金融数学引论答案第二版

合集下载

金融学习题集及参考答案解析word版第二版

金融学习题集及参考答案解析（第二版）金融学习题集（第二版）带★内容为非金融学专业选做题目第一章货币概述一、单项选择题（在每小题列出的四个备选项中只有一个是最符合题目要求的，请将其代码写在题后的括弧内。

）1．金融的本源性要素是【】A. 货币B. 资金C. 资本D. 市场2．商品价值最原始的表现形式是【】A. 货币价值形式B. 一般价值形式C.总和的或扩大的价值形式D. 简单的或偶然的价值形式3．一切商品的价值共同表现在某一种从商品世界中分离出来而充当一般等价物的商品上时，价值表现形式为【】A. 货币价值形式B. 一般价值形式C.总和的或扩大的价值形式D. 简单的或偶然的价值形式4．价值形式的最高阶段是【】A. 货币价值形式B. 一般价值形式C.总和的或扩大的价值形式D. 简单的或偶然的价值形式5．货币最早的形态是【】A. 实物货币B.代用货币C.信用货币D. 电子货币6．最适宜的实物货币是【】A. 天然贝B. 大理石C. 贵金属D. 硬质合金硬币7．中国最早的货币是【】A. 银圆B. 铜钱C. 金属刀币D. 贝币8．信用货币本身的价值与其货币价值的关系是【】A. 本身价值大于其货币价值B.本身价值等于其货币价值C. 本身价值小于其货币价值D. 无法确定9．在货币层次中M0是指【】A. 投放的现金B. 回笼的现金C. 流通的现金D. 贮藏的现金10．从近期来看，我国货币供给量相含层次指标系列中观察和控制的重点是【】A. M0B. M1C. M2D. M0和M111．从中长期来看，我国货币供给量相含层次指标系列中观察和控制的重点是【】A. M0B. M1C. M2D. M0和M112．货币在表现商品价值并衡量商品价值量的大小时，发挥的职能是【】A. 价值尺度B. 流通手段C. 贮藏手段D. 支付手段13．货币在充当商品流通媒介时发挥的职能是【】A. 价值尺度B. 流通手段C. 贮藏手段D. 支付手段14．当货币退出流通领域，被持有者当作独立的价值形态和社会财富的绝对值化身而保存起来时，货币发挥的职能是【】A. 价值尺度B. 流通手段C. 贮藏手段D. 支付手段15．货币在支付租金、赋税、工资等的时候发挥的职能是【】A. 价值尺度B. 流通手段C. 贮藏手段D. 支付手段16．观念货币可以发挥的职能是【】A. 价值尺度B. 流通手段C. 贮藏手段D. 支付手段17．货币最基本、最重要的职能是【】A. 价值尺度B. 流通手段C. 贮藏手段D. 支付手段18．“劣币驱逐良币现象”产生的货币制度背景是【】A. 银本位B. 平行本位C. 双本位D. 金本位19．最早实行金币本位制的国家是【】A. 美国B. 英国C. 中国D. 德国20．人民币是【】A. 实物货币B. 代用货币C. 金属货币D. 信用货币二、多项选择题（在小题列出的五个备选项中，至少有二个是符合题目要求的，请将其代码写在题后的括弧内。

