# 3 - DCF

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```					Discounted Cash Flow
Valuation
FINA 111, Spring 2008, Week 3
RWJ, Chapter 5

1
Chapter 5 learning objectives
• Calculate the future value and present value of
a series of multiple cash flows
• Calculate annuity payments and loan payments
• Calculate the interest rate on an annuity or loan
• Calculate the number of payments in an
annuity or loan
• Explain how mortgage loans are amortized or
paid off
• Calculate the remaining balance, interest paid
and principal repaid on a mortgage loan
• Calculate effective annual interest rates
2
Chapter 5 outline
• Future and present values of multiple
cash flows
• Valuing level cash flows: annuities and
perpetuities
• Comparing interest rates: the effect of
compounding periods
• Loan types and loan amortization

3
Discounted Cash Flow Topics
DCF
valuation

Single cash flows                                             Multiple
(Ch4)                                                  cash flows

Unequal                        Equal cash                  Comparing                    Types of
cash flows                        flows                       rates                       loans

Pure     Interest
Future Value    Present Value   Perpetuities     Annuities     APR               EAR                         Amortized
discount     only

4
FV of unequal cash flows: example
• Suppose you invest \$500 in a mutual fund
today and \$600 in one year. If the fund
pays 9% annually, how much will you have
in two years?
FV = \$500 × 1.092 + \$600 × 1.09 = \$1,248.05

Time      0               1                         2
Line    500            600                       FV?
\$600 × 1.09 =     \$654.00
\$500 × 1.092 =                      594.05
\$1,248.05
5
FV of unequal CFs: text example 5.1
• You have \$7,000 in the bank now, and you put in
\$4,000 more at the end of each of the next three
years. The bank pays 8% interest p.a. How much
will you have in three years? In four years?
• Calculate how much you will have at the end of
each of the three years:
End of year 1:     \$7,000 × 1.08 + \$4,000 = \$11,560.00
End of year 2:    \$11,560 × 1.08 + \$4,000 = \$16,484.80
End of year 3: \$16,484.80 × 1.08 + \$,4000 = \$21,803.58
• The value in 4 years = \$21,803.58 × 1.08
= \$23,547.87
6
Example 5.1 done a different way
• You have \$7,000 in the bank now, and you put in
\$4,000 more at the end of each of the next three
years. The bank pays 8% interest p.a. How much
will you have in three years? In four years?
• First, find the value of each cash flow at the end of
year 3 and then sum these values
Cash flow today: FV      = \$7,000 × 1.083   = \$8,817.98
Cash flow year 1: FV     = \$4,000 × 1.082   =   4,665.60
Cash flow year 2: FV     = \$4,000 × 1.08    =   4,320.00
Cash flow year 3: FV     =                  =   4,000.00
Total value in 3 years                      = \$21,803.58
• The value in 4 years = \$21,803.58 × 1.08
= \$23,547.87
7
Discounted Cash Flow Topics
DCF
valuation

Single cash flows                                            Multiple
(Ch4)                                                 cash flows

Unequal                       Equal cash                  Comparing                    Types of
cash flows                       flows                       rates                       loans

Pure     Interest
Future Value   Present Value   Perpetuities     Annuities     APR               EAR                         Amortized
discount     only

8
PV of unequal CFs: text example 5.3
• What should you pay for an investment that pays
\$200, \$400, \$600 and \$800 in four consecutive years
if you require a 12% rate of return?
• Find the PV of each cash flow and then sum the PVs
PV of CF in year 1: \$200 / 1.121   =   \$178.57
PV of CF in year 2: \$400 / 1.122   =    318.88
PV of CF in year 3: \$600 / 1.123   =    427.07
PV of CF in year 4: \$800 / 1.124   =    508.41
PV of investment               = \$1,432.93
• This is the most you should pay for an investment
offering these cash flows
9
Timeline for example 5.3
0         1        2         3      4    Time (years)

