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**CHAPTER 5 Time Value of Money**

Future value Present value Annuities Rates of return Amortization

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**Last week Objective of the firm Business forms Agency conflicts**

Capital budgeting decision and capital structure decision

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**The plan of the lecture Time value of money concepts**

present value (PV) discount rate/interest rate (r) Formulae for calculating PV of perpetuity annuity Interest compounding How to use a financial calculator

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**Financial choices with time**

Which would you rather receive? $1000 today $1040 in one year Both payments have no risk, that is, there is 100% probability that you will be paid

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**Financial choices with time**

Why is it hard to compare ? $1000 today $1040 in one year This is not an “apples to apples” comparison. They have different units $1000 today is different from $1000 in one year Why? A cash flow is time-dated money

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**Present value To have an “apple to apple” comparison, we**

convert future payments to the present values or convert present payments to the future values This is like converting money in Canadian $ to money in US $.

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Some terms Finding the present value of some future cash flows is called discounting. Finding the future value of some current cash flows is called compounding.

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**What is the future value (FV) of an initial $100 after 3 years, if i = 10%?**

Finding the FV of a cash flow or series of cash flows is called compounding. FV can be solved by using the arithmetic, financial calculator, and spreadsheet methods. FV = ? 1 2 3 10% 100

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**Solving for FV: The arithmetic method**

After 1 year: FV1 = c ( 1 + i ) = $100 (1.10) = $110.00 After 2 years: FV2 = c (1+i)(1+i) = $100 (1.10) =$121.00 After 3 years: FV3 = c ( 1 + i )3 = $100 (1.10) =$133.10 After n years (general case): FVn = C ( 1 + i )n

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**Set up the Texas instrument**

2nd, “FORMAT”, set “DEC=9”, ENTER 2nd, “FORMAT”, move “↓” several times, make sure you see “AOS”, not “Chn”. 2nd, “P/Y”, set to “P/Y=1” 2nd, “BGN”, set to “END” P/Y=periods per year, END=cashflow happens end of periods

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**Solving for FV: The calculator method**

Solves the general FV equation. Requires 4 inputs into calculator, and it will solve for the fifth. 3 10 -100 INPUTS N I/YR PV PMT FV OUTPUT 133.10

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**What is the present value (PV) of $100 received in 3 years, if i = 10%?**

Finding the PV of a cash flow or series of cash flows is called discounting (the reverse of compounding). The PV shows the value of cash flows in terms of today’s worth. 1 2 3 10% PV = ? 100

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**Solving for PV: The arithmetic method**

i: interest rate, or discount rate PV = C / ( 1 + i )n PV = C / ( 1 + i )3 = $100 / ( 1.10 )3 = $75.13

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**Solving for PV: The calculator method**

Exactly like solving for FV, except we have different input information and are solving for a different variable. 3 10 100 INPUTS N I/YR PV PMT FV OUTPUT -75.13

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Solving for N: If your investment earns interest of 20% per year, how long before your investments double? 20 -1 2 INPUTS N I/YR PV PMT FV OUTPUT 3.8

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**Solving for i: What interest rate would cause $100 to grow to $125**

Solving for i: What interest rate would cause $100 to grow to $ in 3 years? 3 -100 125.97 INPUTS N I/YR PV PMT FV OUTPUT 8

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**Now let’s study some interesting patterns of cash flows…**

Annuity Perpetuity

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**ordinary annuity and annuity due**

PMT 1 2 3 i% PMT 1 2 3 i% Annuity Due

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**Value an ordinary annuity**

Here C is each cash payment n is number of payments If you’d like to know how to get the formula below (not required), see me after class.

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**Solving for FV: 3-year ordinary annuity of $100 at 10%**

$100 payments occur at the end of each period. Note that PV is set to 0 when you try to get FV. 3 10 -100 INPUTS N I/YR PV PMT FV OUTPUT 331

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**Solving for PV: 3-year ordinary annuity of $100 at 10%**

$100 payments still occur at the end of each period. FV is now set to 0. 3 10 100 INPUTS N I/YR PV PMT FV OUTPUT

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Example you win the $1million dollar lottery! but wait, you will actually get paid $50,000 per year for the next 20 years if the discount rate is a constant 7% and the first payment will be in one year, how much have you actually won?

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**Solving for FV: 3-year annuity due of $100 at 10%**

$100 payments occur at the beginning of each period. FVAdue= FVAord(1+i) = $331(1.10) = $ Alternatively, set calculator to “BEGIN” mode and solve for the FV of the annuity: BEGIN 3 10 -100 INPUTS N I/YR PV PMT FV OUTPUT 364.10

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**Solving for PV: 3-year annuity due of $100 at 10%**

$100 payments occur at the beginning of each period. PVAdue= PVAord(1+I) = $248.69(1.10) = $ Alternatively, set calculator to “BEGIN” mode and solve for the PV of the annuity: BEGIN 3 10 100 INPUTS N I/YR PV PMT FV OUTPUT

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**What is the present value of a 5-year $100 ordinary annuity at 10%?**

Be sure your financial calculator is set back to END mode and solve for PV: N = 5, I/YR = 10, PMT = 100, FV = 0. PV = $379.08

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**What if it were a 10-year annuity? A 25-year annuity? A perpetuity?**

N = 10, I/YR = 10, PMT = 100, FV = 0; solve for PV = $ 25-year annuity N = 25, I/YR = 10, PMT = 100, FV = 0; solve for PV = $ Perpetuity (N=infinite) PV = PMT / i = $100/0.1 = $1,000.

