Investment Analysis
Lecture 4
Complications in NPV
Investment Analysis
Complication 1
Inflation
How does inflation affect DCF analysis?
CF1
NPV
=
CF2
+
(1+r)
CF3
+
CF4
+
CF5
+
+…
(1+r)2 (1+r)3 (1+r)4 (1+r)5
Discounting Rule
Treat inflation consistently: Discount real cash flows at the real interest
rate and nominal cash flows at the nominal interest rate.
Investment Analysis
Complication 1 (cont’d …)
Terminologies
Cash flows
Nominal
Real
=
=
Actual Cash Flows
Cash Flows expressed in today’s
purchasing power
Real CFt = Nominal CFt / (1 + Interest Rate)t
Discount Rates
Nominal
Real
=
=
Actual Interest Rate
Interest Rate adjusted for Inflation
1 + Real Interest Rate = (1 + Nominal Interest Rate) / (1 + Inflation Rate)
Real Interest Rate = [(1 + Nominal Interest Rate) / (1 + Inflation Rate)] - 1
Investment Analysis
Example
This year you earned $100,000. You expect
your earnings to grow about 2% annually, in
real terms, for the remaining 20 years of your
career. Interest rate is currently 5% and
inflation rate is 2%. What is the present value
of your income?
Investment Analysis
Complication 2
Compounding Frequency
On many investments or loans, interest is often charged or credited more
than once a year.
Examples
Bank Accounts – Daily
Mortgages and Leases – Monthly
Bonds – Semiannually
Implication
Effective Annual Rate (EAR) can be much different that the stated Annual
Percentage Rate (APR).
Investment Analysis
Compounding
- Consider investing $100 at the annual interest rate 10%. How much will your
investment value in 3 years?
> After 1 year you will have:
$100 x 1.1 = $110
> After 2 years you will have:
$100 x 1.12 = $121
> After 3 years you will have:
$100 x 1.13 = $133.1
- This is different from $100 + $30 = $130
- Have to account for interest on interest.
- More generally, investing C dollars at the annual interest rate r gives,
C ( 1 + r )t
dollars after t years.
Investment Analysis
APRs & EARs
Habib Bank quotes you a rate of 7% for your home mortgage. The
mortgage involves monthly payments.
• 7% is not the “true” annual rate, it is only used to derive the monthly rate by:
x = 7% / 12 = 0.583%
• The “true” annual rate is:
(1 + x)12 – 1 = 7.23%
• 7% is the Annual Percentage Rate (APR). It is the quoted rate.
• 7.23% is the Effective Annual Rate (EAR). It is the true rate.
Investment Analysis
APRs & EARs : Compounding Frequencies
Example: Suppose your bank offers a 1-year CD with a 5% APR, what is the EAR
with;
•
•
•
•
•
•
Annual Compounding?
Semiannual Compounding?
Quarterly Compounding?
Monthly Compounding?
Daily Compounding?
Continuous Compounding?
1.05
(1+ 5%/2 )2=1.050625
(1+ 5%/4 )4=1.050945
(1+ 5%/12 )12=1.051161
(1+ 5%/360 )360=1.01267
Lim (1+ 5%/T )T=…
T
∞
General Formula: with T compounding periods in a year
APR
EAR = ( 1 +
)T - 1
T
Investment Analysis
APRs & EARs (cont’d …)
Example
Car Loan
‘Finance charge on an unpaid balance, computed daily, is at the rate of 6.75% per year.
If you borrow $10,000 to be repaid in one year. How much would you owe in a year?
Daily Interest Rate = 6.75 / 365 = 0.0185%
Day 1: balance = 10,000 x 1.00185 = 10,001.85
Day 2: balance = 10,000 x (1.00185)2 = 10,003.70
……….. …………………………………………………………………
Day 4: balance = 10,000 x (1.00185)360 = 10,698.50
EAR = 6.985%
Investment Analysis
Complication 2 (cont’d …)
Discounting Rule
In applications, interest is compounded at the same frequency as
payments.
If, so just divide the APR by number of compounding intervals.
Bonds
Make semiannual payments, interest compounded semiannually, discount
semiannual cash flows by APR / 2.
Mortgages
Make monthly payments, interest compounded monthly, discount
monthly cash flows by APR / 12.
Investment Analysis
Complication 3
Currencies
How foreign currency cash flows are discounted?
CF1
PV = CF0
+
CF2
+
(1+r)
CF3
+
(1+r)2
CF4
+
(1+r)3
+…
(1+r)4
Discounting Rule
Discount each currency at its own interest rate: discount $ at the U.S interest
rate, discount € at the european interest rate.
This gives of PV of each cash flow stream in its own currency.
Convert to domestic currency at the current exchange rate.
Investment Analysis
Currencies (cont’d …)
Logic
You have $1 now. How many Euros can you convert this into
in 1 year? The current exchange rate is 1.6 $/€ and the
European interest rate is 5%.
Today:
$1 = €0.625
1 year:
€0.625 x 1.05 = €0.6563
Implication: $1 today is worth 0.6563 pounds in one year.
Investment Analysis
Example
Your firm just signed a contract to deliver 2,000 batteries in each
of the next 2 years to a customer in Japan, at a per unit price of
¥800. It also signed a contract to deliver 1,500 batteries in each
of the next 2 years to a customer in Britain, at a per unit price of
£6.2. Payment is certain and occurs at the end of the year.
The British interest rate r£ = 5% and the Japanese interest rate is
r¥ = 3.5%. The exchange rates are 118¥/$ and 1.6$/£.
What is the value of each contract?