金融学第二版讲义大纲及课后习题答案详解第八章

CHAPTER 8VALUATION OF KNOWN CASH FLOWS: BONDSObjectives«To show how to value con tracts and securities that promise a stream of cash flows that areknown with certa inty.«To un dersta nd the shape of the yield curve .«To un dersta nd how bond prices and yields cha nge over time.Outline8.1 Us ing Prese nt Value Formulas to Value Known Cash Flows8.2 The Basic Build ing Blocks: Pure Discou nt Bonds8.3 Coupon Bo nds, Curre nt Yield, and Yield to Maturity8.4 Readi ng Bond Listi ngs8.5 Why Yields for the Same Maturity Differ8.6 The Behavior of Bond Prices over TimeSummary* A cha nge in market in terest rates causes a cha nge in the opposite directi on in the market values of all exist ing con tracts promisi ng fixed payme nts in the future.* The market prices of $1 to be received at every possible date in the future are the basic building blocks for valuing all other streams of known cash flows. These prices are inferred from the observed market prices of traded bonds and the n applied to other streams of known cash flows to value them.* An equivale nt valuati on can be carried out by appl ying a discou nted cash flow formula with a differe nt discou nt rate for each future time period.* Differe nces in the prices of fixed-i ncome securities of a give n maturity arise from differe nces in coup on rates, default risk, tax treatme nt, callability, con vertibility, and other features.* Over time the prices of bonds con verge towards their face value. Before maturity, however, bond prices can fluctuatea great deal as a result of cha nges in market in terest rates.Solutions to Problems at End of ChapterBond Valuation with a Flat Term Structure1. Suppose you want to know the price of a 10-year 7% coupon Treasury bond that pays interest annually. a. You have been told that the yield to maturity is 8%. What is the price?b. What is the price if coupons are paid semiannually, and the yield to maturity is 8% per year?c. Now you have been told that the yield to maturity is 7% per year. What is the price? Could you have guessedthe answer without calculating it? What if coupons are paid semiannually?c. Price = 100. When the coup on rate and yield to maturity are the same, the bond sells at par value (i.e. the price equalsthe face value of the bon d).2. Assume six months ago the US Treasury yield curve was flat at a rate of 4% per year (with annualcompounding) and you bought a 30-year US Treasury bond. Today it is flat at a rate of 5% per year. What rate of return did you earn on your initial investment: a. If the bond was a 4% coupon bond? b. If the bond was a zero coupon bond?c. How do your answer change if compounding is semiannual? SOLUTION: a and b.Coupon = 4% 30 4 ? 100 4 PV =100 Zero coupon30 4 ? 100 0 PV =30.83Step 2: Find prices of the bonds today: Coupon = 4% 29.5 5?100 4 84.74 Zero coupon29.5 5 ? 100 0 23.71Step 3: Find rates of retur n:Rate of retur n = (coup on + cha nge in price)/in itial price4% coupon bond: r = (4 + 84.74 —100)/100 = -0.1126 or —11.26%Zero-coupon bon d: r = (0 + 23.71 —30.83)/30.83 = -0.2309 or -23.09%. Note that the zero-coupon bo nd is more sen sitive to yield cha nges tha n the 4% coup on bond. c.Step 1: Find prices of the bonds six mon ths ago:Coup on=4% 60 2 ?100 2 PV =100 Zero coupon 60 2 ? 100 0 PV =30.48 Step 2: Find prices of the bonds today:Coup on=4% 59 2.5? 100 2 84.66 Zero coupon59 2.5 ?10023.30SOLUTION:a. With coup ons paid once a year:Price = 93.29b. With coup ons paid twice a year:Price = 93.20Step 3: Find rates of retur n:Rate of return = (coupon + change in price) / initial price4% coupon bond: r = (2 + 84.66 -100)/100 = -0.1334 or -13.34%Zero coupon bond: r = (0 + 23.30 - 30.48)/30.48 = -0.2356 or -23.56%. Note that the zero-coupon bond is more sen sitive to yield cha nges tha n the 4% coup on bond.Bond Valuatio n With a Non-Flat Term Structure3. Suppose you observe the following prices for zero-coupon bonds (pure discount bonds) that have no risk of default:a. What should be the price of a 2-year coupon bond that pays a 6% coupon rate, assuming coupon paymentsare made once a year starting one year from now?b. Find the missing entry in the table.c. What should be the yield to maturity of the 2-year coupon bond in Part a?d. Why are your answers to parts b and c of this question different?SOLUTION:a. Present value of first year's cash flow = 6 x .97 = 5.82Prese nt value of sec ond year's cash flow = 106 x .90 = 95.4Total prese nt value = 101.22 b^Th^y^^tomaturityon^^^^arzerocoupo^bon^wrt^pr^eof9^an^facevalu^of1^3i^5^^^^^^^^2 I ? I -90 I 100 I 0 1 i = 5.41%c. The yield to maturity on a 2-year 6% coup on bond with price of 101.22 isd. The two bonds are differe nt because they have differe nt coup on rates. Thus they have differe nt yields to maturity.Coupon Stripping4. You would like to create a 2-year synthetic zero-coupon bond. Assume you are aware of the following information: 1-year zero- coupon bonds are trading for $0.93 per dollar of face value and 2-year 7% coupon bonds (annual payments) are selling at $985.30 (Face value = $1,000).a. What are the two cash flows from the 2-year coupon bond?b. Assume you can purchase the 2-year coupon bond and unbundle the two cash flows and sell them.i. How much will you receive from the sale of the first payment?ii. How much do you need to receive from the sale of the 2-year Treasury strip to break even?SOLUTION:a. $70 at the end of the first year and $1070 at the end of year 2.b. i. I would receive .93 x $70 = $65.10 from the sale of the first payment.ii. To break even, I would need to receive $985.30- $65.10 = $920.20 from the sale of the 2-year strip.The Law of One price and Bond Pricing5. Assume that all of the bonds listed in the following table are the same except for their pattern of promised cash flows over time. Prices are quoted per $1 of face value. Use the information in the table and the Law of One Price to infer the values of the missing entries. Assume that coupon payments are annual.6% 2 years 5.5%0 2 years7% 2 years0 1 year $0.95From Bond 1 and Bond 4, we can get the miss ing en tries for the 2-year zero-coup on bond. We know from bond 1 that:2 21.0092 = 0.06/1.055 +1.06/(1.055) . This is also equal to 0.06/(1+z 1) + 1.06/(1+z 2) where z 1 and Z2 are the yields to maturity on on e-year zero-coup on and two-year zero-coup on bonds respectively. From bond 4 , we have z 1, we can find z2.1.0092 -0.06/1.0526 = 1.06/(1+z 2)2, hence z = 5.51%.To get the price P per $1 face value of the 2-year zero-coup on bond, using the same reasoning:1.0092 -0.06x0.95 = 1.06xP, he nee P = 0.8983To find the entries for bond 3: first find the price, then the yield to maturity. To find the price, we can use z 1 and Z2 found earlier: PV of coupon payment in year 1: 0.07 x 0.95 = 0.0665PV of coupon + pri ncipal payme nts in year 2: 1.07 x 0.8983 =0.9612「otal prese nt value of bond 3 二 1.02772 ? 0.07 -1.0277 1 i = 5.50%Hence the table becomes:6% 2 years $1.0092 5.5%0 2 years $0.8983 5.51%SOLUTION:Bond 1:Bond 4:Bond Features and Bond Valuation6. What effect would adding the following features have on the market price of a similar bond which does not have this feature?a. 10-year bond is callable by the company after 5 years (compare to a 10-year non-callable bond);b. bond is convertible into 10 shares of common stock at any time (compare to a non-convertible bond);c. 10-year bond can be “ put back ” to the company after 3 years at par (puttable boiumipare to a 10year non-puttablebond)d. 25-year bond has tax-exempt coupon paymentsSOLUTION:a. The callable bond would have a lower price tha n the non-callable bond to compe nsate the bon dholders for gra nti ng theissuer the right to call the bon ds.b. The con vertible bond would have a higher price because it gives the bon dholders the right to con vert their bonds intoshares of stock.c. The puttable bond would have a higher price because it gives the bondholders the right to sell their bonds back to the issuerat par.d. The bond with the tax-exempt coup on has a higher price because the bon dholder is exempted from pay ing taxes on thecoup ons. (Coup ons are usually con sidered and taxed as pers onal in come).Inferring the Value of a Bond Guarantee7. Suppose that the yield curve on dollar bonds that are free of the risk of default is flat at 6% per year. A 2-year 10% coupon bond (with annual coupons and $1,000 face value) issued by Dafolto Corporation is rates B, and it is currently trading at a market price of $918. Aside from its risk of default, the Dafolto bond has no other financially significant features. How much should an investor be willing to pay for a guarantee against Dafolto ' s defaulting on this bond?The difference between the price of the bond if it were free of default and its actual price (with risk of default) is the value of a guarantee against default: 1073.3-918 = $155.3The implied Value of a Call Provision and Convertibility8. Suppose that the yield curve on bonds that are free of the risk of default is flat at 5% per year. A 20-year default-free coupon bond (with annual coupons and $1,000 face value) that becomes callable after 10 years is trading at par and has a coupon rate of 5.5%.a. What is the implied value of the call provision?b. A Safeco Corporation bond which is otherwise identical to the callable 5.5% coupon bond describedabove, is also convertible into 10 shares of Safeco stock at any time up to the bond ' s maturity. If its yield to maturity is currently 3.5% per year, what is the implied value of the conversion feature?SOLUTION:a. We have to find the price of the bond if it were only free of the risk of default.The bond is traded at par value, hence the differe nee betwee n the value calculated above and the actual traded value is the implied value of the call provisio n: 1062.3 T000 = $62.3Note that the call provisi on decreases the value of the bond.b. We have to find the price of the Safeco Corporati on:This bond has the same features as the 5.5% default free callable bond described above, plus an additional feature: it is con vertible into stocks. Hence the implied value of the con versi on feature is the differe nee betwee n the values of both bonds: 1284.2-1000 = $284.25. Note that the con version feature in creases the value of the bond.Changes in Interest Rates and Bond Prices9. All else being equal, if interest rates rise along the entire yield curve, you should expect that:i. Bond prices will fallii. Bond prices will riseiii. Prices on long-term bonds will fall more than prices on short-term bonds.iv. Prices on long-term bonds will rise more than prices on short-term bondsa. ii and iv are correctb. We can ' t be certain that prices will changec. Only i is correctd. Only ii is correcte. i and iii are correctSOLUTION:The correct an swer is e.Bond prices are in versely proporti onal to yields hence whe n yields in crease, bond prices fall. Lon g-term bonds are more sen sitive to yield cha nges tha n short-term bon ds.。

