future value present value rates of return amortization time value of money
TRANSCRIPT
Future value Present value Rates of return Amortization
Time Value of Money
Time lines show timing of cash flows.
CF0 CF1 CF3CF2
0 1 2 3i%
Tick marks at ends of periods, so Time 0 is today; Time 1 is the end of Period 1; or the beginning of Period 2.
Time line for a $100 lump sum due at the end of Year 2.
100
0 1 2 Yeari%
Time line for an ordinary annuity of $100 for 3 years.
100 100100
0 1 2 3i%
Time line for uneven CFs -$50 at t = 0 and $100, $75, and $50 at the end of Years 1 through 3.
100 50 75
0 1 2 3i%
-50
What’s the FV of an initial $100 after 3 years if i = 10%?
FV = ?
0 1 2 310%
100
Finding FVs is compounding.
After 1 year:
FV1 = PV + INT1 = PV + PV(i)= PV(1 + i)= $100(1.10)= $110.00.
After 2 years:
FV2 = PV(1 + i)2
= $100(1.10)2
= $121.00.
After 3 years:
FV3 = PV(1 + i)3
= 100(1.10)3
= $133.10.
In general,
FVn = PV(1 + i)n.
Four Ways to Find FVs
Solve the equation with a regular calculator.
Use tables. Use a financial calculator. Use a spreadsheet.
Financial calculators solve this equation:
FVn = PV(1 + i)n.
There are 4 variables. If 3 are known, the calculator will solve for the 4th.
Financial Calculator Solution
Here’s the setup to find FV:
Clearing automatically sets everything to 0, but for safety enter PMT = 0.
Set: P/YR = 1, END
INPUTS
OUTPUT
3 10 -100 0N I/YR PV PMT FV
133.10
10%
What’s the PV of $100 due in 3 years if i = 10%?
Finding PVs is discounting, and it’s the reverse of compounding.
100
0 1 2 3
PV = ?
Solve FVn = PV(1 + i )n for PV:
n
nnn
i+11
FV = i+1
FV =PV
PV = $1001
1.10 = $100 PVIF
= $100 0.7513 = $75.13.
i,n
3
.
Financial Calculator Solution
3 10 0 100N I/YR PV PMT FV
-75.13
Either PV or FV must be negative. HerePV = -75.13. Put in $75.13 today, take out $100 after 3 years.
INPUTS
OUTPUT
If sales grow at 20% per year, how long before sales double?
Solve for n:
FVn = 1(1 + i)n; 2 = 1(1.20)n
Use calculator to solve, see next slide.
20 -1 0 2N I/YR PV PMT FV
3.8
Graphical Illustration:
01 2 3 4
1
2
FV
3.8
Year
INPUTS
OUTPUT
Ordinary Annuity
PMT PMTPMT
0 1 2 3i%
PMT PMT
0 1 2 3i%
PMT
Annuity Due
What’s the difference between an ordinary annuity and an annuity due?
What’s the FV of a 3-year ordinary annuity of $100 at 10%?
100 100100
0 1 2 310%
110 121FV = 331
3 10 0 -100
331.00
Financial Calculator Solution
Have payments but no lump sum PV, so enter 0 for present value.
INPUTS
OUTPUTI/YRN PMT FVPV
What’s the PV of this ordinary annuity?
100 100100
0 1 2 310%
90.91
82.64
75.13248.68 = PV
Have payments but no lump sum FV, so enter 0 for future value.
3 10 100 0
-248.69
INPUTS
OUTPUTN I/YR PV PMT FV
Find the FV and PV if theannuity were an annuity due.
100 100
0 1 2 3
10%
100
3 10 100 0
-273.55
Switch from “End” to “Begin.”Then enter variables to find PVA3 = $273.55.
Then enter PV = 0 and press FV to findFV = $364.10.
INPUTS
OUTPUTN I/YR PV PMT FV
What is the PV of this uneven cashflow stream?0
100
1
300
2
300
310%
-50
4
90.91247.93225.39 -34.15530.08 = PV
Input in “CFLO” register:CF0 = 0
CF1 = 100
CF2 = 300
CF3 = 300
CF4 = -50 Enter I = 10, then press NPV
button to get NPV = 530.09. (Here NPV = PV.)
What interest rate would cause $100 to grow to $125.97 in 3 years?