200     400        600   800   Cash flow
178.57

318.88

427.07

508.41
1432.93   PV of all four cash flows

10
Calculator example, PV of unequal CFs
• Your broker offers you an investment opportunity
– Invest today:       \$100
– Receive in 1 year:   \$40
– Receive in 2 years: \$75
• You require a 15% return. Should you invest?
• Calculator solution
Press CF; 2nd ; CLR WORK
CF0 = 0 Enter Press
C01 = 40 Enter Press
F01 = 1 Enter Press
C02 = 75 Enter Press
F02 = 1 Enter Press
Enter NPV
I = 15 Enter Press
NPV appears      Press CPT = \$91.49
• No – to you the investment is worth only \$91.49. This is less
than the amount that must be invested (\$100).             11
• You can use the PV or FV functions in
Excel to find the present value or future
value of a set of cash flows
• Click on the Excel icon for a spreadsheet
example for multiple unequal cash flows
(optional)

12
Discounted Cash Flow Topics
DCF
valuation

Single cash flows                                            Multiple
(Ch4)                                                 cash flows

Unequal                       Equal cash                  Comparing                    Types of
cash flows                       flows                       rates                       loans

Pure     Interest
Future Value    Present Value   Perpetuities    Annuities     APR               EAR                         Amortized
discount     only

13
Perpetuities and annuities
Perpetuity         Annuity
1 Equal cash flows                         Yes              Yes
2 Regular intervals                        Yes              Yes
3 Duration                               Infinite          Finite

• If the first payment occurs at the end of the period, it is
called an ordinary annuity
• If the first payment occurs at the beginning of the period, it
is called an annuity due

Unless otherwise specified, all annuities are
assumed to be ordinary annuities
14
Perpetuity example 1
• Suppose a preferred stock pays a
quarterly dividend of \$2 per share. If an
investor requires an annual return of 8
percent, what price will he or she be
willing to pay for a share of the stock?
• PV = C / r where
C = cash flow (payment) per period
r = rate of return per period
C = \$2 × 4 = \$8 (annual dividends)
PV = \$8 / 0.08 = \$100
15
Perpetuity example 2
• Suppose the Fellini Co. wants to sell preferred stock
at \$100 per share
– A similar issue of preferred stock currently outstanding
sells for \$40 per share and has a quarterly dividend of \$1.
– What dividend will Fellini have to offer if the preferred
stock is going to sell at a fair price?
• Current required return:
PV = C / r    =>     r = C / PV
r = \$1/\$40 = 0.025 or 2.5% per quarter
• Dividend for new preferred stock:
\$100 = C / 0.025
C = \$100 / 0.025 = \$2.50 per quarter
16
PV of annuity: text example 5.5
• You can afford to pay \$632 per month to repay a 48-
month loan. The bank charges interest of 1% per
month on the loan. How much can you borrow?
– you borrow money TODAY so you need to compute the
present value (PV)
Calculator (using TVM function):
Press 2nd , CLR TVM, CE/C
Enter 48, press N
Enter 1, press I/Y
Enter -632, press PMT
Press CPT => PV = \$23,999.54
17
Illustration: saving for retirement
• You want to put some money away for retirement.
Starting 40 years from now, you want to receive five
equal annual payments of \$25,000. How much
should you invest today if you desire a 12% return?
0 1 2     …         39     40   41    42    43     44    Year

0 0 0     …          0    25K 25K 25K        25K 25K CF
PV

1       25 k    25 k   25 k   25 k   25 k 
PV                                              
1.12 39    1.12 1 1.12 2 1.12 3 1.12 4 1.12 5 

PV = 0.0120364  \$90,119.41 = \$1,084.71
What is 0.012? What is 3.6048?                       18
Saving for retirement-use calculator
• You want to put some money away for
retirement. Starting 40 years from now, you
want to receive five equal annual payments of
\$25,000. How much should you invest today if
you desire a 12% return?
• Use cash flow keys:
CF, 2nd , CLR WORK
CF0 = 0 Enter
C01 = 0 Enter
F01 = 39 Enter
C02 = 25000 Enter
F02 = 5 Enter
Press NPV
I = 12 Enter
Press CPT NPV = \$1,084.71
19
FV of annuity example
• Suppose you begin saving for your retirement by
depositing \$2,000 per year in a savings account.
• If the interest rate is 7.5%, how much will you have
in 40 years?
0       1       2     3       …       40    Year
- 2,000 -2,000 2,000      …    -2,000   CF
FV?

