Solutions to Further Problems
Risk Management and Financial
Institutions
Third Edition
1
SM Summary
This manual contains answers to all the Further Questions at the ends of the chapters. A separate
pdf file contains notes on the teaching of the chapters t

611
Appendix G: Taylor Series Expansions
When x = 2 and y = 1
z
= 0.35355
x
2 z
= 0.08839
x2
z
= 0.70711
y
2 z
= 0.35355
y2
2 z
= 0.17678
xy
The first order approximation to z, given by equation (G.3), is
z = 0.35355 0.1 + 0.70711 0.1 = 0.10607
The second

APPENDIX
G
Taylor Series Expansions
C
onsider a function z = F(x). When a small change x is made to x, there is a
corresponding small change z in z. A first approximation to the relationship
between z and x is
z =
dz
x
dx
(G.1)
This relationship is exact

593
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
Rc n
Ae
(

Appendix B: Zero Rates, Forward Rates, and Zero-Coupon Yield Curves
597
Bond Yields
A bonds yield is the discount rate that, when applied to all cash flows, gives a bond
price equal to its market price. Suppose that the theoretical price of the bond we
ha

APPENDIX
A
Compounding Frequencies
for Interest Rates
A
statement by a bank that the interest rate on one-year deposits is 10% per annum
sounds straightforward and unambiguous. In fact, its precise meaning depends on
the way the interest rate is measured.

APPENDIX
E
Valuing European Options
T
he BlackScholesMerton formulas for valuing European call and put options on
an investment asset that provides no income are
c = S0 N(d1 ) KerT N(d2 )
and
p = KerT N(d2 ) S0 N(d1 )
where
d1 =
ln (S0 K) + (r + 2 2)T
T

APPENDIX
D
Valuing Swaps
A
plain vanilla interest rate swap can be valued by assuming that the interest rates
that are realized in the future equal todays forward interest rates. As an example,
consider an interest rate swap that has 14 months remaining a

APPENDIX
F
Valuing American Options
T
o value American-style options, we divide the life of the option into n time steps
of length t. Suppose that the asset price at the beginning of a step is S. At the end
of the time step it moves up to Su with probabil

596
APPENDICES
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

APPENDIX
C
Valuing Forward and
Futures Contracts
T
he forward or futures price of an investment asset that provides no income is
given by
F0 = S0 erT
where S0 is the spot price of the asset today, T is the time to maturity of the forward
or futures contra

606
APPENDICES
At each node:
Upper value = Underlying asset price
Lower value = Option price
Shading indicates where option is exercised.
Strike price = 50
Discount factor per step = 0.9917
Time step, dt = 0.0833 years, 30.42 days
Growth factor per step,

600
APPENDICES
EXAMPLE C.1
Consider a six-month futures contract on the S&P 500. The current value of the index
is 1,200, the six-month risk-free rate is 5% per annum, and the average dividend yield
on the S&P 500 over the next six months is expected to b

598
APPENDICES
8
Zero rate (%)
7
6
5
4
3
2
1
Maturity (years)
0
0
0.5
1
1.5
2
2.5
3
FIGURE B.1 Zero Curve for Data in Table B.3
which gives R = 7.05%. The zero curve is usually assumed to be linear between the
points that are determined by the bootstrap m

APPENDIX
B
Zero Rates, Forward Rates, and
Zero-Coupon Yield Curves
T
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 ye

607
Appendix F: Valuing American Options
We estimate theta from nodes D and C as
Theta =
3.77 4.49
2 0.08333
or 4.30 per year. This is 0.0118 per calendar day. Vega is estimated by increasing the volatility, constructing a new tree, and observing the effe

592
APPENDICES
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)

610
APPENDICES
The first order approximation to z, given by equation (G.1), is
z = 0.35355 0.1 = 0.035355
The second order approximation, given by equation (G.2), is
z = 0.35355 0.1 +
1
(0.08839) 0.12 = 0.034913
2
The third order approximation is
z = 0.3

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DO

Assignment 3
MATBUS 472
Due April 6 at the beginning of class
1. The probability density function for an exponential distribution is e-x where x is the
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Assignment 3 MATBUS 472 § ﬁ'
Due April 2 at the beginning of class
1. The probability densityﬂinctionfor an e.\'ponential distribution is x1e 1“ wheres is the
value of the variable and _ is a parameter. The cumulative probability distribution is l—

Assignment 3
MATBUS 472
Due April 2 at the beginning of class
1. The probability density function for an exponential distribution is e-x where x is the
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