09年SOA北美精算师考试第二门FM官方样题第一部分(主要是金融数学)答案

SAMPLE SOLUTIONS FOR DERIVATIVES MARKETS

Question #1

Answer is D

If the call is at-the-money, the put option with the same cost will have a higher strike price.

A purchased collar requires that the put have a lower strike price. (Page 76)

Question #2

Answer is C

66.59 – 18.64 = 500 – K exp(–0.06) for K = 480 (Page 69)

Question #3

Answer is D

The accumulated cost of the hedge is (84.30-74.80)exp(.06) = 10.09.

Let x be the market price.

If x < 0.12 the put is in the money and the payoff is 10,000(0.12 – x) = 1,200 – 10,000x. The sale of the jalapenos has a payoff of 10,000x – 1,000 for a profit of 1,200 – 10,000x + 10,000x – 1,000 – 10.09 = 190.

From 0.12 to 0.14 neither option has a payoff and the profit is 10,000x – 1,000 – 10.09 = 10,000x – 1,010.

If x >0.14 the call is in the money and the payoff is –10,000(x – 0.14) = 1,400 – 10,000x. The profit is 1,400 – 10,000x + 10,000x – 1,000 – 10.09 = 390.

The range is 190 to 390. (Pages 33-41)

Question #4

Answer is B

The present value of the forward prices is 10,000(3.89)/1.06 + 15,000(4.11)/1.0652 +

20,000(4.16)/1.073 = 158,968. Any sequence of payments with that present value is acceptable. All but B have that value. (Page 248)

Question #5

Answer is E

If the index exceeds 1,025, you will receive x – 1,025. After buying the index for x you will have spent 1,025. If the index is below 1,025, you will pay 1,025 – x and after buying the index for x you will have spent 1,025. One way to get the cost is to note that the forward price is 1,000(1.05) = 1,050. You want to pay 25 less and so must spend 25/1.05 = 23.81 today. (Page 112)

Question #6

Answer is E

In general, an investor should be compensated for time and risk. A forward contract has no investment, so the extra 5 represents the risk premium. Those who buy the stock expect to earn both the risk premium and the time value of their purchase and thus the expected stock value is greater than 100 + 5 = 105. (Page 140)

Question #7

Answer is C

All four of answers A-D are methods of acquiring the stock. The prepaid forward has the payment at time 0 and the delivery at time T. (Pages 128-129)

Question #8

Answer is B

Only straddles use at-the-money options and buying is correct for this speculation. (Page 78)

Question #9

Answer is D

This is based on Exercise 3.18 on Page 89. To see that D does not produce the desired outcome, begin with the case where the stock price is S and is below 90. The payoff is S + 0 + (110 – S) – 2(100 – S) = 2S – 90 which is not constant and so cannot produce the given diagram. On the other hand, for example, answer E has a payoff of S + (90 – S) + 0 – 2(0) = 90. The cost is 100 + 0.24 + 2.17 – 2(6.80) = 88.81. With interest it is 93.36. The profit is 90 – 93.36 = –3.36 which matches the diagram.

Question #10

Answer is D

[rationale-a] True, since forward contracts have no initial premium

[rationale-b] True, both payoffs and profits of long forwards are opposite to short forwards.

[rationale-c] True, to invest in the stock, one must borrow 100 at t=0, and then pay back 110 = 100*(1+.1) at t=1, which is like buying a forward at t=1 for 110. [rationale-d] False, repeating the calculation shown above in part c), but with 10% as a continuously compounded rate, the stock investor must now pay back

100*e.1 = 110.52 at t=1; this is more expensive than buying a forward at t=1

for 110.00.

[rationale-e] True, the calculation would be the same as shown above in part c), but now the stock investor gets an additional dividend of 3.00 at t=.5, which the

forward investor does not receive (due to not owning the stock until t=1). [This is based on Exercise 2-7 on p.54-55 of McDonald]

[McDonald, Chapter 2, p.21-28]

Question #11

Answer is C

Solution: The 35-strike call has future cost (at t=1) of 9.12*(1+.08) = 9.85

The 40-strike call has future cost (at t=1) of 6.22*(1+.08) = 6.72

The 45-strike call has future cost (at t=1) of 4.08*(1+.08) = 4.41

If S1<35, the profits of the 3 calls, respectively, are -9.85, -6.72, and -4.41.

If 35

If 40

If S1>45, the profits of the 3 calls, respectively, are S1-44.85, S1-46.72, and S1-49.41.

The cutoff points for when the relative profit ranking of the 3 calls change are:

S1-44.85=-6.72, S1-44.85=-4.41, and S1-46.72=-4.41, yielding cutoffs of 38.13, 40.44, and 42.31.

If S1<38.13, the 45-strike call has the highest profit, and the 35-strike call the lowest.

If 38.13

If 40.44

If S1<42.31, the 35-strike call has the highest profit, and the 45-strike call the lowest.

We are looking for the case where the 35-strike call has the highest profit, and the 40-strike call has the lowest profit, which occurs when S1 is between 40.44 and 42.31.

[This is based on Exercise 2-13 on p.55-56 of McDonald]

[McDonald, Chapter 2, p.33-37]

Question #12

Answer is B

Solution: The put premium has future value (at t=.5) of 74.20 * (1+(.04/2)) = 75.68 Then, the 6-month profit on a long put position is: max(1,000-S.5,0)-75.68. Correspondingly, the 6-month profit on a short put position is 75.68-max(1,000-S.5,0). These two profits are opposites (naturally, since long and short positions have opposite payoff and profit). Thus, they can only be equal if producing 0 profit. 0 profit is only obtained if 75.68 = max(1,000-S.5,0), or 1,000-S.5 = 75.68, or S.5 = 924.32. [McDonald, Chapter 2, p.39-42]

Question #13

Answer is D

Solution: Buying a call, in conjunction with a short position in a stock index, is a form of insurance called a cap. Answers (A) and (B) are incorrect because they relate to a floor, which is the purchase of a put to insure against a long position in a stock index. Answer (E) is incorrect because it relates to writing a covered call, which is the sale of a call along with a long position in the stock index, so that the investor is selling rather than buying insurance. Note that a cap can also be thought of as ‘buying’ a covered call. Now, let’s calculate the profit:

2-year profit = payoff at time 2 – the future value of the initial cost to establish the position = (-75 + max(75-60,0)) – (-50 + 10)*(1+.03)2 = -75+15+40*(1.03)2 = 42.44-60 = -17.56. Thus, we’ve lost more from holding the short position in the index (since the index went up) than we’ve gained from owning the long call option.