3 -100 0 125.97
8%
$100 (1 + i )3 = $125.97.
INPUTS
OUTPUT
N I/YR PV PMT FV
Will the FV of a lump sum be larger or smaller if we compound more often, holding the stated I% constant? Why?
LARGER! If compounding is morefrequent than once a year--for example, semiannually, quarterly,or daily--interest is earned on interestmore often.
0 1 2 310%
0 1 2 3
5%
4 5 6
134.01
100 133.10
1 2 30
100
Annually: FV3 = 100(1.10)3 = 133.10.
Semiannually: FV6 = 100(1.05)6 = 134.01.
We will deal with 3 different rates:
iNom = nominal, or stated, or quoted, rate per year.
iPer = periodic rate.
EAR= EFF% = .effective annual
rate
iNom is stated in contracts. Periods per year (m) must also be given.
Examples: 8%; Quarterly 8%, Daily interest (365 days)
Periodic rate = iPer = iNom/m, where m is number of compounding periods per year. m = 4 for quarterly, 12 for monthly, and 360 or 365 for daily compounding.
Examples:8% quarterly: iPer = 8%/4 = 2%.
8% daily (365): iPer = 8%/365 = 0.021918%.
Effective Annual Rate (EAR = EFF%):The annual rate that causes PV to grow to the same FV as under multi-period compounding.Example: EFF% for 10%, semiannual: FV = (1 + iNom/m)m
= (1.05)2 = 1.1025.
EFF% = 10.25% because (1.1025)1 = 1.1025.
Any PV would grow to same FV at 10.25% annually or 10% semiannually.
An investment with monthly payments is different from one with quarterly payments. Must put on EFF% basis to compare rates of return. Use EFF% only for comparisons.
Banks say “interest paid daily.” Same as compounded daily.
How do we find EFF% for a nominal rate of 10%, compounded semiannually?
Or use a financial calculator.
%.25.101025.0
0.105.1
0.12
10.01
1m
i1%EFF
2
2
m
Nom
EAR = EFF% of 10%
EARAnnual = 10%.
EARQ = (1 + 0.10/4)4 – 1 = 10.38%.
EARM = (1 + 0.10/12)12 – 1 = 10.47%.
EARD(360) = (1 + 0.10/360)360 – 1 = 10.52%.
Can the effective rate ever be equal to the nominal rate?
Yes, but only if annual compounding is used, i.e., if m = 1.
If m > 1, EFF% will always be greater than the nominal rate.
When is each rate used?
iNom: Written into contracts, quoted by banks and brokers. Not used in calculations or shownon time lines.
iPer:Used in calculations, shown on time lines.
If iNom has annual compounding,then iPer = iNom/1 = iNom.
(Used for calculations if and only ifdealing with annuities where payments don’t match interest compounding periods.)
EAR = EFF%:
Used to compare returns on investments with different payments per year.
FV of $100 after 3 years under 10% semiannual compounding? Quarterly?
= $100(1.05)6 = $134.01.FV3Q = $100(1.025)12 = $134.49.
FV = PV 1 .+ imnNom
mn
FV = $100 1 + 0.10
23S
2x3
What’s the value at the end of Year 3of the following CF stream if the quoted interest rate is 10%, compounded semiannually?
0 1
100
2 35%
4 5 6 6-mos. periods
100 100
Payments occur annually, but compounding occurs each 6 months.
So we can’t use normal annuity valuation techniques.
1st Method: Compound Each CF
0 1
100
2 35%
4 5 6
100 100.00110.25121.55331.80
FVA3 = 100(1.05)4 + 100(1.05)2 + 100= 331.80.
Could you find FV with afinancial calculator?Yes, by following these steps:
a. Find the EAR for the quoted rate:
2nd Method: Treat as an Annuity
EAR = (1 + ) – 1 = 10.25%. 0.10
22
Or, to find EAR with a calculator:
NOM% = 10.
P/YR = 2.
EFF% = 10.25.
EFF% = 10.25P/YR = 1NOM% = 10.25
3 10.25 0 -100 INPUTS
OUTPUT
N I/YR PV FVPMT
331.80
b. The cash flow stream is an annual annuity. Find kNom (annual) whose EFF% = 10.25%. In calculator,
c.
What’s the PV of this stream?
0
100
15%
2 3
100 100
90.7082.27
74.62247.59
Amortization
Construct an amortization schedulefor a $1,000, 10% annual rate loanwith 3 equal payments.
