此条博客将陆续发布2015年春季学期的《金融数学》作业，答案链接会在每章作业提交之后公布。
2015年春《金融数学》课外作业
第1章：利息度量
1.
Jennifer deposites 1000 into a bank account. The bank credits
interest at a nominal annual rate of I convertible semiannually for
the first 7 years and a nominal annual rate of 2i convertible
quarterly for all years thereafter.
The accumulated amount in the account at the end of
5 years is X. The accumulated amount in the account at the end of
10.5 years is 1980.
Calculate X.
2.
Bruce deposits 100 into a bank account. His account is credited
interest at a nominal rate of interest of 4% convertible
semiannually. At the same time, Peter deposits 100 into a separate
account. Peter’s account is credited interest at a force of
interest ofδ.After 7.25 years, the value of each account is the
same.
Calculate δ.
3.
Payments are made to an account at a continuous rate of (8k + tk),
where 0≤t≤10 . Interest is credited at a force of interest
δt = 1/（8 + t）。After 10 years, the account is worth
20,000.Calculate k.
第2章：等额年金
1.
A loan is being repaid with 25 annual payments of 300 each. With
the 10th payment, the borrower pays an extra 1000, and then repays
the balance over 10 years with a revised annual payment. The
effective rate of interest is 8%.
Calculate the amount of the revised annual payment.
2.
A company deposits 1000 at the beginning of the first year and 150
at the beginning of each subsequent year into perpetuity. In return
the company receives payments at the end of each year forever. The
first payment is 100. Each subsequent payment increases by 5%.
Calculate the company’s yield rate for this
transaction. (yield rate=effective rate)
3.
At an annual effective interest rate of i, i > 0%,
the present value of a perpetuity paying `10 at the end of each
3-year period, with the first payment at the end of year 3, is 32.
At the same annual effective rate of i, the present value of
a perpetuity paying 1 at the end of each 4-month period, with first
payment at the end of 4 months, is X.
Calculate X.
第3章：变额年金
1.
An annuity-immediate pays 20 per year for 10 years, then decreases
by 1 per year for 19 years. At an annual effective interest rate of
6%, the present value is equal to X.
Calculate X.
2.
Matthew makes a series of payments at the beginning of each year
for 20 years. The first payment is 100. Each subsequent payment
through the tenth year increases by 5% from the previous payment.
After the tenth payment, each payment decreases by 5% from the
previous payment.
Calculate the present value of these payments at
the time the first payment is made using an annual effective rate
of 7%.
3.
Olga buys a 5-year increasing annuity for X. Olga will
receive 2 at the end of the first month, 4 at the end of the second
month, and for each month thereafter the payment increases by 2.
The nominal interest rate is 9% convertible quarterly.
Calculate X.
第4章：收益率
1.
At the beginning of the year, an investment fund was established
with an initial deposit of 1000. A new deposit of 1000 was made at
the end of 4 months. Withdrawals of 200 and 500 were made at the
end of 6 months and 8 months, respectively. The amount in the fund
at the end of the year is
1560.
Calculate the dollar-weighted (money-weighted) yield rate earned by
the fund during the year.
2.
Hugh invests 1000 in a fund on January 1. On May 1, the fund is
worth 1100 and 600 is withdrawn. On September 1, the fund is worth
400 and 600 is deposited. On January 1 of the following year, the
fund is worth 1200. Hugh’s dollar-weighted rate of return for the
year( computed using simple interest from the date of each payment)
is X. His time-weighted rate of return is Y. Determine X-Y.
3.
The following is a table of interest rates credited under the
investment year method. Unfortunately, a few entries are
missing:
http://s3/mw690/0028pSevzy6Qz4ck2zgf2&690
It is know that an investment of $1000 in 1992 will
accumulate to $1270.15 at the end of five years and that an
investment of $1000 in 1993 will accumulate to $1247.33 at the end
of five years. Calculatehttp://s1/mw690/0028pSevzy6Qz4mbnzi30&690
第5章：债务偿还方法
1.
Lori borrows 10,000 for 10 years at an annual effective interest
rate of 9%. At the end of each year, she pays the interest on the
loan and deposits the level amount necessary to repay the principal
to a sinking fund earning an annual effective interest rate of
8%.The total payments made by Lori over the 10-year period is
X.