《金融学(第二版)》讲义大纲及课后习题答案详解十四章

CHAPTE R 14FORWARD AND FUTURE S PRICE SObjectives∙ To explain the economic role of futures markets∙To show what information can and cannot be inferred from forward and futures prices.Outline14.1 Distinctions Between Forward and Futures Contracts14.2 The Economic Function of Futures Markets14.3 The Role of Speculators14.4 Relation Between Commodity Spot and Futures Prices14.5 Extracting Information from Commodity Futures Prices14.6 Spot-Futures Price Parity for Gold14.7 Financial Futures14.8 The Implied Risk-Free Rate14.9 The Forward Price Is Not a Forecast of the Spot Price14.10 Forward-Spot Parity with Cash Payouts14.11 Implied Dividends14.12 The Foreign-Exchange Parity Relation14.13 The Role of Expectations in Determining Exchange RatesSummary∙ Futures contracts make it possible to separate the decision of whether to physically store a commodity from thedecision to have financial exposure to its price changes.∙ Speculators in futures markets improve the informational content of futures prices and make futures marketsmore liquid than they would otherwise be.∙ The futures price of wheat cannot exceed the spot price by more than the cost of carry:∙ The forward-spot price parity relation for gold is that the forward price equals the spot price times the cost ofcarry:This relation is maintained by the force of arbitrage . ∙One can infer the implied cost of carry and the implied storage costs from the observed spot and forward prices and the risk-free interest rate. ∙ The forward-spot parity relation for stocks is that the forward price equals the spot price times 1 plus the risk-free rate less the expected cash dividend.This relation can therefore be used to infer the implied dividend from the observed spot and forward prices and the risk-free interest rate.∙ The forward-spot price parity relation for the dollar/yen exchange rate involves two interest rates:where F is the forward price of the yen, S is the current spot price, r Y is the yen interest rate, and r $ is the dollarinterest rate.∙If the forward dollar/yen exchange rate is an unbiased forecast of the future spot exchange rate, then one can infer that forecast either from the forward rate or from the dollar-denominated and yen-denominated risk-free interest rates. F S C-≤F S r s =++()1F S r D=+-()1F r S r Y11+=+$Solutions to Problems at End of ChapterForward Contracts and Forward-Spot Parity.1. Suppose that you are planning a trip to E ngland. The trip is a year from now, and you have reserved a hotel room in London at a price of ₤ 50 per day. You do not have to pay for the room in advance. The exchange rate is currently $1.50 to the pound sterling.a.E xplain several possible ways that you could completely hedge the exchange rate risk in this situation.b.Suppose that r₤=.12 and r$=.08. Because S=$1.50, what must the forward price of the pound be?c.Show that if F is $0.10 higher than in your answer to part b, there would be an arbitrage opportunity. SOLUTION:a.Ways to hedge the exchange rate risk:Pay for the room in advanceBuy the pounds you will need in the forward market.Invest the present value of the rental payments in a pound-denominated riskless asset.b. F = S (1+r$)/(1+r£) = $1.50 x 1.08/1.12 = $1.4464 per poundc.If F is $1.55 then arbitrage profits can be made by borrowing dollars, investing in pounds and selling themforward at the inflated forward price. After paying off principle and interest on the dollars borrowed, you would have pure arbitrage profits left over. For example,Borrow $1.50,Convert it into 1 pound,Invest it in pound-denominated bonds to have 1.12 pounds a year from now,Sell 1.12 pounds forward at $1.55 per pound to have $1.736 a year from now,After 1 year, pay off the principle and interest on the loan ($1.50x 1.08 = $1.62).This series of transactions leaves you with $.116 a year from now with no initial outlay of your money.Forward-Spot Parity Relation with Known Cash Payouts2. Suppose that the Treasury yield curve is flat at an interest rate of 7% per year (compounded semiannually).a.What is the spot price of a 30-year Treasury bond with an 8% coupon rate assuming coupons are paidsemiannually?b.What is the forward price of the bond for delivery six months from now?c.Show that if the forward price is $1 lower than in your answer to part b, there should be an arbitrageopportunity.SOLUTION:b. The forward price for delivery six months from now is $1,124.089:F = S(1+r) - C = $1,124.724 x 1.035 - 40 =$1,124.089c. If the forward price is only $1,123.089, then arbitrage profits can be made by selling the bond short and buying itforward at the low forward price. It can be described as follows:Sell short a bond at $1,124.724; buy it forward at $1,123.089; invest the proceeds of the short sale to earn 3.5% for6 monthsAfter 6 months, take delivery of the bond and cover your short saleForward-Spot Parity Relation with Uncertain Dividends3. A stock has a spot price of $100; the riskless interest rate is 7% per year (compounded annually), and the expected dividend on the stock is $3, to be received a year from now.a.What should be the one-year futures price?b.If the futures price is $1 higher than your answer to part a, what might that imply about the expected dividend? SOLUTION:a.S = $100, r = .07, D = $3. F = S ( 1+r) - D = $104b.If F is $105, that might imply that D is really only $2.Storage Costs versus Dividend Yield4. Compare the forward-spot price-parity relation for gold to the one for stocks. Is it fair to say that stocks have a negative storage cost equal to the dividend yield?SOLUTIONOne could definitely say that stocks have a negative storage cost equal to the dividend.5. Suppose you are a distributor of canola seed and you observe the spot price of canola to be $7.45 per bushel while the futures price for delivery one month from today is $7.60. Assuming a $.10 per bushel carrying cost, what would you do to hedge your price uncertainty?SOLUTIONWe see that F> S+C. If you short the futures contract, you can sell your seed at $7.60 per bushel.6. Infer the spot price of an ounce of gold if you observe the price of one ounce of gold for forward delivery in three months is $435.00, the interest rate on a 91-day Treasury bill is 1% and the quarterly carrying cost as a percentage of the spot price is .2%.SOLUTIONDeduce from the futures price parity condition for gold that F = S0 (1 + r + s) so that S0 = $429.84.7. You are a dealer in kryptonite and are contemplating a trade in a forward contract. You observe that the current spot price per ounce of kryptonite is $180.00, the forward price for delivery of one ounce of kryptonite in one year is $205.20, and annual carrying costs of the metal are 4% of the current spot price.a.Can you infer the annual return on a riskless zero-coupon security implied by the Law of One Price?b.Can you describe a trading strategy that would generate arbitrage profits for you if the annual return on theriskless security is only 5%? What would your arbitrage profit be, per ounce of kryptonite?SOLUTIONa.By no-arbitrage, we require that the riskless rate r satisfy:F = S0 (1 + r + s)205.2 = 180 (1 +r +.04) = 187.2 + 180rr = 18/180 = .10 or 10%b.The implicit risk-free rate that you can earn by buying kryptonite, storing it, and selling it forward at $205.2 perounce is 10%. If the riskless borrowing rate is five percent, you should borrow at that rate and invest in hedged kryptonite. If you buy an ounce of kryptonite for $180, you will get $205.2 for it for sure a year from now. If you borrow the $180, you will have to pay principal and interest of $180 x 1.05 plus another .04 x $180 in storage costs.This totals $196.2, thus leaving you with $9 in arbitrage profits.8. Calculate the implicit cost of carrying an ounce of gold and the implied storage cost per ounce of gold if the current spot price of gold per ounce is $425.00, the forward price of an ounce of gold for delivery in 273 days is $460.00, the yield over 91 days on a zero-coupon Treasury bill is 2% and the term structure of interest rates is flat. SOLUTIONFirst, we solve it assuming a simple compounding method for the risk free interest rate. Over 273 days, the Risk free rate is 2%*3=6%. Therefore we have,F = S (1 + r + s )460 = 425 (1.06 + s)s = (460 - 450.5)/425 = 9.5/425 = .02235 for 273 daysThus the carrying costs are roughly 8.24% for 273 days or 10.98% per year.Second, we solve it assuming we need to compound the interest rates. The risk free rate over 273 days will be(1+2%)3-1=6.12%.plug in the above formulae we get s=.021145 for 273 days.Thus the carrying costs are roughly 8.23% for 273 days or 11.13% per year.9. The forward price for a share of stock to be delivered in 182 days is $410.00, whereas the current yield on a 91-day T-bill is 2%. If the term structure of interest rates is fiat, what spot price for the stock is implied by the Law of One Price?SOLUTIONF = $410; r = .02 per quarter.S = F/(1+r)2 = $394.0810. You observe that the one-year forward price of a share of stock in Kramer,Inc.,a New York tour-bus company and purveyor of fine clothing, is $45.00 while the spot price of a share is $41.00. If the riskless yield on a one-year zero-coupon government bond is 5%:a.What is the forward price implied by the Law of One Price?b.Can you devise a trading strategy to generate arbitrage profits? How much would you earn per share?SOLUTIONa.The no-arbitrage value of the forward price is F = $43.05.b.The observed forward price is excessive. Consider short-selling a forward contract and taking a long position ina portfolio consisting of one stock and the sale of a bond with face value of F. Future liabilities for this positionare zero, while the current cash inflow is $1.86.11. Infer the yield on a 273-day, zero-coupon Japanese government security if the spot price of a share of stock in Mifune and Associates is 4,750 yen whereas the forward price for delivery of a share in 273 days is 5,000 yen.SOLUTIONThe implied yield over the 273 day term is r = 5.26%.12. On your first day of trading in Vietnamese forward contracts, you observe that the share price of Giap Industries is currently 54, 000 dong while the one-year forward price is 60, 000 dong. If the yield on a one-year riskless security is fifteen percent, are arbitrage profits possible in this market? If not, explain why not. If so, devise an appropriate trading strategy.SOLUTIONArbitrage profits would seem to be possible, since the no-arbitrage forward price implied by these parameters isF = $62,100.The futures contract is underpriced, relative to this no-arbitrage value. Consider taking a long position in the forward contract and simultaneously selling a share of Giap stock and buying a riskless bond with a face value equal to the observed forward price. The liabilities from these joint positions are zero, while the current cash inflow is $1826.09.13. The share price of Schleifer and Associates, a financial consultancy in Moscow, is currently 10, 000 roubles whereas the forward price for delivery of a share in 182 days is 11,000 roubles. If the yield on a riskless zero-coupon security with term to maturity of 182 days is 15%, infer the expected dividend to be paid by Schleifer and Associates over the next six months.SOLUTIONThe implied dividend is 500 roubles.14. The spot rate of exchange of yen for Canadian dollars is currently 113 yen per dollar but the one-year forward rate is 110 yen per dollar. Determine the yield on a one-year zero-coupon Canadian government security if the corresponding yield on a Japanese government security is 2.21%.SOLUTIONThe implied Canadian rate over this term is approximately 5.00%.。

