此条博客将发布2014年秋季学期的《金融数学》课件和作业。

本课程使用的主要教材和参考资料：

（1）孟生旺，金融数学（第四版），中国人民大学出版社，2014年8月。

（2）Chris Ruckman, Joe Francis. Financial Mathematics, BBP Professional Education. 2005.

 教学大纲：点击查看

 

2014年新版《金融数学》课件（9月10日开始陆续发布）

FM study note by Kevin Shand

第1章（第一讲，第二讲）

第2章（第一讲，PV和FV的应用，第二讲）

第3章（更新）

第4章（下载）

第5章（课件下载，分期偿还表下载，变额分期偿还表）

第6章（更新）

第7章（下载）

第8章（下载）

第9章（下载）

第10章(下载)

2013版《金融数学》课件下载地址：第1—6章（下载），第7—10章（下载）

 

 

2014年《金融数学》课外作业

 

第1章  利息度量

(1.1 )  At time 0, K is deposited into Fund X, which accumulates at a force of interest δ_t = 0.006t². At time m, 2K is deposited into Fund Y ,which accumulates at an annual effective interest rate of 10%. At time n, where n > m, the accumulated value of each fund is 4K. Determine m.

(1.2 ) An investor deposits 20,000 in a bank. During the first 4 years the bank credits an annual effective rate of interest of i. During the next 4 years the bank credits an annual effective rate of interest of i-0.02. At the end of 8 years the balance in the account is 22081.10. What would the account balance have been at the end of 10 years if the annual effective rate of interest were i + 0.01 for each of the 10 years?

(1.3 )  You are given δ_t =2/(1+t). A payment of 300 at the end of 3 years and 600 at the end of 6 years has the same present value as a payment of 200 at the end of 2 years and X at the end of 5 years.

Calculate X.

第2章  等额年金

(2.1)   Jeff deposits 10 into a fund today and 20 fifteen years later.  Interest is credited at a nominal discount rate of d compounded quarterly for the first 10 years, and at a nominal interest rate of 6% compounded semiannually thereafter.  The accumulated balance in the fund at the end of 30 years is 100.  Calculate d.

(2.2)     Ernie makes deposits of 100 at time 0, and X at time 3.  The fund grows at a force of interest δ_t=t²/100. The amount of interest earned from time 3 to time 6 is also X. Calculate X.

（2.3）A perpetuity paying 1 at the beginning of each 6-month period has a present value of 20. A second perpetuity pays X at the beginning of every 2 years. Assuming the same annual effective interest rate, the two present values are equal. Determine X.

第3章  变额年金

(3.1)  1000 is deposited into Fund X, which earns an annual effective rate of 6%.  At the end of each year, the interest earned plus an additional 100 is withdrawn from the fund.  At the end of the tenth year, the fund is depleted. The annual withdrawals of interest and principal are deposited into Fund Y, which earns an annual effective rate of 9%. Determine the accumulated value of Fund Y at the end of year 10.

(3.2)  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 δ_t =1/(8+t). After 10 years, the account is worth 20000. Calculate k.

 (3.3)A perpetuity has payments of $1,$2,$1,$3,$1,$4,$1,$5,…Payments are made at the end of each year. You may assume an annual effective interest rate of 5%. Determine the present value of this perpetuity.

第4章  收益率

(4.1)  An association had a fund balance of 75 on January 1 and 60 on December 31.  At the end of every month during the year, the association deposited 10 from membership fees.  There were withdrawals of 5 on February 28,  25 on June 30,  80 on October 15, and 35 on October 31. Calculate the dollar-weighted (money-weighted) rate of return for the year

(4.2)  You are given the following information about the activity in two different investment accounts. During 1999, the dollar-weighted (money-weighted) return for investment account K equals the time-weighted return for investment account L, which equals i. Calculate i.

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第5章  债务偿还

(5.1)  A loan is amortized over five years with monthly payments at a nominal interest rate of 9% compounded monthly. The first payment is 1000 and is to be paid one month from the date of the loan. Each succeeding monthly payment will be 2% lower than the prior payment. Calculate the outstanding loan balance immediately after the 40th payment is made.

