4.6. APPENDIX: INTEREST RATES, RETURNS, AND BOND YIELDS 155
where e = 2.7182818 is the base of the natural (Neperian) logarithm. Con-
versely, the return is computed from the quoted interest rateρas:1 +rt,T= limm→∞(
1 +
ρ
m)(T−t)m
=eρ(T−t) (4.38)Example 4.24
Let (T−t) = 1/2, and assume the continuously compounded interest rate
equals 9.75803 percent. Then:1 +rt,T=e^0.^0975803 /^2 = 1. 05. (4.39)Note the following link between the continuously and the annually compounded
ratesiandρ:(1 +i) =eln(1+i)⇒(1 +i)T−t=eln(1+i)·(T−t)⇒ln(1 +i) =ρ. (4.40)- Bankers’ discount is yet another way of annualizing a return. This is often
used when the present value is to be computed for T-bills, promissory notes,
and so on—instruments where the time-Tvalue (or “face value”) is the known
variable, not thepvlike in the case of a deposit or a loan. Suppose the time-T
value is 100, the time to maturity is 0.5 years, and the p.a. discount rate is 5
percent. The present value will then be computed as
PV= 100×(1− 0. 05 /2) = 97. 5. (4.41)Conversely, the return is found from the quoted bankers’ discount rate as:1 +rt,T=1
1 −(T−t)×[banker’s discount rate]. (4.42)
Example 4.25
Let (T−t) = 1/2 and thep.a.bankers’ discount rate 9.5238 percent. Then:1 +rt,T=1
1 − 1 / 2 × 0. 095238
= 1. 05. (4.43)
In summary, there are many ways in which a bank can tell its customer that the
effective return is, for instance, 5 percent. It should be obvious that what matters is
the effective return, not the statedp.a.interest rate or the method used to annualize
the effective return. For this reason, in most of this text, we use effective returns.
This allows us to write simply (1 +rt,T). If we had used annualized interest rates,
all formulas would look somewhat more complicated, and would consist of many
versions, one for each possible way of quoting a rate.