Chapter 5 (cont.)
Chapter 5 (cont.)
Discounted Cash Flow Valuation
Discounted Cash Flow Valuation
1
1.. RReevviieew w PPrroobblleemmss 2.
2. MuMultltipiple le CCasash Fh Fllowowss: C: Comompuputtining Fg FVV,, PV, IR, n
PV, IR, n 3.
3. VaValuluiing ng AnAnnunuititieies & s & PePerprpeetutuititieiess 4
First: A Review Problem #1
First: A Review Problem #1
The company President of Acme The company President of Acme
Computers estimates he needs to Computers estimates he needs to
increase sales by 200% before he is increase sales by 200% before he is profitable.
profitable. Sales Sales are are currently currently $150,000.$150,000. His goal is to
His goal is to be profitable 6 years frombe profitable 6 years from now.
now. What What is is the the lowest lowest growth growth rate rate inin sales that is needed to accomplish his sales that is needed to accomplish his goal and be profitable? (Note 200%
First: A Review Problem #1
First: A Review Problem #1
The company President of Acme The company President of Acme
Computers estimates he needs to Computers estimates he needs to
increase sales by 200% before he is increase sales by 200% before he is profitable.
profitable. Sales Sales are are currently currently $150,000.$150,000. His goal is to
His goal is to be profitable 6 years frombe profitable 6 years from now.
now. What What is is the the lowest lowest growth growth rate rate inin sales that is needed to accomplish his sales that is needed to accomplish his goal and be profitable? (Note 200%
Review Problem #2
Review Problem #2
•• The production manager estimates he willThe production manager estimates he will be producing 500,000 units in year 2020. be producing 500,000 units in year 2020. He bases this on the fact
He bases this on the fact that he estimatesthat he estimates his production will increase 7% per year.
his production will increase 7% per year. How many units did he produce in 2002? How many units did he produce in 2002?
Future Value with Multiple Cash Flows:
There are two ways to calculate future values of
multiple cash flows:
1. Compound the accumulated balance forward
one period at a time, or
2. Calculate the future value of each cash flow
and add them up. **
Present Value of Multiple Cash Flows:
There are two ways to calculate present
values of multiple cash flows:
1. Discount the last amount back one period
and add them up as you go, or
2. discount each amount to time 0 and then
add them all up.**
Another Way
–
Annuity Due
• I suggest you do not mess with your calculator. When I ask you to calculate an annuity due:
- decrease N by 1 and calculate PV
- add 1 payment to your calculated PV
An annuity due calls for the first payment up front. So if you buy an annuity that costs $1000 and pays $100 per period you get $100 back right away.
Present Value for Annuity Cash Flows
Ordinary Annuity - multiple, identical cash flows occurring at the end of each period for a fixed number of periods.
Annuity Present Value: APV = C (1 - [1/(1 + r)t ])/r
(Note: this formula is FYI, as you will normally use the calculator for determining PV and FV of annuities).
Example:
If you are willing to make 36 monthly payments of $100 at 1.5% per period, what size loan (APV) can you obtain? APV = C * [(1 - [1/(1 + r)t ])/r].
C = $100 ; t = 36; r = 1.5%
APV = 100 * [(1-[1/(1+.015)36])/.015] =
100*[(1-[1/1.70914])/.015] = 100 * (.4149/.015) = 100 * 27.66
= $2766 Using this formula gets a bit hairy and it’s
easy to make a mistake.
It is much easier to just use the calculator:
Sweepstakes as Annuity Due
• $333,333.33 per year for 30 years
• First payment is up front.
• $10,000,000 in total payments
• 5% interest rate
• Now what is it worth? Reduce N by 1 and calculate the PV then add one payment to your PV.
Sweepstakes as Annuity Due
-Answer
• N = 30 -1 = 29 because first payment is up front; Interest Rate = 5% ;
• PMT = -333,333.33
• Solve for PV 5,047,024.48 after adjust for sign convention. Now add the first payment of 333,333.33 5,380,357.81
• Versus $5,124,150 for Regular Annuity
• Annuity Due is Always worth more, because each payment is earlier.
Annuities
–
the Calculator Method
• 5 possible buttons: you will use only 4 for any problem and have 3 variables given.
• You use either the PV or FV for a problem but not both, so for any problem you will know or solve for N, Interest Rate, PMT (payment), and FV or PV.
Example: If you borrow $400, promising to repay in 4 monthly installments at 1% a month, how much are your payments? APV = C * [(1 - [1/(1 + r)t ])/r]. $400 = C (1 - [1/(1.01)4])/.01 $400 = C 3.90196, so C = $400/3.90196 C = $102.51.