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What is the present value of a four-year annuity of $100 per year that makes its first payment two years from today if the discount rate is 9%? $297.22 $323.97 $ $ $ $100

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**What is the PV of this uneven cash flow stream?**

100 1 300 2 3 10% -50 4 90.91 247.93 225.39 -34.15 = PV

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**Solving for PV: Uneven cash flow stream**

Input cash flows in the calculator’s “CF” register: CF0 = 0 CF1 = 100 CF2 = 300 CF3 = 300 CF4 = -50 Enter I/YR = 10, press NPV button to get NPV = $ (Here NPV = PV.)

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**Detailed steps (Texas Instrument calculator)**

To clear historical data: CF, 2nd ,CE/C To get PV: CF , ↓,100 , Enter , ↓,↓ ,300 , Enter, ↓,2, Enter, ↓, 50, +/-,Enter, ↓,NPV,10,Enter, ↓,CPT “NPV= ”

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**The Power of Compound Interest**

A 20-year-old student wants to start saving for retirement. She plans to save $3 a day. Every day, she puts $3 in her drawer. At the end of the year, she invests the accumulated savings ($1,095=$3*365) in an online stock account. The stock account has an expected annual return of 12%. How much money will she have when she is 65 years old?

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**Solving for FV: Savings problem**

If she begins saving today, and sticks to her plan, she will have $1,487, when she is 65. 45 12 -1095 INPUTS N I/YR PV PMT FV OUTPUT 1,487,262

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**Solving for FV: Savings problem, if you wait until you are 40 years old to start**

If a 40-year-old investor begins saving today, and sticks to the plan, he or she will have $146, at age 65. This is $1.3 million less than if starting at age 20. Lesson: It pays to start saving early. 25 12 -1095 INPUTS N I/YR PV PMT FV OUTPUT 146,001

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Will the FV of a lump sum be larger or smaller if compounded more often, holding the stated i% constant? LARGER, as the more frequently compounding occurs, interest is earned on interest more often. 1 2 3 10% 100 133.10 Annually: FV3 = $100(1.10)3 = $133.10 1 2 3 5% 4 5 6 134.01 100 Semiannually: FV6 = $100(1.05)6 = $134.01

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**What is the FV of $100 after 3 years under 10% semiannual compounding**

What is the FV of $100 after 3 years under 10% semiannual compounding? Quarterly compounding?

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**Classifications of interest rates**

1. Nominal rate (iNOM) – also called the APR, quoted rate, or stated rate. An annual rate that ignores compounding effects. Periods must also be given, e.g. 8% Quarterly. 2. Periodic rate (iPER) – amount of interest charged each period, e.g. monthly or quarterly. iPER = iNOM / m, where m is the number of compounding periods per year. e.g., m = 12 for monthly compounding.

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**Classifications of interest rates**

3. Effective (or equivalent) annual rate (EAR, also called EFF, APY) : the annual rate of interest actually being earned, taking into account compounding. If the interest rate is compounded m times in a year, the effective annual interest rate is

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**Example, EAR for 10% semiannual investment**

An investor would be indifferent between an investment offering a 10.25% annual return, and one offering a 10% return compounded semiannually.

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**EAR on a Financial Calculator**

Texas Instruments BAII Plus keys: description: [2nd] [ICONV] Opens interest rate conversion menu [↑] [C/Y=] 2 [ENTER] Sets 2 payments per year [↓][NOM=] 10 [ENTER] Sets 10 APR. [↓] [EFF=] [CPT] 10.25

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**Why is it important to consider effective rates of return?**

An investment with monthly payments is different from one with quarterly payments. Must use EAR for comparisons. If iNOM=10%, then EAR for different compounding frequency: Annual % Quarterly % Monthly % Daily %

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**If interest is compounded more than once a year**

EAR (EFF, APY) will be greater than the nominal rate (APR).

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**What’s the FV of a 3-year $100 annuity, if the quoted interest rate is 10%, compounded semiannually?**

1 100 2 3 5% 4 5 6 Payments occur annually, but compounding occurs every 6 months. Cannot use normal annuity valuation techniques.

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**Method 1: Compound each cash flow**

1 100 2 3 5% 4 5 6 110.25 121.55 331.80 FV3 = $100(1.05)4 + $100(1.05)2 + $100 FV3 = $331.80

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**Method 2: Financial calculator**

Find the EAR and treat as an annuity. EAR = ( / 2 )2 – 1 = 10.25%. 3 10.25 -100 INPUTS N I/YR PV PMT FV OUTPUT 331.80

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**When is periodic rate used?**

iPER is often useful if cash flows occur several times in a year.

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