Key points:
　　R2-2-1Annuities 的计算：FV, PV, required payment
　　◆   N = number of compounding periods
　　◆   I/Y = interest rate per compounding period
　　◆   PMT = amount of each periodic payment, annuity payments, or constant periodic cash flow
　　◆   FV= future value
　　◆   PV= present value
　　考察方法：计算——N, I/Y, PMT, FV, PV 中任意给定四个，求另外一个
　　        注意要点：
　　1、  正确区分年金模式：普通年金，预付年金还是永续年金。这三种年金模式中，永续年金由于投资期限为正无穷，无法解出N及FV，故其计算公式只有三个变量：I/Y, PMT, PV。PV = PMT  /  (I/Y)。 而普通年金和预付年金的区别在于每一期现金流的流入或流出时点不同，普通年金在每一期的期末，而预付年金则在期初，所以他们的计算结果一般不同。
　　注意现金流的方向：一般我们在计算TVM题时，可以先不区分现金流方向，如若计算器出现ERROR的提示或者计算结果与可选项无一相同，应首先想到是否因现金流方向错误导致。从投资账户的角度考虑，凡是现金流入投资账户，则记为+；流出投资账户，则记为-。
　　        知识点对应的试题如下：
　　R2-2 Time Value of Money
　　Q2-2 Rachel Kelly, 24 years old, is planning for retirement. Kelly’s annual consumptionexpenditures are currently $30,000. She assumes her consumption expenditures willincrease with the rate of inflation, and she expects the average inflation rate is 3% untilshe retires at the age of 68.Given a life expectancy of 93 years and constantexpenditures in retirement, the amount Kelly must accumulate by her retirement date assuming an 8% rate of return on her retirement account, is closest to:
　　A. $423,000.
　　B. $1,176,000.
　　C. $1,552,000.
　　Solution: B
　　
        The annual expense will go to 30,000×1.0344=110,143.57. Since her expenditure is constant after
　　she retires. PMT=‐110,143.57，FV=0，T=25，I/Y=8， CPT PV=1,175,758.
　　Q2-3 A couple plans to pay their child’s college tuition for 4 years starting 18 years from now.The current annual cost of college is C$7,000, and they expect this cost to rise at anannual rate of 5 percent. In their planning, they assume that they can earn 6% annually.How much must they put aside each year, starting next year, if they plan to make 17
　　equal payments?
　　A. C$2,221.58.
　　B. C$2,212.58.
　　C. C$2,231.58.
　　Solution: A
　　        Step1: calculate the CF since 18,
　　CF18=7000*(1+5%)18=16846, CF19=7000*(1+5%)19=17689,…
　　        Step 2: PV of tuition (4 CF since 18) at year 17
　　C$16,846 (1,06)‐1 + C$17,689 (1.06)‐2 + C$18,573 (1.06)‐3 + C$19,502 (1.064 )‐4=$62,677
　　        Step 3: Calculate PMT
　　Input FV=C$62,677, PV=0, I/Y=6, N=17, then CPT PMT= C$2,221.58
　　In summary, the couple will need to put aside C$2,221.58 each year if they start next year and
　　make 17 equal payments.
　　Q2-4For planning purposes, an individual wants to be able to spend €80,000 per year, at theend of each year, for an anticipated 25 years in retirement. In order to fund thisretirement account, he will make annual deposits of €6,608 at the end of each of hisworking years. What is the minimum number of such deposits he will need to make tofund his desired retirement? Use 6% interest compounded annually for all calculations.
　　A. 29 payments
　　B. 40 payments
　　C. 51 payments
　　Solution: B
　　B is correct. Using a financial calculator, first calculate the needed funds at retirement:N = 25, I/Y = 6, PMT = 80,000, FV = 0; calculate PV to be 1,022,668.Then use 1,022,668 as the FV of the accumulation phase annuity as per:I/Y = 6, PV = 0, PMT = 6,608, FV = 1,022,668; calculate N. N is 40.
　　Q2-5An analyst expects that a company's net sales will double and the company's net incomewill triple over the next five‐year period starting now. Based on the analyst'sexpectations, which of the following best describes the expected compound annualgrowth?
　　A. Net sales will grow 15% annually and net income will grow 25% annually.
　　B. Net sales will grow 20% annually and net income will grow 40% annually.
　　C. Net sales will grow 25% annually and net income will grow 50% annually.
　　Solution: A
　　A is correct.
　　Using the general time value of money formula, for sales, solve for r in the equation 2 = 1 * (1 + r)5. For income, solve 3 = 1 * (1 + r)5. Alternatively, using a financial calculator, for sales, enter ：NT=5. PV=l, PMT=O, IN= -2 and compute 1/Y. For income, change the FV to -3 and again solvefor I/Y. The solution for sales is 11.87%; and for income is 24.57%.
        高顿正能量：在一个崇高的目标支持下，不停地工作，即使慢，也一定会获得成功。

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