金融学第二版课后习题答案

金融学第二版课后习题答案金融学第二版课后习题答案金融学是一门研究金融市场、金融机构和金融工具的学科，它对于理解和解决现代金融问题具有重要意义。

而课后习题则是帮助学生巩固所学知识、提高解决问题能力的重要工具。

本文将为读者提供金融学第二版课后习题的答案，以帮助读者更好地理解金融学的概念和理论。

第一章：金融的基本概念和职能1. 金融的基本概念是指金融的定义和范围。

金融的定义是指金融活动和金融制度的总称。

金融的范围包括金融市场、金融机构和金融工具等。

2. 金融的职能是指金融对于经济发展和社会进步的作用。

金融的主要职能包括储蓄和融资、支付和结算、风险管理和信息中介等。

第二章：金融市场1. 金融市场的分类包括货币市场、资本市场和衍生品市场等。

货币市场是指短期资金融通的市场，资本市场是指长期资金融通的市场，衍生品市场是指金融衍生品交易的市场。

2. 金融市场的功能包括资源配置、风险管理和信息传递等。

资源配置是指将资金从供给者转移给需求者的过程，风险管理是指通过金融市场进行风险的转移和分散，信息传递是指金融市场通过价格和交易信息传递经济信息。

第三章：金融机构1. 金融机构的分类包括银行、非银行金融机构和金融市场机构等。

银行是最重要的金融机构，它包括商业银行、中央银行和政策性银行等。

2. 金融机构的职能包括储蓄和融资、支付和结算、风险管理和信息中介等。

储蓄和融资是指金融机构接受存款并提供贷款的过程，支付和结算是指金融机构提供支付和结算服务的过程，风险管理是指金融机构通过风险评估和风险转移来管理风险，信息中介是指金融机构通过收集、加工和传递信息来提供金融服务。