Use calculator
40 N; 7.5 I/Y; -2000 PMT; CPT FV = \$454,513.04

20
• The PV and FV Excel spreadsheet functions
may be used to calculate the present value and
future value of annuities
• The PMT function may be used to calculate
annuity payments
• Click on the Excel icon to see spreadsheet
annuity examples

(optional)

21
Finding an annuity payment
• Suppose you pay \$200,000 for an income
annuity that pays 8% per year, compounded
monthly (8% ÷ 12 = 0.66667% per month). If
you receive 48 monthly payments, what is the
amount of each payment?

Use calculator
48 N       8 ÷ 12 = 0.66667 I/Y;
-200000 PV      CPT PMT = \$4,882.58

22
Finding the number of annuity
payments: text example 5.6
• You charge \$1,000 to your credit card. The bank
charges 1.5% per month on unpaid card balances.
If you make the minimum payment of \$20 per
month, how long will it take to pay off the \$1,000

Use calculator
1.5 I/Y     -20 PMT
1000 PV       CPT N = 93.11 months

• And this is only if you don’t charge anything
more on the card!
23
Finding the number of loan
payments: illustration
• Suppose you borrow \$2,000 at 5% and
repay it by making annual payments of
\$734.42. How long will it be before you pay
off the loan?
Use calculator
5 I/Y          2000 PV
-734.42 PMT            CPT N = 3.0

It will take 3 years to pay off the loan
24
Finding the rate on an annuity
• Suppose you borrow \$10,000. You agree to pay
\$207.58 per month for 60 months. What is the
monthly interest rate? The annual interest rate?
0             1        . . .            60
10,000       207.58                    207.58

Use calculator
60 N;     -207.58 PMT;      10000 PV;
CPT I/Y = 0.75% per month (0.75% × 12 = 9% per annum)

25
• Other TVM functions can be found in
(optional)
–   PMT(rate,nper,pv,fv)
–   NPER(rate,pmt,pv,fv)
–   RATE(nper,pmt,pv,fv)
–   The same sign convention holds here as for
the PV and FV functions
• Click on the Excel icon for spreadsheet
examples of these annuity functions
26
PV of annuity due
0        1        2         3             0       1          2          3
100      100       100           100     100        100

Ordinary Annuity                           Annuity Due
100       100       100                            100       100
PV0                                    PV0  100             
1  r  1  r 2 1  r 3                      1  r 1 1  r 2

Note that each cash flow is discounted for 1 more period in an
ordinary annuity compared to an annuity due.
Therefore, we have the following annuity PV relation:

PVannuity due = (1 + r) PVordinary annuity
27
Annuities due and the calculator
• You can use the PMT key on the calculator
for the equal payments
• The sign convention still holds
• Ordinary annuity versus annuity due
– you can switch your calculator between the two types by
pressing 2nd BGN 2nd Set on the TI BA-II Plus
calculator
– if you see “BGN” or “Begin” in the display of your
calculator, you have it set for an annuity due

Most problems are ordinary annuities
28
PV of annuity due—example
• Suppose you rent a flat for three years. You pay
\$10,000 today when you sign the contract and then
pay the same amount at the beginning of each of
the next two years. What is the present value of
this rental agreement if the interest rate is 10%?
0        1         2       3
-10,000   -10,000   -10,000
PV = ?

Use calculator (press: 2nd BGN; 2nd SET; 2nd QUIT)
3 N; 10 I/Y; -10000 PMT; CPT PV = \$27,355.37

29
Annuity PV: sweepstakes example
• Suppose you win the ‘Mark Six 六合彩’ \$10 million
sweepstakes and the money is paid in equal annual
installments of \$333,333.33 over 30 years. If the
appropriate discount rate is 5% and the first installment is
paid today, how much is the sweepstakes actually worth
today?

0      1     2     …      29      Years
PV    333,333               …    333,333      r = 5%, t = 30

Use calculator (2nd BGN; 2nd SET; 2nd QUIT)
30 N; 5 I/Y; -333333.33 PMT; CPT PV = \$5,380,357.81

30
FV of annuity due
0         1         2        3                0         1        2         3
100       100      100              100       100      100

Ordinary Annuity                              Annuity Due
FV3  1001  r   1001  r   100   FV3  1001  r   1001  r   1001  r 
2            1                           3             2

Note that each cash flow is compounded for 1 more period in
an annuity due compared to an ordinary annuity.