(5.2)  A 20-year loan of 20,000 may be repaid under the following two methods:  i) amortization method with equal annual payments at an annual effective rate of 6.5%.   ii) sinking fund method in which the lender receives an annual effective rate of 8% and the sinking fund earns an annual effective rate of j  Both methods require a payment of X to be made at the end of each year for 20 years. Calculate j.

(5.3)A $6,000 loan is being repaid with regular payments of X at the end of each year for as long as necessary plus a smaller payment one year after the final regular payment. Immediately after the ninth payment, the outstanding principal is three times the size of regular payment (that is, 3X). If the annual interest rate i is 10%, what is the value if X?

 

第6章  债券定价

(6.1)  You have decided to invest in Bond X, an n-year bond with semi-annual coupons and the following characteristics:

Par value is 1000.

The ratio of the semi-annual coupon rate to the desired semi-annual yield rate , is 1.03125.  The present value of the redemption value is 381.50.

Given vⁿ= 0.5889, what is the price of bond X?

(6.2)  Toby purchased a 20-year par value bond with semiannual coupons at a nominal annual rate of 8% convertible semiannually at a price of 1722.25. The bond can be called at par value 1100 on any coupon date starting at the end of year 15.  What is the minimum yield that Toby could receive, expressed as a nominal annual rate of interest convertible semiannually? 

(6.3)Common stock X pays a dividend of 50 at the end of first year, with each subsequent annual dividend being 5% greater than the preceding one. John purchase the stock at a theoretical price to earn an expected annual effective yield of 10%. Immediately after receiving the 10^th dividend, John sells the stock for a price P. His annual effective yield over the 10-year period was 8%. Calculate P.

第7章  远期、期货和互换

(7.1)   The current price of a stock is 200, and the continuously compounded interest rate is 4%.  A dividend will be paid every quarter for the next 3 years, with the first dividend occurring 3 months from now.  The amount of the first dividend is 1.50, but each subsequent dividend will be 1% higher than the one previously paid.   Calculate the fair price of a 3-year forward contract on this stock?

(7.2)   A farmer expects to sell 50 tons of pork bellies at the end of each of the next 3 years.  Suppose that the pork bellies forward price for delivery in 1 year is 1,600 per ton.  For delivery in 2 years, the forward price is 1,700 per ton.   Also, for delivery in 3 years, the forward price is 1,800 per ton. Suppose that interest rates are determined from the following table: 

   Years to Maturity      Zero-Coupon Bond Yield

1                     5.0%

2                     5.5%

3                     6.0%

If the farmer uses a commodity swap to hedge the price for selling pork bellies, what is the level amount he would receive each year (i.e. – the swap price) for all 50 tons?

(7.3)A stock has a current spot price of $90, and a nine-month forward price of $95. The continuously-compounded annual interest rate is 10%. Find the stock’s annualized continuous dividend yield which is consistent with forward price.

 

第8章  期权

(8.1)  You are given the following information:

• The current price to buy one share of XYZ stock is 500.
• The stock does not pay dividends.
• The risk-free interest rate, compounded continuously, is 6%.
• A European call option on one share of XYZ stock with a strike price of K that expires in one year costs 66.59.
• A European put option on one share of XYZ stock with a strike price of K that expires in one year costs 18.64.

Using put-call parity, determine the strike price, K.

(8.2)  Consider a European put option on a stock index without dividends, with 6 months to expiration, and a strike price of 1,000.  Suppose that the nominal annual risk-free rate is 4% convertible semiannually, and that the put costs 74.20 now.  What price must the index be in 6 months so that being long the put would produce the same profit as being short the put?

 

第9章  利率风险

(9.1)  John purchased three bonds to form a portfolio as follows:

• Bond A has semi-annual coupons at 4%, a duration of 21.46 years, and was purchased for 980.
• Bond B is a 15-year bond with a duration of 12.35 years and was purchased for 1015.
• Bond C has a duration of 16.67 years and was purchased for 1000.

Calculate the duration of the portfolio at the time of purchase.

(9.2)  An insurance company accepts an obligation to pay 10,000 at the end of each year for 2 years. The insurance company purchases a combination of the following two bonds at a total cost of X in order to exactly match its obligation:

• 1-year 4% annual coupon bond with a yield rate of 5%.
• 2-year 6% annual coupon bond with a yield rate of 5%.

Calculate X.

第10章  利率的期限结构

(10.1)  You are given the following term structure of spot interest rates:

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.