[McDonald, Chapter 3, p.59-65]

Question #14

Answer is A

Solution: This consists of standard applications of the put-call parity equation on p.69. Let C be the price for the 40-strike call option. Then, C + 3.35 is the price for the 35-strike call option. Similarly, let P be the price for the 40-strike put option. Then, P – x is the price for the 35-strike put option, where x is what we’re trying to find. Using put-call parity, we have:

(C + 3.35) + 35*e-.02 - 40 = P – x (this is for the 35-strike options)

C + 40*e-.02 – 40 = P (this is for the 40-strike options)

Subtracting the first equation from the second, 5*e-.02 – 3.35 = x = 1.55.

[McDonald, Chapter 3, p.68-69]

Answer is C

Solution: The initial cost to establish this position is 5*2.78 – 3*6.13 = -4.49. Thus, you are receiving 4.49 up front. This grows to 4.49*e .08*.25 = 4.58 after 3 months. Then, the following payoff/profit table can be constructed at T=.25 years:

S T : 5*max(S T – 40, 0) – 3*max(S T – 35, 0) + 4.58 = Profit

S T <

35 0 - 0 + 4.58 = 4.58 35 <= S T <= 40 0 - 3*(S T – 35) + 4.58 = 109.58-3S T S T > 40 5*(S T -40) - 3*(S T – 35) + 4.58 = 2S T -90.42

Thus, the maximum profit is unlimited (as S T increases beyond 40, so does the profit) Also, the maximum loss is 10.42 (occurs at S T = 40, where profit = 109.58-120 = -10.42)

[Notes] The above problem is an example of a ratio spread.

[McDonald, Chapter 3, p.73]

Question #16

Answer is D

Solution: The ‘straddle’ consists of buying a 40-strike call and buying a 40-strike put. This costs 2.78 + 1.99 = 4.77 at t=0, and grows to 4.77*e .02 = 4.87 at t=.25. The ‘strangle’ consists of buying a 35-strike put and a 45-strike call. This costs 0.44 + 0.97 = 1.41 at t=0, and grows to 1.41*e .02 = 1.44 at t=.25. For S T <40, the ‘straddle’ has a profit of 40-S T -4.87 = 35.13, and for S T >=40, the ‘straddle’ has a profit of S T -40-4.87 = 44.87. For S T <35, the ‘strangle’ has a profit of 35-S T -1.44 = 33.56, and for S T >45, the ‘strangle’ has a profit of S T -45-1.44 = 46.44. However, for 35<=S T <=45, the ‘strangle’ has a profit of -1.44 (since both options would not be exercised). Comparing the payoff structures between the ‘straddle’ and ‘strangle,’ we see that if S T <35 or if S T >45, the ‘straddle’ would outperform the ‘strangle’ (since 35.13 > 33.56, and since -44.87 > -46.44). However, if 35<=S T <=45, we can solve for the two cutoff points for S T , where the ‘strangle’ would outperform the ‘straddle’ as follows:

-1.44 > 35.13 – S T, and -1.44 > S T - 44.87. The first inequality gives S T > 36.57, and the second inequality gives S T < 43.43. Thus, 36.57 < S T < 43.43.

[McDonald, Chapter 3, p.78-80]

Answer is B

[rationale I] Yes, since Strategy I is a bear spread using calls, and bear spreads perform better when the prices of the underlying asset goes down.

[rationale II] Yes, since Strategy II is also a bear spread – it just uses puts instead! [rationale III] No, since Strategy III is a box spread, which has no price risk; thus, the payoff is the same (1,000-950 = 50), no matter what the price of the

underlying asset.

[Note]: An alternative, but much longer, solution is to develop payoff tables for all 3 strategies.

[McDonald, Chapter 3, p.70-73]

Question #18

Answer is B

Solution: First, let’s calculate the expected one-year profit without using the forward. This would be .2*(700+150-750) + .5(700+170-850) + .3*(700+190-950) = 20 + 10 - 18 = 12. Next, let’s calculate the expected one-year profit when buying the 1-year forward for 850. This would be 1*(700+170-850) = 20. Thus, the expected profit increases by 20 - 12 = 8 as a result of using the forward.

[This is based on Exercise 4-7 on p.122 of McDonald]

[McDonald, Chapter 4, p.98-100]

Question #19

Answer is D

Solution: There are 3 cases, one for each row in the above probability table.

For all 3 cases, the future value of the put premium (at t=1) = 100*e.06 = 106.18.

In Case 1, the 1-year profit would be: 750 - 800 - 106.18 + max(900-750,0) = -6.18

In Case 2, the 1-year profit would be: 850 - 800 - 106.18 + max(900-850,0) = -6.18

In Case 3, the 1-year profit would be: 950 - 800 - 106.18 + max(900-950,0) = 43.82 Thus, the expected 1-year profit = .7 * -6.18 + .3 * 43.82 = -4.326 + 13.146 = 8.82.

[This is based on Exercise 4-3 on p.121 of McDonald]

[McDonald, Chapter 4, p.92-96]

Answer is B

Solution: This is an example of pricing a forward contract using discrete dividends. Thus, we need the future value of the current stock price minus the future value of each of the 12 dividends, where the valuation date is T=3. Thus, the valuation equation is: Forward price = 200*e.04(3) – [1.50*e.04(2.75) + 1.50*1.01*e.04(2.5) + 1.50*(1.01)2*e.04(2.25) + …

1.50*(1.01)12] = 200*e.12 - 1.50*e.11{[1-(1.01*e-.01)12]/[1-(1.01*e.01)]}, using the geometric series formula from interest theory. This simplifies numerically to 225.50 -

1.67442*11.99666 = 205.41.

[This problem combines the material from interest theory and derivatives, although one could also simplify the above expression by brute force (instead of geometric series), since there are only 12 dividends to accumulate forward to T=3.]

[McDonald, Chapter 5, p.133-134]

Question #21

Answer is E

Solution: Here, the fair value of the forward contract is given by S0 * e(r-d)T =

110 * e(.05-.02).5 = 110 * e.015 = 111.66. This is 0.34 less than the observed price. Thus, one could exploit this arbitrage opportunity by selling the observed forward at 112 and buying a synthetic forward at 111.66, making 112-111.66 = 0.34 profit.