Step 1: Find the required payments.
PMT PMTPMT
0 1 2 310%
-1,000
3 10 -1000 0 INPUTS
OUTPUT
N I/YR PV FVPMT
402.11
Step 2: Find interest charge for Year 1.
INTt = Beg balt (i)INT1 = $1,000(0.10) = $100.
Step 3: Find repayment of principal in Year 1.
Repmt = PMT – INT = $402.11 – $100 = $302.11.
Step 4: Find ending balance afterYear 1.
End bal = Beg bal – Repmt = $1,000 – $302.11 = $697.89.
Repeat these steps for Years 2 and 3to complete the amortization table.
Interest declines. Tax implications.
BEG PRIN ENDYR BAL PMT INT PMT BAL
1 $1,000 $402 $100 $302 $698
2 698 402 70 332 366
3 366 402 37 366 0
TOT 1,206.34 206.34 1,000
$
0 1 2 3
402.11Interest
302.11
Level payments. Interest declines because outstanding balance declines. Lender earns10% on loan outstanding, which is falling.
Principal Payments
Amortization tables are widely used--for home mortgages, auto loans, business loans, retirement plans, etc. They are very important!
Financial calculators (and spreadsheets) are great for setting up amortization tables.
On January 1 you deposit $100 in an account that pays a nominal interest rate of 10%, with daily compounding (365 days).
How much will you have on October 1, or after 9 months (273 days)? (Days given.)
iPer = 10.0% / 365= 0.027397% per day.
FV = ?
0 1 2 273
0.027397%
-100
Note: % in calculator, decimal in equation.
FV = $100 1.00027397 = $100 1.07765 = $107.77.
273273
...
273 -100 0
107.77
INPUTS
OUTPUT
N I/YR PV FVPMT
iPer = iNom/m= 10.0/365= 0.027397% per day.
Enter i in one step.Leave data in calculator.
Now suppose you leave your money in the bank for 21 months, which is 1.75 years or 273 + 365 = 638 days.
How much will be in your account at maturity?
Answer: Override N = 273 with N = 638.FV = $119.10.
iPer = 0.027397% per day.
FV = 119.10
0 365 638 days
-100
FV = $100(1 + .10/365)638
= $100(1.00027397)638
= $100(1.1910)= $119.10.
......
You are offered a note that pays $1,000 in 15 months (or 456 days) for $850. You have $850 in a bank that pays a 7.0% nominal rate, with 365 daily compounding, which is a daily rate of 0.019178% and an EAR of 7.25%. You plan to leave the money in the bank if you don’t buy the note. The note is riskless.
Should you buy it?
3 Ways to Solve:
1. Greatest future wealth: FV2. Greatest wealth today: PV3. Highest rate of return: Highest EFF%
iPer = 0.019178% per day.
1,000
0 365 456 days
-850
......
1. Greatest Future Wealth
Find FV of $850 left in bank for15 months and compare withnote’s FV = $1,000.
FVBank = $850(1.00019178)456
= $927.67 in bank.
Buy the note: $1,000 > $927.67.
456 -850 0
927.67
INPUTS
OUTPUT
N I/YR PV FVPMT
Calculator Solution to FV:
iPer = iNom/m= 7.0/365= 0.019178% per day.
Enter iPer in one step.
2. Greatest Present Wealth
Find PV of note, and comparewith its $850 cost:
PV = $1,000/(1.00019178)456
= $916.27.
456 .019178 0 1000
-916.27
INPUTS
OUTPUT
N I/YR PV FV
7/365 =
PV of note is greater than its $850 cost, so buy the note. Raises your wealth.
PMT
Find the EFF% on note and compare with 7.25% bank pays, which is your opportunity cost of capital:
FVn = PV(1 + i)n
$1,000 = $850(1 + i)456
Now we must solve for i.
3. Rate of Return
456 -850 0 1000
0.035646% per day
INPUTS
OUTPUT
N I/YR PV FVPMT
Convert % to decimal:
Decimal = 0.035646/100 = 0.00035646.
EAR = EFF% = (1.00035646)365 – 1 = 13.89%.
Using interest conversion:
P/YR = 365.
NOM% = 0.035646(365) = 13.01.
EFF% = 13.89.
Since 13.89% > 7.25% opportunity cost,buy the note.