第四章：金融工具1. 金融工具的分类包括货币工具、债券、股票和衍生品等。

货币工具是指短期借贷和短期投资的金融工具，债券是指借款人向债权人发行的债务凭证，股票是指公司向股东发行的所有权凭证，衍生品是指衍生自其他金融资产的金融工具。

2. 金融工具的特点包括流动性、收益性和风险性等。

金融数学引论答案 .docx

第一章习题答案1.设总量函数为A(t) = t2 + 2/ + 3 o试计算累积函数a(t)和第n个吋段的利息【仇°解：把t =()代入得4(()) = 3于是：4(t) t? + 2t + 3啲=丽=3In = 4(北)一A(n一1)=(n2 + 2n + 3) — ((n — I)2 + 2(n — 1) + 3))= 2n+l2.对以下两种情况计算从t时刻到冗(£ < n)时刻的利息：(1)厶(0 < r < n);(2)/r =2r(0<r <n).解：(1)I = A(n) - A(t)—In + in-1+ • • • + A+l n(n + 1) t(t + 1)=2 2I = A(n) - A(t)n n=乞h = 土hk=t+l A：=t+13.已知累积函数的形式为：Q(t) = at2 +几若0时刻投入的100元累积到3吋刻为172元,试计算：5时刻投入的10()元在10时刻的终值。

解：由题意得。

(0) = 1, «(3) = = L72=> a = 0.0& 6=14(5) = 100>1(10) = 4(0) • «(10) = 4⑸• W = 100 x 3 = 300.a(5)4.分别对以下两种总量函数计算订和讪:(1) A(t) = 100 + 5t; (2) A(t) = 100(1 + 0.1尸・解：(1)_ 4(5) - 4(4)5 _ 4(4)5二面-.17% . 4(10)-4(9)210 =—4(9)—5=—^ 3.45%145⑵_ 4(5) - 4(4)5 - 4⑷_ 100(1 + 0.1)5 - 100(1 + 0.1)4 = 100(1+ 0.1)4=10%. 4(10) —4(9)皿=_ 100(1+ O.1)10-100(1+ 0.1)9 = 100(1 + 0.1)9=10%5•设4(4) = 1000, i n = O.Oln.试计算4(7)。

金融学第二版讲义大纲及课后习题答案详解第十章

CHAPTER 10AN OVERVIEW OF RISK MANAGEMENTObjectives« To explore how risk affects finan cial decisi on-mak ing.« To provide a con ceptual framework for the man ageme nt of risk.«To explain how the financial system facilitates the efficient allocation of risk-bearing.Outline10.1 What Is Risk?10.2 Risk and Econo mic Decisi ons10.3 The Risk Ma nageme nt Process10.4 The Three Dime nsions of Risk Tran sfer10.5 Risk Tran sfer and Econo mic Efficie ncy10.6 In stituti ons for Risk Man ageme nt10.7 Portfolio Theory: Quan titative An alysis for Optimal Risk Man ageme nt10.8 Probability Distributions of ReturnsSummary* Risk is defined as uncertainty that matters to people. Risk management is the process of formulating the benefit- cost trade-offs of risk-reduction and deciding on a course of action to take. Portfolio theory is the quantitative analysis of those trade-offs to find an optimal course of action.* All risks are ultimately borne by people in their capacity as consumers, stakeholders of firms and other econo mic orga ni zati ons, or taxpayers.* The risk in ess of an asset or a tra nsacti on cannot be assessed in isolati on or in the abstract; it depe nds on the specific frame of refere nee. In on e con text, the purchase or sale of a particular asset may add to one ' s risk exposure; in another, the same transaction may be risk-reducing.* Speculators are in vestors who take positi ons that in crease their exposure to certa in risks in the hope of in creas ing their wealth. In con trast, hedgers take positi ons to reduce their exposures. The same pers on can be a speculator on some exposures and a hedger on others.* Many resource-allocation decisions, such as saving, investment, and financing decisions, are significantly in flue need by the prese nee of risk and therefore are partly risk-ma nageme nt decisi ons.* We disti nguish among five major categories of risk exposures for households: sick ness, disability, and death job loss; consumer-durable asset risk ; liability risk ; and financial asset risk .* Firms face several categories of risks: production risk , price risk of outputs , and price risk of in puts .* There are five steps in the risk-management process: risk identification, risk assessment, selection of riskman ageme nt tech ni ques, impleme ntati on, review.* There are four techniques of risk management: r isk avoidanee, loss prevention and control, risk retention, risk tra nsfer.* There are three dimensions of risk transfer: hedging , insuring , and diversifying .* Diversificati on improves welfare by spread ing risks among many people, so that the existi ng un certa inty matters less. * From society ' s perspective-n^ageme nt in stituti ons con tribute to econo mic efficie ncy in two importa nt ways. First, they shift risk away from those who are least willing or able to bear it to those who are most willing to bear it. Second, they cause a reallocation of resources to production and consumption in accordance with the new distribution of risk-bearing.By allowing people to reduce their exposure to the risk of undertaking certain bus in ess ven tures, they may en courage en trepre neurial behavior that can have a ben efit to society.* Over the cen turies, various econo mic orga ni zati ons and con tractual arra ngeme nts have evolved to facilitate a more efficient allocation of risk-bearing by expanding the scope of diversification and the types of risk that are shifted.* Among the factors limit ing the efficie nt allocati on of risks are tra nsacti ons costs and problems of adverse selecti on and moral hazard.Solutions to Problems at End of ChapterOn the Nature of Risk and Risk Management1. Suppose that you and a friend have decided to go to a movie together next Saturday. You will select any movie for which tickets are available when you get to the theater. Is this a risky situation for you? Explain. Now suppose that your friend has already purchased a ticket for a movie that is going to be released this Saturday. Why is this a risky situation? How would you deal with the risk?SOLUTION:No, the uncertainty doesn ' t represienncteriysokusdo not care which movie you see. However, if your friend has a ticket already, and if you wait till Saturday to buy yours, the show may be sold out. To eliminate the risk that you may not be able to sit with your friend and see the same movie, you might buy your ticket in advance.2. Suppose you are aware of the following investment opportunity: You could open a coffee shop around the corner from your home for $25,000. If business is strong, you could net $15,000 in after-tax cash flows each year over the next 5 years.a. If you knew for certain the business would be a success, would this be a risky investment?b. Now assume this is a risky venture and that there is a 50% chance it is a success and a 50% chance you gobankrupt within 2 years. You decide to go ahead and invest. If the business subsequently goes bankrupt, did you make the wrong decision based on the information you had at the time? Why or why not?SOLUTION:a. No, this investment would not be risky.b. No, you did not make a “ wrong ” decision. When you made your decision, you did not know for certain that thecompany would go bankrupt. You decided to invest for many reasons, including the possibility of making a lot of money.Given your tolerance for risk and the fact that you based our decision on the information available at the time, your decision was not wrong and may have been optimal at the time.3. Suppose you are a pension fund manager and you know today that you need to make a $100,000 payment in 3 months.a. What would be a risk-free investment for you?b. If you had to make that payment in 20 years instead, what would be a risk free investment?c. What do you conclude from your answers to Parts a and b of this question?SOLUTION:a. A risk-free investment for you would be a Treasury Bill (default risk free) which matures in exactly 3 months.b. A risk-free investment would be a zero coupon U.S. Treasury security maturing in 20 years and which would have thesame single payment of $100,000.c. Because risk is dependent upon circumstances, what is risk-free for one individual may be risky for another too. There canbe any number of risk-free investments depending upon circumstances. Your investment time horizon is critical tochoosing the best risk-free investment (so payments in can exactly match payments out so that you are left with no risk).4. Is it riskier to make a loan denominated in dollars or in yen?SOLUTION:It depends on the context. For people whose income and expenses are denominated in dollars (perhaps because they live in the U.S), denominating a loan in yen would be riskier than denominating it in dollars. But for someone whose income and expenses are denominated in yen, denominating the loan in yen would be less risky than in dollars.5. Which risk management technique has been chosen in each of the following situations?« Installing a smoke detector in your home« Investing savings in T-bills rather than in stocks« Deciding not to purchase collision insurance on your car« Purchasing a life insurance policy for yourselfSOLUTION:« Loss preve nti on and con trol.・Risk avoida nee« Risk rete nti on・Risk tran sfer6. You are considering a choice between investing $1,000 in a conventional one-year T-Bill offering an interest rate of 8% and a one-year Index 丄inked Inflation Plus T-Bill offering 3% plus the rate of inflation.a. Which is the safer investment?b. Which offers the higher expected return?c. What is the real return on the Index 丄inked Bond?SOLUTION:a. The inflation-indexed T-Bill offers a fixed real rate of return of 3% over the life of the investment. The realreturn on the conventional T- Bill ' s real return depends upon the expected rate of inflation over the life of thein vestme nt. The safer in vestme nt is the In flati on Plus T-Bill.b. The real rate of return on the conventional T-Bill depends upon the expected rate of inflation over the life of thein vestme nt. You do not know which expected retur n is higher unl ess you know what in flati on is expected to be.c. The real retur n on the in dex-l in ked T-Bill is 3%.Hedging and Insurance7. Suppose you are interested in financing your new home purchase. You have your choice of a myriad financing options. You could enter into any one of the following agreements: 8% fixed rate for 7 years, 8.5% fixed rate for 15 years, 9% fixed for 30 years. In addition, you could finance with a 30-year variable rate that begins at 5% and increases and decreases with the prime rate, or you could finance with a 30year variable rate that begins at 6% with ceilings of 2% per year to a maximum of 12% and no minimum.a. Suppose you believe that interest rates are on the rise. If you want to completely eliminate your risk of risinginterest rates for the longest period of time, which option should you choose?b. Would you consider that hedging or insuring? Why?c. What does you r risk management decision “ cost ” you in terms of quoted interest rates during the firstyear?SOLUTION:a. You would choose the 30-year fixed rate at 9%.b. That would be a hedge because you have elim in ated both the upside (decli ning rates) or dow nside ( rising rates).c. This costs me at least 4% since I could get a variable rate loa n at 5%.8. Referring to the information in problem 7, answer the following:a. Suppose you believe interest rates are going to fall, which option should you choose?b. What risk do you face in that transaction?c. How might you insure against that risk? What does that cost you (in terms of quoted interest rates?). SOLUTION:a. You would want one of the variable rate options, in particular the variable loan tied to the prime rate, currently equal to5%.b. You face the risk of rising rates.c. You could in sure aga inst that risk by purchas ing the opti on to have a 12% ceil ing on the rate (2% in crease per year.This option cost you 1% (the difference between 6% and 5%).9. Suppose you are thinking of investing in real estate. How might you achieve a diversified real estate investment?SOLUTION:« You could own several differe nt build ings in the same gen eral area.« You could own several differe nt build ings in differe nt geographic areas.« You could sell some of your equity own ership to other owners to lower your own in dividual exposure to decli ning market values.10. Suppose the following represents the historical returns for Microsoft and Lotus Development Corporation:Historical ReturnsYear MSFT LOTS110%9%215%12%3-12%-7%420%18%57%5%a. What is the mean return for Microsoft? For Lotus?b. What is the standard deviation of returns for Microsoft? For Lotus?c. Suppose the returns for Microsoft and Lotus have normally distributed returns with means and standarddeviations calculated above. For each stock, determine the range of returns within one expected standard deviation of the mean and within two standard deviations of the mean.SOLUTION:a. Mea n return Microsoft: 8.0%; Lotus: 7.4%b. If you use the formula for the sta ndard deviati on based on a sample of size n:You find that the standard deviations are: MSFT: 10.94%; Lotus: 8.357%.However, if you use the formula for the population standard deviation:You find that the standard deviations are: MSFT 12.23% and LOTS 9.34%.c. Range of returns within 1 standard deviation Microsoft: -2.94% to +18.94% Range of returns within 1 standarddeviation Lotus: -0.957% to + 15.76% Range of returns within 2 standard deviations Microsoft: -13.88% to+29.88% Range of returns within 1 standard deviation Lotus: -9.31% to + 24.11%。