[This is based on Exercise 5-8 on p.163-164 of McDonald]

[McDonald, Chapter 5, p.135-138]

Answer is B

Solution: First, we must determine the present value of the forward contracts. On a per ton basis, this is: 1,600/1.05 + 1,700/(1.055)2 + 1,800/(1.06)3 = 4,562.49.

Then, we must solve for the level swap price, which is labeled x below, as follows:

4,562.49 = x/1.05 + x/(1.055)2 + x/(1.06)3 = x*[1/1.05 + 1/(1.055)2 + 1/(1.06)3] =

2.69045*x.

Thus, x = 4,562.49 / 2.69045 = 1,695.81.

Thus, the amount he would receive each year is 50*1,695.81 = 84,790.38. [McDonald, Chapter 8, p.247-248]

Question #23

Answer is E

Solution: First, note that the notional amount and the future 1-year LIBOR rates (not given) do not factor into the calculation of the swap’s fixed rate. All we need at the various zero-coupon bond prices P(0, t) for t=1,2,3,4,5, along with the 1-year implied forward rates, which are given by r0(t-1,t), for t=1,2,3,4,5. These calculations are shown in the following table:

t 1 2 3 4 5

P(0,t) (1.04)-1(1.045)-2 (1.0525)-3 (1.0625)-4 (1.075)-5

=.96154 =.91573 =.85770 =.78466 =.69656 r0(t-1,t) s1[1.0452/1.04]-1 [1.05253/1.0452]-1 [1.06254/1.05253]-1 [1.0755/1.06254]-1 =.04000 =.05002 =.06766 =.09307 =.12649 Thus, the fixed swap rate = R = [(.96154)*(.04)+…+(.69656)*(.12649)] / [.96154 +…+

.69656]

= [.03846 + .04580 + .05803 + .07303 + .08811]/[.96154 + .91573 + .85770 + .78466 +

.69656]

= .30344 / 4.21619 = .07197 = 7.20% (approximately).

[Note: This is much less calculation-intensive if you realize that the numerator (.30344) for R can also be obtained by taking 1- P(0,n) = 1 – P(0,5) = 1 - .69656 = .30344. Then, you would not need to calculate any of the implied forward rates!]

[McDonald, Chapter 8, p.255-258]

Answer is D

[rationale-a] True, hedging reduces the risk of loss, which is a primary function of derivatives.

[rationale-b] True, derivatives can be used the hedge some risks that could result in bankruptcy.

[rationale-c] True, derivatives can provide a lower-cost way to effect a financial

transaction.

[rationale-d] False, derivatives are often used to avoid these types of restrictions. [rationale-e] True, an insurance contract can be thought of as a hedge against the risk of loss.

[McDonald, Chapter 1, p.2-3]

Question #25

Answer is C

[rationale-a] True, both types of individuals are involved in the risk-sharing process. [rationale-b] True, this is the primary reason reinsurance companies exist.

[rationale-c] False, reinsurance companies share risk by issuing rather than investing in catastrophic bonds. In effect, they are ceding this excess risk to the

bondholder.

[rationale-d] True, it is diversifiable risk which is reduced or eliminated when risks are shared.

[rationale-e] True, this is a fundamental idea underlying risk management and

derivatives.

[McDonald, Chapter 1, p.5-6]

Question #26

Answer is B

[rationale-I] True, the forward seller has unlimited exposure if the underlying asset’s price increases.

[rationale-II] True, the call issuer has unlimited exposure if the underlying asset’s price rises.

[rationale-III] False, the maximum loss on selling a put is FV(put premium) – strike price. [McDonald, Chapter 2, p.43 (Table 2.4)]

Answer is A

[rationale-I] True, as prices go down, the value of holding a put option increases.

can

be thought of as a put option.

insurance

Homeowner’s

[rationale-II] False, returns from equity-linked CDs are zero if prices decline, but positive if prices rise. Thus, owners of these CDs benefit from rising prices. [rationale-III] False, a synthetic forward consists of a long call and a short put, both of which benefit from rising prices (so the net position also benefits as such). [McDonald, Chapter 2, p.44-48]

Question #28

Answer is E

[rationale-a] True, derivatives are used to shift income, thereby potentially lowering taxes.

[rationale-b] True, as with taxes, the transfer of income lowers the probability of

bankruptcy.

[rationale-c] True, hedging can safeguard reserves, and reduce the need for external financing, which has both explicit (e.g. – fees) and implicit (e.g. –

reputational) costs.

[rationale-d] True, when operating internationally, hedging can reduce exchange rate risk. [rationale-e] False, a firm that credibly hedges will reduce the riskiness of its cash flows, and will be able to increase debt capacity, which will lead to tax savings,

since interest is deductible.

[McDonald, Chapter 4, p.103-106]

Question #29

Answer is A

Solution: If S0 is the price of the stock at time-0, then the following payments are required: Outright purchase – payment at time 0 – amount of payment = S0.

Fully leveraged purchase – payment at time T – amount of payment = S0*e rT.

Prepaid forward contract – payment at time 0 – amount of payment = S0*e-dT.

Forward contract – payment at time T – amount of payment = S0*e(r-d)T.

Since r>d>0, it must be true that S0*e-dT < S0 < S0*e(r-d)T < S0*e rT.

Thus, the correct ranking is given by choice (A).

[McDonald, Chapter 5, p.127-134]

Answer is C

[rationale-a] True, marking to market is done for futures, and can lead to price

differences relative to forward contracts.

[rationale-b] True, futures are more liquid; in fact, if you use the same broker to buy and sell, your position is effectively cancelled.

[rationale-c] False, forwards are more customized, and futures are more standardized. [rationale-d] True, because of the daily settlement, credit risk is less with futures (v.

forwards).

