A. Compounding Frequencies for Interest Rates
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RISK MANAGEMENT AND FINANCIAL INSTITUTIONS
TABLE A.1
Effect of the Compounding Frequency on the
Value of $100 at the End of One Year When the Interest Rate
is 10% per Annum
Compounding
Frequency
Value of $100 at
End of Year ($)
Annually (m = 1)
Semiannually (m = 2)
Quarterly (m = 4)
Monthly (m = 12)
Weekly (m = 52)
Daily (m = 365)
110.00
110.25
110.38
110.47
110.51
110.52
If the rate is compounded m times per annum, the terminal value of the investment is
A 1+
R
m
mn
(A.1)
When m = 1 the rate is sometimes referred to as the equivalent annual interest rate.
Continuous Compounding
The limit as the compounding frequency, m, tends to infinity is known as continuous
compounding.1 With continuous compounding, it can be shown that an amount A
invested for n years at rate R grows to
Ae Rn
(A.2)
where e = 2.71828. The function e x , which is also written exp(x), is built into most
calculators, so the computation of the expression in equation (A.2) presents no
problems. In the example in Table A.1, A = 100, n = 1, and R = 0.1, so that the
value to which A grows with continuous compounding is
100e0.1 = $110.52
This is (to two decimal places) the same as the value with daily compounding.
For most practical purposes, continuous compounding can be thought of as being
equivalent to daily compounding. Compounding a sum of money at a continuously
compounded rate R for n years involves multiplying it by e Rn . Discounting it at a
continuously compounded rate R for n years involves multiplying by e−Rn .
1
Actuaries sometimes refer to a continuously compounded rate as the force of interest.
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Appendix A: Compounding Frequencies for Interest Rates
Suppose that Rc is a rate of interest with continuous compounding and Rm is the
equivalent rate with compounding m times per annum. From the results in equations
(A.1) and (A.2), we have
Ae Rc n = A 1 +
Rm
m
mn
or
e Rc = 1 +
Rm
m
m
This means that
Rc = m ln 1 +
Rm
m
(A.3)
and
Rm = m(e Rc /m − 1)
(A.4)
These equations can be used to convert a rate with a compounding frequency of m
times per annum to a continuously compounded rate and vice versa. The function ln
is the natural logarithm function and is built into most calculators. It is defined so
that if y = ln x, then x = e y .
EXAMPLE A.1
Consider an interest rate that is quoted as 10% per annum with semiannual compounding. From equation (A.3), with m = 2 and Rm = 0.1, the equivalent rate with
continuous compounding is
2 ln 1 +
0.1
2
= 0.09758
or 9.758% per annum.
EXAMPLE A.2
Suppose that a lender quotes the interest rate on loans as 8% per annum with
continuous compounding, and that interest is actually paid quarterly. From equation
(A.4), with m = 4 and Rc = 0.08, the equivalent rate with quarterly compounding is
4(e0.08/4 − 1) = 0.0808
or 8.08% per annum. This means that on a $1,000 loan, interest payments of $20.20
would be required each quarter.
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APPENDIX
B
Zero Rates, Forward Rates, and
Zero-Coupon Yield Curves
he n-year zero-coupon interest rate is the rate of interest earned on an investment
that starts today and lasts for n years. All the interest and principal is realized
at the end of n years. There are no intermediate payments. The n-year zero-coupon
interest rate is sometimes also referred to as the n-year spot interest rate, the n-year
zero rate, or just the n-year zero. The zero rate as a function of maturity is referred
to as the zero curve. Suppose a five-year zero rate with continuous compounding
is quoted as 5% per annum. (See Appendix A for a discussion of compounding
frequencies.) This means that $100, if invested for five years, grows to
T
100 × e0.05×5 = 128.40
A forward rate is the future zero rate implied by today’s zero rates. Consider the
zero rates shown in Table B.1. The forward rate for the period between six months
and one year is 6.6%. This is because 5% for the first six months combined with
6.6% for the next six months gives an average of 5.8% for the two years. Similarly,
the forward rate for the period between 12 months and 18 months is 7.6%, because
this rate when combined with 5.8% for the first 12 months gives an average of 6.4%
for the 18 months. In general, the forward rate F for the period between times T1
and T2 is
F =
R2 T2 − R1 T1
T2 − T1
(B.1)
where R1 is the zero rate for maturity of T1 and R2 is the zero rate for maturity T2 .
This formula is exactly true when rates are measured with continuous compounding
and approximately true for other compounding frequencies. The results from using
this formula on the rates in Table B.1 are in Table B.2. For example, substituting
T1 = 1.5, T2 = 2.0, R1 = 0.064, and R2 = 0.068 gives F = 0.08 showing that the
forward rate for the period between 18 months and 24 months is 8.0%.
Investors who think that future interest rates will be markedly different from
forward rates have no difficulty in finding trades that reflect their beliefs. Consider
an investor who can borrow or lend at the rates in Table B.1. Suppose the investor
thinks that the six-month interest rates will not change much over the next two years.
The investor can borrow six-month funds and invest for two years. The six-month
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RISK MANAGEMENT AND FINANCIAL INSTITUTIONS
TABLE B.1
Zero Rates
Maturity
(years)
Zero Rate (%)
(cont. comp.)
0.5
1.0
1.5
2.0
5.0
5.8
6.4
6.8
borrowings can be rolled over at the end of 6, 12, and 18 months. If interest rates do
stay about the same, this strategy will yield a profit of about 1.8% per year because
interest will be received at 6.8% and paid at 5%. This type of trading strategy is
known as a yield curve play. The investor is speculating that rates in the future will
be quite different from the forward rates shown in Table B.2.
Robert Citron, the Treasurer at Orange County, used yield curve plays similar to
the one we have just described very successfully in 1992 and 1993. The profit from
Mr. Citron’s trades became an important contributor to Orange County’s budget
and he was re-elected. In 1994, he used the same strategy more aggressively. If shortterm interest rates had remained the same or declined, he would have done very
well. As it happened, interest rates rose sharply during 1994. On December 1, 1994,
Orange County announced that its investment portfolio had lost $1.5 billion and
several days later it filed for bankruptcy protection.
Bond Pricing
Most bonds provide coupons periodically. The bond’s principal (which is also known
as its par value or face value) is received at the end of its life. The theoretical price
of a bond can be calculated as the present value of all the cash flows that will be
received by the owner of the bond. The most accurate approach is to use a different
zero rate for each cash flow. To illustrate this, consider the situation where zero rates
are as in Table B.1. Suppose that a two-year bond with a principal of $100 provides
coupons at the rate of 6% per annum semiannually. To calculate the present value
of the first coupon of $3, we discount it at 5.0% for six months; to calculate the
present value of the second coupon of $3, we discount it at 5.8% for one year; and
so on. The theoretical price of the bond is therefore
3e−0.05×0.5 + 3e−0.058×1.0 + 3e−0.064×1.5 + 103e−0.068×2.0 = 98.39
or $98.39.
TABLE B.2
Forward Rates for Zero Rates in Table B.1
Period
(years)
Forward Rate (%)
(cont. comp.)
0.5 to 1.0
1.0 to 1.5
1.5 to 2.0
6.6
7.6
8.0