Therefore, we have the following annuity FV relation:

FVannuity due = (1 + r) FVordinary annuity
31
Discounted Cash Flow Topics
DCF
valuation

Single cash flows                                        Multiple
(Ch4)                                             cash flows

Unequal                   Equal cash                 Comparing                    Types of
cash flows                   flows                      rates                       loans

Pure     Interest
FV                PV   Perpetuities     Annuities     APR              EAR                         Amortized
discount     only

32
Annual percentage rate (APR)
• This is the annual rate that is usually quoted by the
lender
– e.g. 12% p.a.
• By definition, APR = the interest rate charged per
period rate × the number of compounding periods
per year
• Consequently, to get the period rate we can rearrange
the APR equation:
Period rate = APR ÷ number of periods per year
12 periods per year => Period rate = 12% ÷ 12 = 1%

33
Computing APRs
• What is the APR if the monthly rate is
0.5%?
0.5% × 12 = 6%
• What is the APR if the semiannual rate is
0.5%?
0.5% × 2 = 1%
• What is the monthly (periodic) rate if the
APR is 12% with monthly compounding?
12% ÷ 12 = 1%
34
Things to remember
• You ALWAYS need to make sure that the interest
rate and the time periods match
– if you have annual periods, you need an annual rate

– if you have monthly periods, you need a monthly rate

• If you have an APR based on monthly
compounding, you have to use monthly periods for
payments

35
Effective annual rate (EAR)
• This is the actual interest rate paid (or received)
after taking into account any compounding that
occurs during the year
– It is the interest rate expressed as if it were compounded
once per year
• If you want to compare two alternative
investments with different compounding periods,
you need to compute their EARs and use those
rates for comparison
You should NEVER divide the effective rate by the number
of periods per year – it will NOT give you the period rate
36
Computing EARs - examples
• Suppose you can earn 1% per month on \$1
invested today
– What is the APR? 1% × 12 = 12%
– How much are you effectively earning annually?
• FV = \$1 × 1.0112 = \$1.12683
• EAR = (\$1.12683 / \$1) - \$1 = 0.12683 = 12.683%
• Suppose you put \$1 in another account where
you can earn 3% per quarter
– What is the APR? 3% × 4 = 12%
– How much are you effectively earning annually?
• FV = \$1 × 1.034 = \$1.12551
• EAR = (\$1.12551 / \$1) - \$1 = 0.12551 = 12.551%
37
EAR decisions: example
• You are looking at two savings accounts. One
pays 5.25%, with daily compounding. The other
pays 5.30% with semiannual compounding.
Which account should you choose?
– First account:
EAR = (1 + 0.0525/365)365 – 1 = 0.05390 = 5.390%
– Second account:
EAR = (1 + 0.053/2)2 – 1 = 0.05370 = 5.370%
• Which account should you choose and why?

38
EAR decisions: example, cont
• Let’s verify the choice. Suppose you invest \$100
in each account. How much will you have in
each account in one year?
– First account:
• Daily rate = 0.0525 / 365 = 0.00014383562
• FV = \$100 × 1.00014383562365 = \$105.38987 = \$105.39
– Second account:
• Semiannual rate = 0.0530 / 2 = 0.0265
• FV = \$100 ×1.02652 = \$105.37023 = \$105.37

• You will have more money in the first account
39
Computing APRs from EARs
• If you have an effective rate, how can you
compute the APR? Rearrange the EAR
equation and you get:
m
    APR 
EAR  1                 1
     m 

APR  m (1  EAR) 1m - 1

                

m: number of compounding periods per year

No need to memorize these formulas   40
APR: example
• Suppose you want to earn an effective rate
of 12% and your money is compounded on
a monthly basis. What APR must be paid?

APR  12 (1  .12 )
1 / 12

 1  .113 865
or 11.39%

• Now, redo the problem assuming semi-
annual compounding
                       
APR  2 (1  .12 )1/ 2  1  .116601
or 11.66%
41
APR, EAR and the calculator
• The TI BA II Plus contains a built-in function
to convert APR to EAR and vice versa
• e.g. What is the EAR when the APR is 12%
compounded monthly?
Note: the calculator uses NOM instead of APR
(NOM stand for “nominal”, a term typically applied to interest rates
that are not adjusted for inflation)

Use calculator
2nd ICONV NOM = 0.00
Enter 12, press ENTER, press , press
C/Y = Enter 12, press ENTER, press
press CPT: EFF = 12.68%
42
Computing payments with APRs
• Suppose you want to buy a new computer
and the store will allow you to make
monthly payments. The computer costs
\$3,500, the loan period is for 2 years, and
the APR is 16.9%. What is your monthly
payment?