[rationale-e] True, futures markets, like stock exchanges, do have daily price limits. [McDonald, Chapter 5, p.142]

金融数学附答案

金融数学附答案文件排版存档编号：[UYTR-OUPT28-KBNTL98-UYNN208]

1、给定股票价格的二项模型，在下述情况下卖出看涨期权 S 0 S u S d X r τ 股数 50 60 40 55 1/2 1000 （1）求看涨期权的公平市场价格。 （2）假设以公平市场价格+美元卖出1000股期权，需要买入多少股股票进行套期保值，无风险利润是多少 答案：（1）d u d r S S S e S q --=τ0=56.040 6040505.005.0=--??e （2）83.2>73.2，τr e S V -?+?='00 83.2> τr e S -?+?'0 40 6005--=--=?d u S S D U =25.0股 104025.00'-=?-=?-=?d S D 753.9975.0105.005.0'-=?-=??-e 美元 则投资者卖空1000份看涨期权，卖空250股股票，借入9753美元 所以无风险利润为1.85835.005.0=?e 美元 2、假定 S 0 = 100，u=，d=，执行价格X=105，利率r=，p=，期权到期时间t=3， 请用连锁法则方法求出在t=0时该期权的价格。(答案见课本46页) 3、一只股票当前价格为30元，六个月期国债的年利率为3%，一投资者购买一份执行价格为35元的六个月后到期的美式看涨期权，假设六个月内股票不派发红利。波动率σ为. 问题：（1）、他要支付多少的期权费【参考N （=；N （）= 】 {提示：考虑判断在不派发红利情况下，利用美式看涨期权和欧式看涨期权的关系}

解析：在不派发红利情况下，美式看涨期权等同于欧式看涨期权！所以利用Ｂ—Ｓ公式，就可轻易解出来这个题！同学们注意啦，N（d1）=N（），N（d2）=N （）。给出最后结果为 4、若股票指数点位是702，其波动率估计值σ=，指数期货合约将在3个月后到期，并在到期时用美元按期货价格计算，期货合约的价格是715美元。关于期货的看涨期权时间与期货相同，执行价是740美元，短期利率位7%，问这一期权的理论价格是多少（N（）=，N）= *= 解：F=715，T-t=,σ=,X=740,r= F/X=715/740=，σ(T-t)=*= d1=ln/+2= d2== G=**740) =美元 5、根据看涨期权bs定价公式证明德尔塔等于N（d1）（答案见课本122页）

金融数学第一章练习试题详解

金融数学第一章练习题详解 第 1 章 利息度量 1.1 现在投资$600，以单利计息，2 年后可以获得$150 的利息。如果以相同的复利利率投资$2000，试确定在 3 年后的累积值。 65.2847%)5.121(2000% 5.1215026003=+=?=?i i 1.2 在第 1 月末支付 314 元的现值与第 18 月末支付 271 元的现值之和，等于在第 T 月末支付 1004 元的现值。年实际利率为 5% 。求 T 。 58 .1411205.1ln /562352.0ln 562352.0ln 05.1ln 12 562352.01004/)05.127105.1314(05.105.1%)51()1(271314100412/1812/112/12 /1812/112/=?-==-=?+?==+=+=+=------T T i v v v v T t t t t T 两边取对数，其中 1.3 在零时刻，投资者 A 在其账户存入 X ，按每半年复利一次的年名义利率 i 计息。同时，投资者Ｂ在另一个账户存入 2X ，按利率 i （单利）来计息。 假设两人在第八年的后六个月中将得到相等的利息，求 i 。 094588 .02)12(2)2 1(2 )21()21()21())2 1()21((2 12:))21()21((:215/11515151615161516=?-==+?+=+-+==+-+=??+-+i i i i i i i Xi i i X Xi i X B i i X A i A 两边取对数 ，的半年实际利率为 1.4 一项投资以 δ 的利息力累积，27.72 年后将翻番。金额为 1 的投资以每两年复利一次的名义利率 δ 累积 n 年，累积值将成为 7.04。求 n 。 () 80 2)05.1ln /04.7(ln 04 .7)21025 .072.27/2ln 2 )1()(1ln 2/5.072.27=?==+=====+=+=n i e e i t a i n t t δδ δδδδ（

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材让你的学习事半功倍……在课堂教学和报考服务、工作推荐等工作上形成一套特色体系，使人才培养质量达到更高的水平。 在此，我们对在此次考试中取得优异成绩的学员表示祝贺。同时，希望他们再接再厉，在新的起点上，奋发图强，坚持不懈，争取更大的辉煌。也希望其他学员能够学习总结师兄师姐成功经验、锲而不舍，在自己的SOA考试中取得优秀的成绩。 来源：宏景AICPA 原创分享，转载请联系授权，未经授权禁止转载。文中图片部分来自于网络，版权归原作所有，如有侵权行为请联系删除。

金融数学试卷及答案

一、填空（每空4分，共20分） 1．一股股票价值100元，一年以后，股票价格将变为130元或者90元。假设相应的衍生产 品的价值将为U=10元或D=0元。即期的一年期无风险利率为5%。则t=0时的衍生产品 的价格_______________________________。（利用博弈论方法) 2．股票现在的价值为50元，一年后，它的价值可能是55元或40元，一年期利率为4%， 则执行价为45元的看跌期权的价格为___________________。（利用资产组合复制方法） 3．对冲就是卖出________________， 同时买进_______________。 4．Black-Scholes 公式_________________________________________________。 5．我们准备卖出1000份某公司的股票期权，这里.1,30.0,05.0,40,500=====T r X s σ 因此为了对我们卖出的1000份股票期权进行对冲，我们必须购买___________股此公司 的股票。（参考8643.0)100.1(,8554.0)060.1(==N N ） 1．（15分）假设股票价格模型参数是：.120,8.0,7.10===S d u 一个欧式看涨期权到期时间,3=t 执行价格,115=X 利率06.0=r 。请用连锁法则方法求出在0=t 时刻期权的价格。 2．（15分）假设股票价格模型参数是：85.0.100,9.0,1.10====p S d u 一个美式看跌期权到期时间,3=t 执行价格,105=X 利率05.0=r 。请用连锁法则方法求出在0=t 时刻期权的价格。 3.（10分）利用如下图的股价二叉树，并设置向下敲出的障碍为跌破65元，50=X 元，.06.0=r 求0=t 时刻看涨期权的价格。 4.（15分）若股票指数点位是702，其波动率估计值,4.0=σ指数期货合约将在3个月后到期，并在到期时用美元按期货价格结算。期货合约的价格是715美元。若执行价是740美元，短期利率为7%，问这一期权的理论价格应是多少？（参考