《金融数学》(第二版)练习题(修订版)

⎧kt,
1.18
假设利息力为 δt
=
⎪
⎨ ⎪⎩
1 25
kt
2
,
0<t ≤5 ，期初存入单位 1 在第 10 年末将会累积到 2.7183。试求 k。
5 < t ≤ 10
1.19
已知利息力为 δt
=
1 2+t
，一笔金额为
1
的投资从 t=0
开始的前
n
年赚取的总利息是
8。试求
n。
1.20 1996 年 1 月 1 日，某投资者向一个基金存入 1000，该基金在 t 时刻的利息力为 0.1(t −1)2 ，求 1998 年 1 月 1 日的累积值。
1.7 基金 A 以每月复利一次的名义利率 12 %累积。基金 B 以 δt = t / 6 的利息力累积。在零时刻，分别存入 1 到两个基金中。请问何时两个基金的金额
1
将相等。
1.8
基金
A
以δ t
=
a
+ bt
的利息力累积。基金
B
以δ t
=
g
+
ht
的利息力累积。基金
A
与基金
B
在零时刻和
n
时刻相等。已知 a
3
2.2 某人将在 10 年后退休。他打算从现在开始每年初向一种基金存入 2000 元，如果基金的收益率为 6%，试计算他在退休时可以积存多少退休金。 2.3 某人从 2000 年 3 月 1 日起，每月末可以领取 200 元，2010 年 5 月末是最后一次领取。如果每月复利一次的年名义利率为 6%，试计算：（1）年金的现值；（2）年金的终值；（3）年金在 2005 年 12 月 31 日的值。 2.4 某人在今后的 20 年内，每年初向一基金存入 10000 元。从第 30 年开始，每年末可以领取一笔退休金。该基金的收益率为 6%。（1）如果限期领取 20 年，每次可以领取多少？（2）如果无限期地领下去（当他死亡后，由其继承人领取），每次可以领取多少？ 2.5 某人留下了 10 万元的遗产，遗嘱规定，该笔遗产前 5 年的利息收入由其长子领取，第二个 5 年的利息由其次子领取，从第 11 年开始，剩余遗产全部归第三个儿子。如果年实际利率为 8%，试计算三个儿子在该笔遗产中分别占多大份额？ 2.6 如果年实际利率为 i，那么一笔在 36 年内每年末支付 4000 元的年金，与另一笔在 18 年内每年末支付 5000 元的年金将有相等的现值。试计算 1000 元的投资在年实际利率为 i 时，经过多长时间可以翻番。 2.7 借款人原计划在每月末偿付 1000 元，用 5 年的时间还清贷款。每月复利一次的年名义利率为 12%。如果他现在希望一次性支付 60000 元还清贷款，他应该在何时偿还？ 2.8 投资者每月初往基金存入一笔款项，5 年后可以积存到 60000 元。如果前 2 年每次存入 1000 元，后 3 年每次存入 500 元。试计算每月复利一次的名义利率。 2.9 投资者每年末向一基金存入 2000 元，如果在前 2 年的投资按 6%的年实际利率计算，在后两年的投资按 5%的年实际利率计算，投资者在第 4 年末可以积存多少价值？ 2.10 一项 10 年期的年金，在前 5 年的每季度末付款 1000 元，后 5 年的每季度末付款 2000 元。如果年实际利率为 5%，试计算该项年金的现值。 2.11 一项每 3 年末支付 1 元的永续年金，其现值为 125/91，试确定年实际利率是多少？ 2.12 某人将一笔遗产（每年末可以领取的永续年金）捐赠给了四家慈善机构 A，B，C 和 D。在前 n 年，每次领取的款项由 A、B、C 三家平均分享，n 年以后，剩余部分均由 D 领取。试确定当(1 + i) n 为多少时，A、B、C、D 四家在该遗产中享有的现值相等。假设年实际利率为 8%。 2.13 一项永续年金在每月初付款 1 元，如果每年结转四次利息的年名义利率为 4%，试计算该项年金的现值。