Or the easy way on the calculator:
Example of calculating n for an annuity:
How many $100 payments will pay off a $5,000 loan at 1% per period?
Answer
1. $5000 Loan; $100 payments ; 1% per month
2. PV = $5000 ; Interest Rate = 1%; PMT = -$100 3. Compute N 69.66
Example: A finance company offers to loan you $1,000 today if you will make 48 monthly
payments of $32.60. What rate is implicit in the loan?
Answer:
1. $1000 today ; 48 payments of $32.60. 2. PV = 1000; N = 48; PMT = -32.60
Future Values for Annuities:
1. Method 1: Discount the payments, then find the
future value: Annuity future value AFV = APV
(1 + r)t.
2. Method 2: AFV = C ((1 + r)t - 1)/r I would
recommend the calculator = Method 3
3. Method 3: Use the calculator: You know n, IR, payment; solve for FV.
Summary of Annuities
• Fixed Payment for a set time period, not forever like a perpetuity.
• You know 3 of 4 things and have to figure out the fourth. You also have to decide if you are dealing with PV or FV.
• Use your calculator to solve. No good method to check other that testing by plugging in your
Perpetuities = Series of level cash flows forever.
Perpetuity present value: PPV=
C/r
,
since
PPV
r
must give payment,
C
.
Preferred stock is an important
example of a perpetuity.
Sample Perpetuity Problem #1
Preferred stock pays a perpetual dividend. If the current market rate is 8% for an investment in the
risk category of ABC Company’s preferred stock,
the price of its common stock is $1000, and the
annual dividend on the preferred stock is $100 with the next dividend payable a year from now, what is
the appropriate market price for ABC’s preferred
Answer #1
• Price of common stock has no impact on preferred stock; I often will give you
extraneous info, which you should ignore.
• PPV = C/r
Sample Perpetuity Problem #2
• Preferred stock pays a perpetual dividend. If the current market price of the preferred stock is $95, the price of its common stock is $75, and the annual dividend (coupon) on the preferred stock is $6.75 with the next dividend payable a year from now,
what is the interest rate for ABC’s
Answer #2
• Price of common stock has no impact on preferred stock; I often will give you
extraneous info, which you should ignore.
• PPV = C/r
Annuity Due: The difference between Annuity Due cash flows and
Ordinary Annuity cash flows is that the Annuity Due cash flows are always made or received at the beginning of each period.
3 ways to solve for the PV or FV of an annuity due.
1. The first method is to use a special annuity key on a financial calculator, and then provide the same inputs as for an ordinary annuity. This is the method I do not recommend as it involves changing calculator settings.
2. The second method called multiplying by the base. To multiply by the base, first solve the entire problem as if it were an ordinary
annuity, and then multiply the solution by (1+r).
3. The third method** recognizes that an Annuity Due is somewhat of a silly concept in that you pay PV for the annuity and then immediately get the first payment back; so its value = the value of Ordinary
Example:
You are leasing an SUV for 36 months at $400 per
month, payable on the first day of each month. If the appropriate interest rate is 0.5% per month, what is the present value of this lease to the finance company?
Let’s try this 2 ways.
First, do it by solving for a regular annuity and then multiplying by (1+r). Second, do it by reducing N by
Multiplication by the base, first set up a financial
calculator solution for an ordinary annuity solution: 36 N, -400 PMT, .5 I/Y. Then compute PV
$13,148.41. Multiply this by 1.005 (which is 1+r)
$13,214.15
Reduce N by 1: 35 = N; -400 = PMT; .5 = interest rate; Compute PV $12,814.15; add $400
Stated or quoted interest rate - rate before
considering any compounding effects, such as 10% compounded quarterly . This is the APR (Annual Percentage Rate). It is also called the nominal
interest rate.
Effective annual interest rate (EAR) - rate, on an annual basis, that reflects compounding effects, e.g., 10% compounded quarterly is effective rate of 10.38% = (1.025)4 - 1.
T5.13 Compounding Periods, EARs, and APRs
Compounding Number of times Effective
period compounded annual rate
Year 1 10.00000% Quarter 4 10.38129 Month 12 10.47131 Week 52 10.50648 Day 365 10.51558 Hour 8,760 10.51703
Calculating and Comparing Effective Annual Rates (EAR)
To get the effective rate, divide the quoted annual rate by
number of periods in a year (semi-annual = 2, quarterly = 4, monthly = 12, etc.), add 1, raise to the number of periods
power, then subtract 1. That is,
EAR = [1 + (quoted rate or APR)/m]m – 1
where m = number of periods per year
This formula is in the Table. (This is one reason that it was helpful to learn the yx key.)