金融数学附答案定稿版

金融数学附答案精编 W O R D版 IBM system office room 【A0816H-A0912AAAHH-GX8Q8-GNTHHJ8】

1、给定股票价格的二项模型，在下述情况下卖出看涨期权 S 0 S u S d X r τ 股数 50 60 40 55 0.55 1/2 1000 （1）求看涨期权的公平市场价格。 （2）假设以公平市场价格+0.10美元卖出1000股期权，需要买入多少股股票进行套期保值，无风险利润是多少 （3） 答案：（1）d u d r S S S e S q --=τ0=56.040 6040505.005.0=--??e （2）83.2>73.2，τr e S V -?+?='00 83.2> τr e S -?+?'0 406005--=--= ?d u S S D U =25.0股 104025.00'-=?-=?-=?d S D 753.9975.0105.005.0'-=?-=??-e 美元 则投资者卖空1000份看涨期权，卖空250股股票，借入9753美元 所以无风险利润为1.85835.005.0=?e 美元

2、假定 S0 = 100，u=1.1，d=0.9，执行价格X=105，利率r=0.05，p=0.85，期权到期时间t=3，请用连锁法则方法求出在t=0时该期权的价格。(答案见课本46页) 3、一只股票当前价格为30元，六个月期国债的年利率为3%，一投资者购买一份执行价格为35元的六个月后到期的美式看涨期权，假设六个月内股票不派发红利。波动率σ为0.318. 问题：（1）、他要支付多少的期权费【参考N（0.506)=0.7123；N（0.731）=0.7673 】{提示：考虑判断在不派发红利情况下，利用美式看涨期权和欧式看涨期权的关系} 解析：在不派发红利情况下，美式看涨期权等同于欧式看涨期权！所以利用Ｂ—Ｓ公式，就可轻易解出来这个题！同学们注意啦，N（d1）=N（-0.506），N（d2）=N（-0.731）。给出最后结果为0.608 4、若股票指数点位是702，其波动率估计值σ=0.4，指数期货合约将在3个月后到期，并在到期时用美元按期货价格计算，期货合约的价格是715美元。关于期货的看涨期权时间与期货相同，执行价是740美元，短期利率位7%，问这一期权的理论价格是多少（ N（-0.071922）=0.4721，N(-0.2271922）=0.3936 e-0.07*0.25=0.98265 解：F=715，T-t=0.25,σ=0.4,X=740,r=0.07 F/X=715/740=0.9622，σ(T-t)=0.4*0.5=0.2 d1=ln(0.9662)/0.2+0.2/2=-0.071922 d2=d1-0.2=-0.071922

金融数学(利息理论)复习题练习题

1. 某人借款1000元，年复利率为9%，他准备利用该资金购买一张3年期，面值为1000元的国库券，每年末按息票率为8%支付利息，第三年末除支付80元利息外同时偿付1000元的债券面值，如果该债券发行价为900元，请问他做这项投资是否合适 2. 已知：1） 16 565111-++=+))(()()()(i i m i m 求?=m 2） 1 65 65111--- =- ))(()()()(d d m d m 求?=m 由于i n n i m m i n m +=+=+111)()() ()( 由于d n n d m m d n m -=-=- 111)()() ()( 3. 假设银行的年贷款利率12%，某人从银行借得期限为1年，金额为100元的贷款。银行对借款人的还款方式有两种方案：一、要求借款人在年末还本付息；二、要求借款人每季度末支付一次利息年末还本。试分析两种还款方式有何区别哪一种方案对借款人有利 4. 设1>m ，按从小到大的顺序排列δ,,,,)() (m m d d i i 解：由 d i d i ?=- ? d i > )()(m m d d >+1 ? )(m d d < )()(n m d i > ? )()(m m i d < )()(m m i i <+1 ? i i m <)( δδ+>=+11e i ， δ==∞ →∞ →)()(lim lim m m m m d i ? i i d d m m <<<<)()(δ 5. 两项基金X,Y 以相同的金额开始，且有：（1）基金X 以利息强度5%计息；（2） 基金Y 以每半年计息一次的名义利率j 计算；（3）第8年末，基金X 中的金额是基金Y 中的金额的倍。求j.

北美精算师考试官方样题2015-12-exam-fm-syllabus

Financial Mathematics Exam—December 2015 The Financial Mathematics exam is three-hour exam that consists of 35 multiple-choice questions and is administered as a computer-based test. For additional details, please refer to Exam Rules The goal of the syllabus for this examination is to provide an understanding of the fundamental concepts of financial mathematics, and how those concepts are applied in calculating present and accumulated values for various streams of cash flows as a basis for future use in: reserving, valuation, pricing, asset/liability management, investment income, capital budgeting, and valuing contingent cash flows. The candidate will also be given an introduction to financial instruments, including derivatives, and the concept of no-arbitrage as it relates to financial mathematics. The Financial Mathematics Exam assumes a basic knowledge of calculus and an introductory knowledge of probability. The following learning objectives are presented with the understanding that candidates are allowed to use specified calculators on the exam. The education and examination of candidates reflects that fact. In particular, such calculators eliminate the need for candidates to learn and be examined on certain mathematical methods of approximation. Please check the Updates section on this exam's home page for any changes to the exam or syllabus. Each multiple-choice problem includes five answer choices identified by the letters A, B, C, D, and E, only one of which is correct. Candidates must indicate responses to each question on the computer. Candidates will be given three hours to complete the exam. As part of the computer-based testing process, a few pilot questions will be randomly placed in the exam (paper and pencil and computer-based forms). These pilot questions are included to judge their effectiveness for future exams, but they will NOT be used in the scoring of this exam. All other questions will be considered in the scoring. All unanswered questions are scored incorrect. Therefore, candidates should answer every question on the exam. There is no set requirement for the distribution of correct answers for the multiple-choice preliminary examinations. It is possible that a particular answer choice could appear many times on an examination or not at all. Candidates are advised to answer each question to the best of their ability, independently from how they have answered other questions on the examination. Since the CBT exam will be offered over a period of a few days, each candidate will receive a test form composed of questions selected from a pool of questions. Statistical scaling methods are used to ensure within reasonable and practical limits that, during the same testing period of a few days, all forms of the test are comparable in content and passing criteria. The methodology that has been adopted is used by many credentialing programs that give multiple forms of an exam. The ranges of weights shown in the Learning Objectives below are intended to apply to the large majority of exams administered. On occasion, the weights of topics on an individual exam may fall outside the published range. Candidates should also recognize that some questions may cover multiple learning objectives.