Calculate X.
2.
A loan is repaid with level annual payments based on an annual
effective interest rate of 7%. The 8th payment consists
of 789 of interest and 211 of principal.
Calculate the amount of interest paid in the 18th payment.
3.
John borrows 10,000 for 10 years at an annual effective interest
rate of 10%. He can repay this loan using the amortization method
with payments of 1,627.45 at the end of each year. Instead, John
repays the 10,000 using a sinking fund that pays an annual
effective interest rate of 14%. The deposits to the sinking fund
are equal to 1,627.45 minus the interest on the loan and are made
at the end of each year for 10 years.
Determine the balance in the sinking fund
immediately after repayment of the loan.
第6章：证券定价
1.
A ten-year 100 par value bond pays 8% coupons semiannually. The
bond is priced at 118.20 to yield an annual nominal rate of 6%
convertible semiannually.
Calculate the redemption value of the bond.
2.
The dividends of a common stock are expected to be 1 at the end of
each of the next 5 years and 2 for each of the following 5 years.
The dividends are expected to grow at a fixed rate of 2% per year
thereafter. Assume an annual effective interest rate of 6%.
Calculate the price of this stock using the dividend discount
model.
3.
A company buys an annual coupon bond maturing at 105 in 25 years.
The company pays P to get a yield to maturity of 4% effective. The
write down on the bond in the 10th year is 1.00.
Calculate P.
第7章：利率风险
1.
Calculate the Macaulay duration of an eight-year 100 par value bond
with 10% annual coupons and an effective rate of interest equal to
8%.
2.
|
Bond face
$100
|
|
Coupon
4%
semiannual
|
|
Term to maturity
3 years
|
|
Yield to maturity
5% annual
|
|
effective.Duration
2.854
|
|
Convexity
10.205
|
|
Bond price
97.41
|
Calculate the bond updated price if the yield to
maturity changes to 6%
annual effective.
3.
You are given the following information:
Projected liability cash flows ：
Year 1 Year 2
Year 3 Year 4
Year 5
43
123
214
25
275
Available assets for investment:
• 2-year bond with annual coupon of 5%
• 3-year bond with annual coupon of 8%
• 5-year bond with annual coupon of 10%
Face amount of the bond: 100
Current market yield curve: 7% for all
durations
Calculate the initial cost to cash-flow match the
projected liability cash
flows utilizing the assets listed above.
第8章：利率的期限结构
1.
You are given the following term structure of spot interest
rates:
Term(in
years)
Spot interest rate
1
5.00%
2
5.75%
3
6.25%
4
6.50%
A three-year annuity-immediate will be issued a
year from now with annual payments of 5000.
Using the forward rates, calculate the present value of this
annuity a year from now.
2.
The following are the current prices of $1000 zero-coupon
bonds:
Term to Maturity
Price
1
$943.4
2
X
3
805.08
If the one-year forward rate for year 2 (i.e., the
one-year effective rate during year 2) is 8%, determine X.
第9章：远期、期货和互换的定价
1.
You are given the following information:
i) The current price of stock A is 50.
ii) Stock A will not pay any dividends in the next
year.
iii) The annual effective risk-free interest rate
is 6%.
iv) Each transaction costs 1.
v) There are no transaction costs when the
forward
is settled.
Based on no arbitrage, calculate the maximum price of a one-year
forward.
2.
Two interest rate forward contracts are available for interest
payments due 1 and 2 years from now. The forward interest rates in
these contracts are based on a one-year spot rate of 5% and a
2-year spot rate of 5.5%.X is the level swap interest rate in a
2-year interest rate swap contract that is equivalent to the two
forward contracts. Determine X.
第10章：期权定价
1. A stock with a current price of $82 will pay no
dividends in the coming year. The premium for a one-year European
call is $10.424 and the premium for the corresponding put is
$8.993. The risk-free interest rate is 5.5% effective per annum. X
is the strike price of both the call and the put. Determine X to
the nearest $1.00
习题答案链接：
第一章答案
第二章答案
第三章答案
第四章答案
第五章答案
第六章答案
第七章答案
第八章答案
第九、十章答案
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(2015-03-09 17:04:25)