Ch 6. Discounted cash flow
valuation
1. FV and PV with multiple cash flows
1) FV with multiple cash flows: Two methods
(1) Rolling over FV year by year
(2) FV=FV1+FV2+FV3….
Ex) Deposit $100 every year for 3 yrs. And 10%
interest rate. FV?
2) PV with multiple cash flows: Two
method
(1) Rolling back year by year
(2) PV=PV1+PV2+…..
Ex) You are supposed to need $1000 in one year
and $2000 in the second year. If you can
earn 9% on your money, how much you have
to put up today?
2. Annuities and Perpetuities
1) Def of Annuity:
Constant cash flows for a fixed period of time
Ex) car loan
Ex) Assets with promised to pay $500 at the end
of the each of the next three years. What is
the price of the asset now if a discount rate
is 10%?
Answer:
500/(1.1)+500/(1.1^2)+500/(1.1^3)= 1243.43
2) Formula for Annuity Present Value
C  [1  1 /(1  k )t ]
PV 
k
C
C
C
PV 

 ..... 
2
(1  k ) (1  k )
(1  k ) t
C
C
C
PV  (1  k )  C 

 ..... 
2
(1  k ) (1  k )
(1  k ) t 1
C
PV  PV  (1  k ) 
C
t
(1  k )
1
PV (  k )  C[
 1]
t
(1  k )
1
C[1 
]
t
(1  k )
Therefore PV 
k
Ex) You stop by a car dealer shop and find a
really good car. The sticker price of the car is
$15000. But you don’t have money now. So,
want installment payment over 4 yrs. Over
conversation, the dealer suggests $632 per
month for 48 month at 1% per month.
How much is going to be your PV of total
installments?
2-2) Finding C
Ex) You stop by a car dealer shop and find a
really good car. The sticker price of the car is
$15000. But you don’t have money now. So, if
you want installment payment over 4 yrs, how
much you have to pay monthly? (Here interest
rate is 12%)
2-3) Finding rate
Ex) an insurance company offers to pay you
$1000 per year for 10 years if you pay $6710
up front. What rate is used in this annuity?
3) Def of perpetuities:
An annuity in which the cash flow continues
forever
4) Formula for PV of perpetuities
PV=C/k
Ex) Preferred stock – promised fixed dividend every
period forever.
A company want to sell preferred stock at $100 per
share. How much of dividend it has to pay.
Currently the similar preferred stock is sold at $40
with $1 dividend.
i) Calculate r:
R= 1/40 = 0.025
ii) Calculate C:
100 = C/0.025. Then, C=2.5
5) FV for Annuities
FV  C  [(1  k )  1] / k
t
FV  C  C (1  k )  C (1  k )  ......  C (1  k )
2
t 1
FV  (1  k )  C (1  k )  C (1  k ) 2  C (1  k )3  .....  C (1  k )t
FV  FV (1  k )  C  C (1  k )t
FV (k )  C[1  (1  k ) ]
t
C[(1  k )t  1]
FV 
k
Ex) $2000 annuity for 30 years and k= 0.08. What is
the annuity future value?
6) Annuities due
Def: annuity for which the cash flows occur at the
beginning of the period
Annuity due value =
ordinary annuity value * (1+k)
• 7) Uneven Cash Flows;
• Summing PV and FV of each cash flows
• Using the cash flow patterns to apply formula
• 8) Growing annuities: payment growth by g%.
1 g t
1[
]
1 k ]
PV  C  [
kg
• 9) Growing perpetuity: payment grows by g%
forever.
PV  C /( k  g )
3. Rate
Q1. 10% compounded semi-annually is the same as
10% per year in compounding?
No! here, 10% is APR, (annual percentage rate) and
actually, 10.25% (=(1+0.05)*(1+0.05)-1) is the
effective annual rate.
To compare to other rates, we need to convert APR
into the effective rates
3-1) APRs (Annual Percentage Rate)
Def: interest rate charged per period (periodic rate)
multiplied by the number of periods per year
APR =EAR?
No!!!!
So, APR is a quoted rate and need to be converted
to the EAR
EAR(Effect ive Annual Rate)
 [1  (APR/m)] m  1
unlimited interest calculatio n
EAR  e k  1
e  2.71828
Ex)
One credit card company selling a card by telemarketing. The company said the card will
benefit its cardholders with semi-annual
15%APRs, compared to the other credit card
with 16% EAR.
Do you agree or not?
6.Loan types and loan Amortization
1) Pure discount loan:
Receive money today and repay a single lump sum
in future
What is the price of loan that you will pay $25,000 in 5
years? A lender wants to apply 12% interest rate.
14,186 = 25000/(1.12)^5
2) Interest only loan:
Pay interest each period and repay the entire
principal at some point in the future
• A three year, 10% interest only loan of $1000.
• A borrower has to pay $100 at the end of first
and second year. At the end of third year, he
or she has to pay $100 and $1000.
• 3) Amortized loan:
Repay parts of the loan amount over time
• Borrow $5000 for 5 years. An interest rate is
9%. Annual payment happens.
• 3-1) constant principal payment
Year
Begin
Pay
Interest
paid
Principa End
l
1
5000
1450
450
1000
4000
2
4000
1360
360
1000
3000
3
3000
1270
270
1000
2000
4
2000
1180
180
1000
1000
5
1000
1090
90
1000
0
6350
1350
5000
total
3-2) Fixed payment
• 5000 = c*[1-1/(1+0.09)^5]/0.09
• C=1285.46
Year
begin
pay
interest
principal End
1
5000
1285.46
450
835.46
4164.54
2
4164.54
1285.46
374.81
910.65
3253.88
3
3253.88
1285.46
292.85
992.61
2261.27
4
2261.27
1285.46
203.51
1081.95
1179.32
5
1179.32
1285.46
106.14
1179.32
6427.30
1427.31
5000
total