金融数学附答案

1、给定股票价格的二项模型，在下述情况下卖出看涨期权 S 0 S u S d X r τ 股数 50 60 40 55 1/2 1000 （1）求看涨期权的公平市场价格。 （2）假设以公平市场价格+美元卖出1000股期权，需要买入多少股股票进行套期保值，无风险利润是多少 答案：（1）d u d r S S S e S q --=τ0=56.040 6040505.005.0=--??e （2）83.2>73.2，τr e S V -?+?='00 83.2> τr e S -?+?'0 40 6005--=--=?d u S S D U =25.0股 104025.00'-=?-=?-=?d S D 753.9975.0105.005.0'-=?-=??-e 美元 则投资者卖空1000份看涨期权，卖空250股股票，借入9753美元 所以无风险利润为1.85835.005.0=?e 美元 2、假定 S 0 = 100，u=，d=，执行价格X=105，利率r=，p=，期权到期时间 t=3，请用连锁法则方法求出在t=0时该期权的价格。(答案见课本46页) 3、一只股票当前价格为30元，六个月期国债的年利率为3%，一投资者购买一份执行价格为35元的六个月后到期的美式看涨期权，假设六个月内股票不派发红利。波动率σ为. 问题：（1）、他要支付多少的期权费【参考N （=；N （）= 】 {提示：考虑判断在不派发红利情况下，利用美式看涨期权和欧式看涨期权的关系}

解析：在不派发红利情况下，美式看涨期权等同于欧式看涨期权！所以利用Ｂ—Ｓ公式，就可轻易解出来这个题！同学们注意啦，N（d1）=N（），N（d2）=N（）。给出最后结果为 4、若股票指数点位是702，其波动率估计值σ=，指数期货合约将在3个月后到期，并在到期时用美元按期货价格计算，期货合约的价格是715美元。关于期货的看涨期权时间与期货相同，执行价是740美元，短期利率位7%，问这一期权的理论价格是多少（N（）=，N）= *= 解：F=715，T-t=,σ=,X=740,r= F/X=715/740=，σ(T-t)=*= d1=ln/+2= d2== G=**740) =美元 5、根据看涨期权bs定价公式证明德尔塔等于N（d1）（答案见课本122页）

金融数学附答案

金融数学附答案 Prepared on 24 November 2020

1、给定股票价格的二项模型，在下述情况下卖出看涨期权 S 0 S u S d X r τ 股数 50 60 40 55 1/2 1000 （1）求看涨期权的公平市场价格。 （2）假设以公平市场价格+美元卖出1000股期权，需要买入多少股股票进行套期保值，无风险利润是多少 答案：（1）d u d r S S S e S q --=τ0=56.040 6040505.005.0=--??e （2）83.2>73.2，τr e S V -?+?='00 83.2> τr e S -?+?'0 40 6005--=--=?d u S S D U =25.0股 104025.00'-=?-=?-=?d S D 753.9975.0105.005.0'-=?-=??-e 美元 则投资者卖空1000份看涨期权，卖空250股股票，借入9753美元 所以无风险利润为1.85835.005.0=?e 美元 2、假定 S 0 = 100，u=，d=，执行价格X=105，利率r=，p=，期权到期时间 t=3，请用连锁法则方法求出在t=0时该期权的价格。(答案见课本46页) 3、一只股票当前价格为30元，六个月期国债的年利率为3%，一投资者购买一份执行价格为35元的六个月后到期的美式看涨期权，假设六个月内股票不派发红利。波动率σ为. 问题：（1）、他要支付多少的期权费【参考N （=；N （）= 】 {提示：考虑判断在不派发红利情况下，利用美式看涨期权和欧式看涨期权的关系}

解析：在不派发红利情况下，美式看涨期权等同于欧式看涨期权！所以利用Ｂ—Ｓ公式，就可轻易解出来这个题！同学们注意啦，N（d1）=N（），N（d2）=N（）。给出最后结果为 4、若股票指数点位是702，其波动率估计值σ=，指数期货合约将在3个月后到期，并在到期时用美元按期货价格计算，期货合约的价格是715美元。关于期货的看涨期权时间与期货相同，执行价是740美元，短期利率位7%，问这一期权的理论价格是多少（N（）=，N）= *= 解：F=715，T-t=,σ=,X=740,r= F/X=715/740=，σ(T-t)=*= d1=ln/+2= d2== G=**740) =美元 5、根据看涨期权bs定价公式证明德尔塔等于N（d1）（答案见课本122页）

PAK Study Manual QF-北美精算师(QFIQF)

Intro-Maths-Fin-1 Financial Derivatives (A Brief Introduction ) Background This chapter deals with the two basic building blocks of financial derivatives: 1. Options 2. Forwards and futures. We briefly introduce the third class of derivative: swap. We see how a complex swap can be decomposed into a number of forwards and options. Definitions Derivatives securities are financial contracts that ‘derive’ their value from cash market instruments such as stocks, bonds, currencies and commodities. At the time of the maturity of the derivative contract, denoted by T , the price F(T) of the derivative asset is completely determined by the market price of the underlying asset (S T ). For instance, the value at maturity (T ) of a long position in a call option of strike (K) written on an asset (S T ) is: Max [S T ?K ;0] Also, the value of time T of a long position in a forward contract of forward price (F) written on an underlying asset worth (S(T) at time T is given by: S (T )?F Types of derivatives We group derivatives into three general headings: 1. Futures, Forwards, Repos, Reverse Repos and Flexible Repos (Basic building blocks ) 2. Options and 3. Swaps Many of these instruments will be discussed in other parts of the syllabus for the QF Exam. The underlying asset : We let (S t ) represent the price of the relevant cash instrument, which we call the underlying asset . The five main groups of underlying asset : We list five main groups of underlying assets:

北大版金融数学引论第二章答案,DOC

版权所有，翻版必究 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 20 ?p 7%+Xs 10 ?p 7% X= 50000?1000s 20 ?p 7% s 10 ?p7% =651.72 2．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 48 ?p1.5% 解得 X=1489.36 3．设有n 年期期末年金，其中年金金额为n ，实利率i=1 。试计算该年金的现值。 解： PV = na?n pi 1?v n n = n 1 n = (n+1)n n 2 ?n n +2 (n+1)n 4．已知：a?n p =X ，a 2 ?n p =Y 。试 用X 和Y 表示d 。 解：a 2 ?n p =a?n p +a?n p (1?d)n 则 Y ?X 1 d=1?( X )n 5．已知：a?7 p =5.58238,a 11 ?p =7.88687,a 18 ?p =10.82760。计算i 。 解： a 18 ?p =a ?7p +a 11 ?p v 7 解得 6.证明： 1 1?v 10 = s 10?p +a ∞?p 。 s 10?p i=6.0% 北京大学数学科学学院金融数学系 第1页