国际金融第二版课后答案(全)

国际金融第二版课后答案（全）国际金融习题答案第一章国际收支本章重要概念国际收支：国际收支是指一国或地区居民与非居民在一定时期内全部经济交易的货币价值之和。

它体现的是一国的对外经济交往，是货币的、流量的、事后的概念。

国际收支平衡表：国际收支平衡表是将国际收支根据复式记账原则和特定账户分类原则编制出来的会计报表。

它可分为经常项目、资本和金融项目以及错误和遗漏项目三大类。

丁伯根原则：1962年，荷兰经济学家丁伯根在其所著的《经济政策：原理与设计》一书中提出：要实现若干个独立的政策目标，至少需要相互独立的若干个有效的政策工具。

这一观点被称为“丁伯根原则”。

米德冲突：英国经济学家米德于1951年在其名著《国际收支》当中最早提出了固定汇率制度下内外均衡冲突问题。

米德指出，如果我们假定失业与通货膨胀是两种独立的情况，那么，单一的支出调整政策（包括财政、货币政策）无法实现内部均衡和外部均衡的目标。

分派原则：这一原则由蒙代尔提出，它的含义是：每一目标应当指派给对这一目标有相对最大的影响力，因而在影响政策目标上有相对优势的工具。

自主性交易：亦称事前交易，是指交易当事人自主地为某项动机而进行的交易。

国际收支失衡：国际收支失衡是指自主性交易发生逆差或顺差，需要用补偿性交易来弥补。

它有不同的分类，根据时间标准进行分类，可分为静态失衡和动态失衡；根据国际收支的内容，可分为总量失衡和结构失衡；根据国际收支失衡时所采取的经济政策，可分为实际失衡和潜在失衡。

复习思考题1．一国国际收支平衡表的经常账户是赤字的同时，该国的国际收支是否可能盈余，为什么？答：可能，通常人们所讲的国际收支盈余或赤字就是指综合差额的盈余或赤字.这里综合差额的盈余或赤字不仅包括经常账户，还包括资本与金融账户，这里，资本与金融账户和经常账户之间具有融资关系。

但是，随着国际金融一体化的发展，资本和金融账户与经常账户之间的这种融资关系正逐渐发生深刻变化。

一方面，资本和金融账户为经常账户提供融资受到诸多因素的制约。

1、下载文档前请自行甄别文档内容的完整性，平台不提供额外的编辑、内容补充、找答案等附加服务。
2、"仅部分预览"的文档,不可在线预览部分如存在完整性等问题,可反馈申请退款(可完整预览的文档不适用该条件!)。
3、如文档侵犯您的权益，请联系客服反馈,我们会尽快为您处理(人工客服工作时间：9:00-18:30)。

金融数学引论答案第二版【篇一：北大版金融数学引论第二章答案】>第二章习题答案1．某家庭从子女出生时开始累积大学教育费用5万元。

如果它们前十年每年底存款1000元，后十年每年底存款1000+x 元，年利率7%。

计算x 。

解：s = 1000s?7%+xs?7%20p10p20px = 50000 ? 1000s?7% = 651.72s?p7%102．价值10,000元的新车。

购买者计划分期付款方式：每月底还250元，期限4年。

月结算名利率18%。

计算首次付款金额。

解：设首次付款为x ，则有10000 = x + 250a?p1.5%48解得x = 1489.3613．设有n年期期末年金，其中年金金额为n，实利率i =n解：p v = na?npi= 1nn+2 =(n + 1)nn2n4．已知：a?pn= x，a?p2n= y 。

试用x和y 表示d 。

解： a?p2n= a?pn+ a?p (1 ? d)则nny ? xd = 1 ? ( x ) n5．已知：a?p7= 5.58238, a?= 7.88687, a?= 10.82760。

计算i。

11p18p解：a?p = a?p + a?p v718711解得=i = 6.0%10?p +a∞?p6.证明： 11?v10s。

解：p v = 100a?+ 100a20?8p3% p3% = 2189.7168．某人现年40岁，现在开始每年初在退休金帐号上存入1000元，共计25年。

然后，从65岁开始每年初领取一定的退休金，共计15年。

设前25年的年利率为8%，后15年的年利率7%。

计算每年的退休金。

解：设每年退休金为x，选择65岁年初为比较日=解得x = 8101.658。

1解： d = 10%，则 i=1?d? 1 =981 ? v8nnv；nnnn1n1n1ni+ 1? vnn1+i所以nn(1+nni)n(1+i)n?1=(1+i)?1nd=? 1i1+ii+ (1 + i)n所以nn版权所有，翻版必究12.从1980年6月7日开始，每季度年金100元，直至1991年12月7日，季结算名利率6%，计算：1）该年金在1979年9月7日的现值；2）该年金在1992年6月7日的终值。

解：p v = 100a49?p1.5% ? 100a?2p1.5% = 3256.88av = 100s?1.5% ? 100s?p1.5% = 6959.37p213.现有价值相等的两种期末年金a和b。