Use calculator
24 N       16.9 ÷ 12 = 1.40833 I/Y
3500 PV       CPT PMT = -\$172.88
43
Future values with monthly
compounding
• Suppose you deposit \$50 per month into an
account that has an APR of 9%, based on
monthly compounding. How much will you
have in the account in 35 years?

Use calculator
35 × 12 = 420 N   9 ÷ 12 = 0.75 I/Y
-50 PMT            CPT FV = \$147,089.22

44
Present value with daily compounding

• You need \$250,000 in 3 years. If you can
deposit money into an account that pays an
APR of 5.5% based on daily compounding,
how much would you need to deposit?

Use calculator
3 × 365 = 1095 N   5.5 ÷ 365 = 0.015068 I/Y
250000 FV   CPT PV = -\$211,976.06

45
Discounted Cash Flow Topics
DCF
valuation

Single cash flows                                        Multiple
(Ch4)                                             cash flows

Unequal                   Equal cash                  Comparing                    Types of
cash flows                   flows                       rates                       loans

Pure     Interest
FV                PV   Perpetuities     Annuities     APR               EAR                         Amortized
discount     only

46
Types of loans
Pure discount loans        Interest-only loans     Amortized loans

Payments are equal
The entire principal
in size
amount is paid at
Principal and interest
maturity
Payments   are paid together at the                          Payments contain
same time, at maturity                            both interest and
Interest is paid
repayment of
periodically
principal

• Zero coupon bond
• Coupon bond          • Mortgage loan
Examples   • Government treasury
• Bank term loan       • Automobile loan
bill

47
Treasury bill: example
• Treasury bills are excellent examples of pure
discount loans
– the principal amount is repaid at some future date,
without any periodic interest payments
• If a T-bill promises to repay \$10,000 in 1 year
and the market interest rate is 7 percent, how
much will the bill sell for in the market?
0    1     2     …     12                    r = 7%
PV = ?                10000
PV = \$10,000 ÷ 1.07 = \$9,345.79
Investor pays \$9,345.79 today and receives \$10,000 in 1 year
48
Interest-only loan: example
• Consider a 4-year, interest-only loan with a 7%
interest rate. The principal amount is \$10,000
and interest is paid annually.
– What would the stream of cash flows be?
• Years 1–3: interest payments of 0.07 × \$10,000 = \$700
• Year 4: interest + principal = \$700 + \$10,000 = \$10,700
• This cash flow stream is similar to the cash flow
stream on corporate bonds that we will talk
0     1    2                 3      4
10,000 -700 -700              -700 -10,700
49
Amount borrowed            Loan payments
Amortization
• Amortization refers to the process of reducing
the amount of the loan outstanding over a
period of time
– a variety of loan repayment schedules can be used
• Typically, mortgage loans are fully amortized
– this means that payments are equal in size and
made at regular time intervals (e.g. monthly)
– each payment is composed of
• interest on the outstanding loan amount
• repayment of a portion of the loan

50
Amortization example
• Consider a 4-year fully amortized loan
with payments made at the end of each
year for which the interest rate is 7% and
the amount borrowed (principal) is
\$10,000.
• What is the annual payment?
\$10,000 = C[1 – 1 / 1.074] / 0.07
C = \$2,952.28
Note that this is the formula for finding an annuity payment

51
Amortization table for previous example
7,747.72 x 0.07 = 542.34                             2,952.28 – 542.34 = 2,409.94

D=B                         F=B
A           B            C                      E=CD
0.07                         E
Beginning       Total      Interest      Principal      Ending
Year
balance       payment       paid         repaid        balance
1     10,000.00      2,952.28     700.00        2,252.28     7,747.72
2      7,747.72      2,952.28     542.34        2,409.94     5,337.78
3      5,337.78      2,952.28     373.64        2,578.64     2,759.14
4      2,759.14      2,952.28     193.14        2,759.14        0.00
Totals       11,809.12                10,000.00
1,809.12 7,747.72 – 2,409.94 = 5,337.78