版权所有，翻版必究 证明： s 10 ?p +a ∞?p (1+i)10 ?1+1 1 s 10?p = i (1+i)10 ?1 i i = 1?v 10 7．已知：半年结算名利率6%，计算下面10年期末年金的现值：开始4年每半 年200元，然后减为每次100元。 解： PV =100a?8p3% +100a 20?p 3% =2189.716 8．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 1000¨25?p8%=X¨15?p7% 解得 9．已知贴现率为10%，计算¨?8 p 。 X=8101.65 解：d=10%，则 i=1 10.求证： (1)¨?n p =a?n p +1?v n ； 1?d ?1=1 9 ¨?8 p =(1+i) 1?v 8 i =5.6953 (2)¨?n p =s??n p 1+(1+i)n 并给出两等式的实际解释。 证明：(1)¨?n p =1 ? d v n =1 ?i v n =1 ?v n i +1?v n 所以 (2)¨?n p = (1+ i)n ?1 1+i ¨?n p =a?n p +1?v n (1+i )n ?1=(1+i)n ?1 n ?1 d = i 1+i i +(1+i) 所以 ¨?n p =s??n p 1+(1+i) n

金融数学 练习题详解

金融数学第一章练习题详解 第1章利息度量 1.1 现在投资$600，以单利计息，2年后可以获得$150的利息。如果以相同的复利利率投资$2000，试确定在3年后的累积值。 1.2 在第1月末支付314元的现值与第18月末支付271元的现值之和，等于在第T月末支付1004元的现值。年实际利率为5%。求T。 1.3 在零时刻，投资者A在其账户存入X，按每半年复利一次的年名义利率i计息。同时，投资者Ｂ在另一个账户存入2X，按利率i （单利）来计息。假设两人在第八年的后六个月中将得到相等的利息，求i。 1.4 一项投资以δ的利息力累积，27.72年后将翻番。金额为1的投资以每两年复利一次的名义利率δ累积n年，累积值将成为7.04。求n。 1.5 如果年名义贴现率为6%，每四年贴现一次，试确定$100在两年末的累积值。 1.6 如果)(m i=0.1844144，)(m d=0.1802608，试确定m。 1.7 基金A以每月复利一次的名义利率12%累积。基金B以 =t/6 t 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将相等。

1.8 基金A 以t δ=a+bt 的利息力累积。基金B 以t δ=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>g>0，h>b>0。求n 。 1.9 在零时刻将100存入一个基金。该基金在头两年以每个季度贴现一次的名义贴现率?支付利息。从t=2开始，利息按照t t +=11δ的利息力支付。在t=5时，存款的累积值为260。求δ。 1.10在基金A 中，资金1的累积函数为t+1，t>0；在基金B 中，资金1的累积函数为1+t 2。请问在何时，两笔资金的利息力相等。 1.11已知利息力为t t +=12δ。第三年末支付300元的现值与在第六年末支付600元的现值之和，等于第二年末支付200元的现值与在第五年末支付X 元的现值。求X 。 82 .315))51/(())21(200-)61(600)31(300() 5()2(200)6(600)3(300)1()()1()(22-2211112 12)1ln(212 0=++?+?++?=??+?=?+?+=?+==?=---------++X a X a a a t t a t e e t a t dt t t 1.12已知利息力为1003t t =δ。请求)3(1-a 。 1.13资金A 以10%的单利累积，资金B 以5%的单贴现率累积。请问在何时，两笔资金的利息力相等。 1.14某基金的累积函数为二次多项式，如果向该基金投资1年，在上半年的名义利率为5%（每半年复利一次），全年的实际利率为7%，试确定5.0δ。

【SOA】关于北美精算师,你必须知道的入门级知识——Exam P

关于北美精算师，你必须知道的入门级知识——Exam P 成为一名北美准精算师（ASA）必须要经历五门SOA的准精算师考试，而其中最简单也是大部分人最先开始学习准备的就是Exam P，即probability。顾名思义，Exam P考察的就是最基本的数理统计与概率问题。下面我们就来了解一下Exam P的考试形式与内容。 考试目的 考生可以掌握用于定量评估风险的基本的概率方法，并着重于用这些方法应用解决精算学中遇到的问题。参加这门考试的考生应具有一定的微积分基础，并了解基本的概率、保险和风险管理的概念。 考试形式 Exam P采用机考的形式，总共30道单项选择题，考试时间为3个小时。每道选择题共有5个选项，其中只有一个正确选项。 与SAT考试不同的是，Exam P考试答错并不会额外扣分，也就是说考生一定不要空任何一道题。Exam P中会随机分布几道“pilot question”，这些题目是主办方用来分析从而改进将来的考试而出现的，它们的正确与否并不会影响到考生的实际分数。但是由于考生并无法分辨出这些题目，所以对每一道题目，考生都要同样认真地对待。 考试内容

概率（占总分10%－20%） 最基本的事件概率计算问题。包括集合方程与表示（sat functions）、互斥事件（mutually exclusive events）、事件独立性（independence of events）、组合概率（Combinatorial probability）、条件概率（Conditional probability）以及贝叶斯定理（Bayes theorem）等。 拥有单因素概率分布的随机变量（占总分35%－45%） 连续分布或离散分布的单因素随机变量的研究。包括PDF&CDF（Probability density functions and Cumulative distribution functions）、独立随机事件的和的分布、众数（Mode）、中位数（Median）、百分位数（Percentile）、动差（Moment）、方差（Variance）以及变形等问题。 拥有多因素概率分布的随机变量（占总分35%－45%） 包括联合PDF&CDF、中心极限定理（central limit theorem）、条件与边缘概率分布与动差（Conditional and marginal probability distributions and moments）、条件与边缘概率分布的方差、协方差与概率系数（Covariance and correlation coefficients）以及变换与顺序统计（Transformation and order statistics）等。 提醒：众所周知，2018年7月1日起，SOA新课程体系将正式生效，其中Exam P科目不变，考试大纲有变动，具体有那些变化？？？后台回复“Exam P”免费获取Exam P最新考试大纲。 考试时间