年金a在第1－10年和第21－30年中每年1元，在第11－20年中每年2元；年金b在第1－10年和第21－30年中每年付款金额为y ，在第11－20年中没有。

已知：v=，计算y 。

102解：因两种年金价值相等，则有a?i+a?iv10=y a? ?iy a10?piv1030p10p30p所以 y =31030.814.已知年金满足：2元的2n期期末年金与3元的n期期末年金的现值之和为36；另外，递延n年的2元n 期期末年金的现值为6。

计算i。

1+v10?2v30= 1解：由题意知，2a?pi+ 3a?pi = 362nn2a?pivn= 6n解得73xi = 8.33%yzp a?p a?p + s?= 15.已a?p a?p + s?p 。

求x，y和z。

知解：由题意得=1 ? v11 (1 + i)z ? vy解得x = 4, y = 7, z = 4117x3153016.化简a15?p (1 + v+ v)。

4.5%解：年金在4月1日的价值为p =2000 = 46444.44 ，则1+4pp v =(1 + i)2+= 41300.657318.某递延永久年金的买价为p ，实利率解：设递延时间为t，有1 p = i vtln解得t = ? ln(1+i)19.从现在开始每年初存入1000元，一直进行20年。

从第三十年底开始每年领取一定的金额x，直至永远。

计算x。

解：设年实利率为i，由两年金的现值相等，有x ?=i29解得x = 1000((1 + i)? (1 + i))301020.某人将遗产以永久年金的方式留给后代a、b、c、和d：前n年，a、b和c三人平分每年的年金，n年后所有年金由d一人继承。

如果四人的遗产份额的现值相同。

计算(1 + i)。

n解： i，那么a,b,c得到的遗产的现值为 i ，而d得到遗产的现值为v。

由题意得 3?pinn1 ? v= v 3nn所以(1 + i)= 4n21.永久期末年金有a、b、c、和d四人分摊，a接受第一个n年，b接受第二个n年，c接受第三个n 年，d接受所有剩余的。

已知：c与a的份额之比为0.49，求b与d的份额之比。

计算最后一次还款的数量和时间。

vnp4.5%41000 100a?解：100an+1?p4.5%v4100016解得 n = 172列价值方程解得+100a?p4.5%xv1 = 1000x = 146.0723.36年的期末年金每次4元，另有18年的期末年金每次5元；两者现值相等。

如果以同样的年利率计算货币的价值在n年内将增加一倍，计算n。

由题意， (1 + i)= 2 解得 n = 91836pn24.某借款人可以选择以下两种还贷方式：每月底还100元，5年还清；k个月后一次还6000元。

已知月结算名利率为12%，计算k。

解：由题意可得方程100a?p1% = 6000(1 + i)?k60解得k = 2925.已知a?pi= 1.75，求i。

2解：由题意得1 ? v= 1.75i2解得i = 9.38%26.某人得到一万元人寿保险赔付。

如果购买10年期末年金可以每年得到1538元，20年的期末年金为每年1072元。

计算年利率。

解：【篇二：金融数学引论北大版第4章答案】现有1000 元贷款计划在5 年内按季度偿还。

已知季换算名利率6%，计算第2 年底的未结贷款余额。

解：设每个季度还款额是r ，有ra(4)5p6%￢ = 1000解得r ，代入b2 的表达式b2 = ra(4)3p6%￢= 635.32 元2 设有10000 元贷款，每年底还款2000 元，已知年利率12% ，计算借款人的还款总额等于原贷款额时的未结贷款余额。

解：n =100002000= 5= 4917.72 元3 某贷款在每季度末偿还1500 元，季换算名利率10% ，如果已知第一年底的未结贷款余额为12000 元，计算最初的贷款额。

解：以季度为时间单位，i = 2.5% 。

b0 = b1 ? v + 1500a4pi ￢= 16514.4 元4 某贷款将在15 年内分期偿还。

前5 年每年底还4000 元，第二个5 年每年底还 3000 元，最后5 年每年底还2000 元。

计算第二次3000 元还款后的未结贷款余额的表达式。

如果已知第4 次还款后的未结贷款余额为5000 元，计算原始贷款金额。

解：设原始贷款额为l ，每次还款为r ，以半年为时间单位，有 ???5000 = ra3p4% ￢l = ra7p4% ￢整理得：l = 5000 ? a￢7pa￢3p= 10814.16 元6 现有20000 元贷款将在12 年内每年底分期偿还。

若(1+i)4 = 2 ，计算第4 次还款后的未结贷款余额。

解：设第4 次还款后的未结贷款余额为l ，每次还款为r ，有 ???20000 = r ? a12pi ￢l = r ? a8pi ￢把(1 + i)4 = 2 代入整理得：l = 5000 ? 1 ? (1 + i)?81 ? (1 + i)?12= 17142.86 元7 20000 元抵押贷款将在20 年内每年分期偿还，在第5 次还款后，因资金短缺，随后的两年内未进行正常还贷。

若借款人从第8 年底重新开始还贷，并在20 年内还清。

计算调整后的每次还款额。

解：设正常每次还款为r ，调整后每次还款x ，以当前时间和第5年底为比较日，有???20000 = ra2￢0pxa1￢3p ? v2 = ra1￢5p整理得：x = 20000 ? a15p ￢a2￢0p? (1 + i)2a1￢3p8 某贷款l 原计划在25 年内分年度等额还清。

但实际上从第6 次到第10 次的还款中每次多付k 元，结果提前5 年还清贷款。

试证明： k =a2￢0p ? a1￢5pa2￢5p a￢5p l证：以第20 年年底为比较日，设每次还款为r ，有???l = ra2￢5pks￢5p (1 + i)10 = ra￢5p整理即得。

9 设bt 表示未结贷款余额，证明：(1) (bt ? bt+1)(bt+2 ? bt+3) = (bt+1 ? bt+2)2;(2) bt + bt+3 bt+1 + bt+2证： (1)(bt ? bt+1)(bt+2 ? bt+3) = (r + bt+11 + i? bt+1) ? (bt+2 ? ((1 + i)bt+2 ? r))=r ? ibt+11 + i? (r ? ibt+2)= (r ? ibt+1) ? r ? i((1 + i)bt+1 ? r)1 + i= (r ? ibt+1)2= (bt+1 ? bt+2)2(2)bt ? bt+1 = r ? ibtr ? ibt+2= bt+2 ? bt+3) bt + bt+3 bt+1 + bt+2默认每次还款额是相同的！10 某贷款按季度分期偿还。

每次1000 元，还期5 年，季换算名利率12%。

计算第6 次还款中的本金量。

解：p6 = b5 ? b6= 1000a20?5p3% ￢ ? 1000a20?6p3% ￢= 641.86 元11 n 年期贷款，每年还款1元。