The amount borrowed (\$10,000) is repaid in part each period, with the
result that at the loan is completely repaid by the end of the amortization
period. Interest paid each year declines while principal repaid increases.
52
Interest-only vs. amortization
• Note that in the previous examples
– the loan amount (PV) is the same (\$10,000)
– the interest rate is the same (7%)
– the duration of the loans is the same (4 years)

• The two loans differ only in their repayment
patterns
– the interest-only loan has interest payments each period
plus repayment of principal in the last period
– the amortized loan has equal payments each period that
contain both interest and repayment of principal

53
Calculator and previous example
•   Given:
– amount borrowed: \$10,000
– interest rate: 7%
– amortization period: 4 years
•   Find the annual mortgage payment
4 N; 7 I/Y;    10000 PV;     CPT PMT = -\$2,952.28
•   Find the principal remaining after two years and
the interest paid and principal repaid in year 2
Press 2nd AMORT
P1 = 2 (beginning of specified period); ENTER
P2 = 2 (end of specified period); ENTER
BAL = \$5,337.77 (principal remaining at end of period)
PRN = -2,409.94 (principal repaid during period)
INT = -\$542.34 (interest paid during period)
54
Another example
• What is the monthly payment on a loan of \$1,000,000 at 6%
interest that is amortized over 25 years?
– N = 25 × 12 = 300          PV = 1000000
– I/Y = 6 / 12 = 0.500                CPT PMT: -6,433.01
– Note: you can enter 300 directly or use the calculator (enter 25 × 12 =
300 and press N)
– The same holds for I/Y (enter 6 ÷ 12 = 0.5 and press I/Y)
• What is the outstanding loan amount at the end of the first
year and how much interest is paid and principal repaid in the
first year?
–   Press 2nd AMORT           P1 is the first period and P2 is the last
–   P1 = 1 Enter              period in the amortized life of the loan
–   P2 = 12 Enter             for which interest paid (INT) and
–   BAL= 982,199.64           principal repaid (PRN) are calculated.
–   PRN = -17,800.36          The remaining loan balance (BAL) is
–   INT = -59,515.76          calculated as at P2.

55
Mortgage characteristics
• Mortgages are used to finance real
property (land and buildings)
• Mortgages are important in Hong Kong
– Hong Kong has a high volume of property
– mortgages account for approximately 25% of
all loans made by Hong Kong financial
institutions for which the borrowed money is
used in Hong Kong

56
Types of mortgage loans in HK
• Conventional mortgages
– are prime rate-based
• E.g. mortgage rate = prime rate – 2.50%
• rates change with changes in the prime rate
– are fully amortized
– the mortgage rate is fixed for 1, 2 or 3 years after
which it changes with changes in the prime rate

57
The prime rate
• Is set by the bank itself and not in a market
• In the past, the prime rate was viewed as the loan rate
charged to the banks “best” business borrowers
• Today, it is viewed as a reference rate used in setting other
interest rates
– some borrowers may pay less, some more than the prime rate
• The prime rates tend to move with U.S. interest rates due to
the HKD-USD exchange rate peg
• The prime rate is currently 5.75% for larger banks and 6%
for smaller banks
– it has declined from 7.5% in September 2007 to its present level of
5.75% (February 2008)

58
Mortgage loan rates in HK
• The Hong Kong mortgage market is very competitive
– lenders tend to charge similar rates
• The prime rate is the most common base rate
– this rate is posted for everyone to see
• A “spread” is subtracted from the prime rate to get the
mortgage rate
– spreads, which currently range between 2.50% and 3.25%,
are not posted
– spreads narrow (mortgage rate rises) as HIBOR rates
increase and as the size of the loan increases
– spreads may also reflect the perceived riskiness of the
borrower and the value placed by the bank on the
customer relationship
• Suppose a mortgage rate = 3.25% (prime minus spread)
– this mortgage rate is less than the inflation rate of 3.8%
59
• Consider a 4 year loan with annual
payments. The interest rate is 7% and
the principal amount is \$10,000.
• Each payment covers the interest
expense and reduces the principal
• Click on the Excel icon to see the
amortization table
(optional)

60

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