Example:
What is the present value of $100 in two years at 10% compounded quarterly? What is the EAR? We need to figure out the rate at each period being compounded. 10% compounded quarterly = 10%/4 each quarter = 2.5% each quarter. There are 8 quarters in 2 years, so we are compounding 8 times.
PV = FV/(1+r)t
PV = 100/(1+.025)8 PV = $82.07.
Another Example
• 15% interest rate compounded monthly for 3 years. If we start with $200, what do we end with? What is the EAR?
For problems:
• Use the EAR rate & the # of years; or
• Use the APR/m & the appropriate # of periods (m * # of years);
Another Example
• 15% APR interest rate compounded monthly for 3 years. If we start with $200, what do we end with? What is the EAR?
• We are compounding monthly, so the monthly rate is 15%/12 = 1.25%.
• FV = PV (1+r)t
• FV = $200 (1.0125)36 = 200(1.5639) =
$312.79
Annual Percentage Rate (APR) = simply the rate per period number of periods per year, making it a
quoted or stated rate.
APR is the rate quoted in most leases, mortgages, etc. Why is that? (EAR is normally not quoted on these.) However, it understates the effective rate if there is more than 1 period per year.
EAR vs. APR Formula
• EAR = (1+APR/m)m – 1; m = # of times compounded in a year.
• Essentially APR is a quoted annual rate.
To convert it to EAR, you have to know the type of compounding. A 10% APR can be compounded daily, monthly, quarterly,
EAR vs. APR
• APR is a legal term used in contracts. You
normally have to convert it to the rate and time period being compounded. Comparing APRs is inexact. Example: 15% APR compounded daily = 16.18% [calculated as (1+.15/365)365] so it is a
higher rate than 16% compounded annually.
• EAR is the “real” financial rate. Comparing
Loan Types and Loan Amortization
Pure Discount Loans: Borrower pays a single lump sum (principal and interest) at maturity. These are types of
problems we solved for already (4 button method and checked with yx method.)
Example: A U.S. Treasury bill
Interest-Only Loans: Borrower pays interest only each period and entire principal at maturity.
Example: A typical corporate bond
Amortized Loans
Borrower repays part or all of the principal over the life of the loan. Two methods are:
1) fixed amount of principal to be repaid each period, which results in uneven payments, and
2) fixed payment (i.e., an annuity), which results in uneven principal reduction. (Interest decreases and principal
increases as the loan “amortizes” or is paid down with each
payment.)
A traditional automobile loan or fixed-rate home mortgage. These normally have a fixed payment per month (with
Problem: 30-year mortgage
• $100,000 loan
• Fixed rate of APR 6%, compounded
monthly.
• 30 year mortgage
• What is the EAR? What is the monthly payment?
More Loan Problems
Problem #1: You are considering buying a car. The car’s
total cost is $17,500. The salesperson quotes a monthly interest rate of 1% based on 36 monthly payments with the first payment due in 30 days. Excluding taxes and any other fees, what should your monthly payment be?
Problem #2: You are evaluating a car loan. The car’s cost
is $20,500. On a 60-month loan, the quoted payment is $396.32, what is the monthly interest rate? What is the APR? What is the EAR?
Chapter 5
–
Topics to
Remember
• Present Value of MultipleCash Flows (take PV of each one and add up the PVs)
• Annuity vs. Perpetuity
• Annuities – how to find the 4th variable if know 3
• Annuity Due (in advance) vs. Regular Annuity –
how to find the PV of Annuity Due
• What makes an PV or FV
• Perpetuity – 3 variables, find the answer if you know 2; PPV = C/r
• EAR vs. APR
• Be able to compute EARs and APRs
• Types of Loans - Lump Sum - Interest Only
-Amortized Loans – use annuity method for fixed payments
Answer to Review Problem #1
• N=6; PV=150,000;
• FV= PV(1+200%) =150000*(3.00) =
450000
• Compute Interest Rate = 20.09%
• Check using yx method: (1.2009)6 = 3 and 3*150,000 = 450,000
Answer to Review Problem #2
• N= 2020 – 2002 = 18; PV= ?; Interest rate = 7%; FV = 500000
• Compute PV = 147932
• Check using yx method: (1.07)18 = 3.380 and 3.380*147932 = 500,000
Solution to 30-year Mortgage
• $100,000 loan for 30 years, 6% APR,
compounded monthly, what is the monthly payment?
• EAR = (1+.06/12)12 = 1.00512 = 6.17%
• PV = -100,000 ; N = 360 ; 0.5% = interest
