000 I/Y = 7.0 => N = 10.2448 Rule of 72 = 72 / 7 = 10.2857
You made a deposit of $2,500 6 years ago today, at the annual interest rate of 2% compounded annually. Today, you are told that the interest rate for your deposit has increased, and you will get $4,586 after 10 years. What is the new annual interest rate?
PV=2815.41 n=10 FV=4586 r=? r=5%
FV ordinary annuity vs annuity due
ordinary - lump sum occurs at the same time as the last payment occurs both occur at same time - so the annuity amounts are the same number BUT it is not the same value because ordinary is due at a later time than annuity due, would need to multiple the annuity due by 1 + r to get the real value
