Lecture 5. Discounted cash flow applications
1) Effective annual rate
Both businesses and investors need to make objective comparisons of loan costs or investment returns
over different compounding periods. In order to put interest rates on a common basis, to allow
comparison, we distinguish between nominal and effective annual rates.
The nominal, or stated, annual rate is the contractual annual rate of interest charged by a lender or
promised by a borrower. The effective, or true, annual rate (EAR) is the annual rate of interest actually
paid or earned. The effective annual rate reflects the impact of compounding frequency, whereas the
nominal annual rate does not.
Using the notation introduced earlier, we can calculate the effective annual rate, EAR, by substituting
values for the nominal annual rate, i, and the compounding frequency, m, into Equation:
EAR  (1 
i m
) 1
m
(1)
Example. Fred Moreno wishes to find the effective annual rate associated with an 8% nominal annual
rate (i = 0.08) when interest is compounded (1) annually (m = 1); (2) semiannually (m = 2); and (3)
quarterly (m = 4). Substituting these values into Equation, we get
For annual compounding:
EAR  (1 
0.08 1
)  1  (1  0.08)1  1  1  0.08  1  0.08  8%
1
For semiannual compounding:
EAR  (1 
0.08 2
)  1  (1  0.04)2  1  1.0816  1  0.0816  8.16%
2
For quarterly compounding:
EAR  (1 
0.08 4
)  1  (1  0.02)4  1  1.0824  1  0.0824  8.24%
4
The effective annual rate increases with increasing compounding frequency, up to a limit that occurs
with continuous compounding.
The effective annual rate for this extreme case can be found by using the following equation:
EAR (continuous compounding)  e i  1
For the 8% nominal annual rate (i = 0.08), substitution into Equation results in an effective annual rate
of
EAR  e
0.08
 1  1.0833  1  0.0833  8.33%
At the consumer level, “truth-in-lending laws” require disclosure on credit card and loan agreements of
the annual percentage rate (APR). The APR is the nominal annual rate found by multiplying the
periodic rate by the number of periods in one year. For example, a bank credit card that charges 1.5%
percent per month (the periodic rate) would have an APR of 18% (1.5% per month*12 months per
year).
“Truth-in-savings laws,” on the other hand, require banks to quote the annual percentage yield (APY)
on their savings products. The APY is the effective annual rate a savings product pays. For example, a
savings account that pays 0.5 percent per month would have an APY of 6.17 percent [(1.005)12 - 1].
2) Internal rate of return (IRR)
Internal rate of return is the discount rate that sets the sum of the present value of the positive cash
flows equal to the sum of the present value of the negative cash flows.
- the discount rate at which NPV = 0
Interpretation: IRR is the expected compound return when all intervening cash flows in the project
can be reinvested at the IRR and the investment will be held until maturity.
Calculation: In practice, use a spreadsheet, financial software, or financial calculator to determine
IRR.
Decision: Accept projects for which IRR > hurdle rate, cost of capital → increases shareholder wealth.
N
NPV  
t 0
CFt
0
(1  IRR)t
(2)
3) money-weighted and time-weighted rate of return
The internal rate of return calculated on a portfolio is called the «money-weighted rate of return».
- money-weighted rate of return accounts for the timing and amount of dollar flows into and out of
the portfolio.
- money-weighted rate of return place greater weight on those periods in which investment is higher
and, therefore, give a «better» picture of the actual investor experience.
Time-weighted rate of returns is the investment measure that is not sensitive to the additions and
withdrawals of funds. Time-weighted returns remove the effect of inflows and outflows to the
portfolio and are, therefore, a better indicator of managerial skill.
Instruments that mature in less than a year are known as money market instruments.
There are a number of different conventions for calculating yields on money market instruments.
- bank discount yield;
- holding period yield;
- effective annual yield;
- money market yield;
Bank discount yield is a percentage of face value instead of price and assumes 360 days in a year
rBD 
where
D 360
Par  Pr ice 360
(
)
(
)
F t
Par
t
(3)
HPY measures the (unannualizied) return over the remaining life of the investment when P1 is the
return of face value.
HPY 
P1  P0  D1
P0
(4)
The annualized expression of the holding period yield is
EAY  (1  HPY)
365
t
1
(5)
Generally, the money market yield will be equivalent to the holding period yield adjusted to a 360day basis.
rMM 
360(rBD )
360  t (rBD )
(6)
Given any one of HPY, rMM, or rBD, we can calculate the others.
HPY (
360
F
)  rMM  rBD
t
P0
(7)