数学 《金融数学》期末试卷A参考答案与评分标准

浙江外国语学院 2013～2014学年第一学期期末考试 （参考答案及评分标准） 课程名称 金融数学 课程编号3040702003试卷类型A 一、单项选择题（本大题共6小题，每小题3分，共18分.） 题号 1 2 3 4 5 6 答案 B C B D A A 二．填空题（本大题共10小题，每小题 3分，共30分.） 1. 已知总量函数为2 ()33A t t t =++，则利息4I = 10 。 2. 已知1000元存入银行，在两年后可以得到1100元，银行按季进行结算，则 季挂牌名利率为 4.79% 。 3. 设有2年期2000元的贷款，月换算名利率为6%，如果按等额摊还方式在每 月底还款，则每次的还款金额为 88.64 元。 4. 已知标准永久期初年金的现值是26，则利率i = 4% 。 5. 某投资者第1年初投资3000元，第2年初投资2000元，而第2年至第4年 末均回收4000元。则利率为9%时的现金流现值为 4454.29 元。 6. 如果现在投资300元，第二年末投资100元，则在第四年末将积累到500元， 则实际利率为 6.54% . 7. 设有1000元贷款，每季度还款100元，已知季挂牌名利率为16%，则第4 次还款中本金有 67.49 元。 8. 设有1000元贷款，月换算挂牌利率为12%，期限一年，按偿债基金方式还款， 累积月实利率0.5%，则第4次还款中利息有 10 元。 9. 现有2年期面值为100元的债券，每半年付息一次，名息率8%，如果以名收 益率10%认购，则认购价格为 96.45 元。 10. 现有3年期面值为1000元的无息票债券，如果认购价格为850元，则收益率 为 5.57% 。 三．计算题（本大题共4小题，每小题10分，共40分.） 1. 现有某商品两种等价的付款方式：(1)按低于零售价10%的价格付现款;(2)在 半年和一年后按零售价的48%分别付款两次，求隐含的年利率。

北美精算师(SOA)考试 FM 2001 November 年真题

November 2001 Course 2 Interest Theory, Economics and Finance Society of Actuaries/Casualty Actuarial Society

1.Ernie makes deposits of 100 at time 0, and X at time 3 . The fund grows at a force of interest 2 100 t t δ=, t > 0 . The amount of interest earned from time 3 to time 6 is X. Calculate X. (A)385 (B)485 (C)585 (D)685 (E)785

2.The production of a good requires two inputs, labor and capital. At its current level of daily output, a competitive firm employs 100 machine hours of capital and 200 labor hours. The marginal product of machine hours is 10 units. The marginal product of labor hours is 5 units. The rental rate, or “price,” of capital is 20 per machine hour. If the firm minimizes its costs, what is the hourly wage rate? (A) 2.5 (B) 5.0 (C)10.0 (D)20.0 (E)40.0

上传版--金融数学期末考试A卷(统计)

金融学院《金融数学》课程期末考试试卷（A）卷 学院、专业、班级学号姓名 一、填空题（每小题4分，共20分） 1.如果现在投资2，第二年末投资1，则在第四年末将积累5，则实际利率等于； 2.某年金每年初付款1000元，共8年，各付款利率为8%，各付款所得利息的再投资利率为6%。则第8年末的年金积累值为元； 3.甲在银行存入2万元，计划分4年支取完，每半年末支取一次，每半年计息一次的年名义利率为7%，则每次支取的额度为元； 4.有一期末变化年金，其付款额从10开始，每年增加5，直到50，若利率为6%，求该变化年金的现值为； 5.某人的活期账户年初余额为1000元，其在4月底存入500元，又在6月底和8月底分别提取200元和100元，到年底账户余额为1236元．用资本加权法近似计算该账户的年利率为． 二、选择题（每小题5分，共30分） 1. 某人于2002年1月1日向某企业投资20万元，希望从2007年至2011年中每年1月1日以相等的金额收回资金。若年复利率为8%，则其每年应收回的资金为（）元。 A.63100.59 B.68148.64 C.73600.53 D.79488.57 2.某人在第1年、第2年初各投资1000元到某基金，第1年末积累额为1200元，第2年末积累额为2200元。根据时间加权法计算年收益率为（） A.10.5% B.9.5% C.8.5% D.7.5% 3.某单位计划用10年时间每年初存入银行一笔固定金额建立基金，用于从第10年末开始每年2000元的永久资励支出。假设存款年利率为12%，则每年需要存入的金额为（）元。 A.847.98 B.851.98 C.855.98 D.861.98 4.现有1000元贷款通过每季度还款100元偿还，且已知季换算挂牌利率为16%．计算第4次还款中的本金量为（）元。 A. 62.49 B.57.49 C.52.49 D.67.49 5.设每季度计算一次的年名义贴现率为12%，则5年后积累值为20000元的投资在开始时的本金为（）元。 A.10774.9 B.10875.9 C.10976.9 D.11077.9 6. 某人将收到一项年金支付，该年金一共有5次支付，每次支付100元，每3年支付一次，第一次支付发生在第7年末，假设年实际利率为5%，则该年金的现值为（）元。A．234 B.256 C. 268 D. 298 三、解答题（每小题10分，共50分） 1.李某每年年初存入银行1000元，前4年的年利率为6%，后6年由于通货膨胀率的提高，年利率升到10%。求第10年末时的存款积累值。 2. 某人继承了一笔遗产：从现在开始每年得到10000元．该继承人以年利率10%将每年的遗产收入存入银行．第5年底，在领取第6次遗产收入之前，他将剩余的遗产领取权益转卖给他人，然后，将所得的转卖收入与前5年的储蓄收入合并，全部用于年收益率为12%的某种投资．若每年底的投资回报是相同的，且总计30年，试计算每年底的回报金额。 3.年初建立一项投资基金，1月1日初始存款为10万元；5月1日基金账户价值为11.2万，再增加3万元的投资；11月1日账户价值为12.5万元，取出 4.2万；第二年1月1日投资基金价值变为10万元。分别用资本加权法和时间加权法计算基金投资收益率。 4. 某贷款的还贷方式为：前5年每半年还2000元，后5年每半年还1000元．如果半年换算的挂牌利率为10%，分别用预期法和追溯法计算第6次还贷后的贷款余额． 5. 某保险受益人以年金形式从保险公司分期领取10万元死亡保险金，每年末领取一次，共领取25年，年利率为3%，在领取10年后，考虑未来通货膨胀，保险公司决定通过调整利率至5%来增加后面15年的受益人的年领取额，求后15年里受益人每年可领取的金额。