Key points:
　　R2-2-1Annuities 的計(jì)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：計(jì)算——N, I/Y, PMT, FV, PV 中任意給定四個(gè)，求另外一個(gè)
　　        注意要點(diǎn)：
　　1、  正確區(qū)分年金模式：普通年金，預(yù)付年金還是永續(xù)年金。這三種年金模式中，永續(xù)年金由于投資期限為正無(wú)窮，無(wú)法解出N及FV，故其計(jì)算公式只有三個(gè)變量：I/Y, PMT, PV。PV = PMT  /  (I/Y)。 而普通年金和預(yù)付年金的區(qū)別在于每一期現(xiàn)金流的流入或流出時(shí)點(diǎn)不同，普通年金在每一期的期末，而預(yù)付年金則在期初，所以他們的計(jì)算結(jié)果一般不同。
　　注意現(xiàn)金流的方向：一般我們?cè)谟?jì)算TVM題時(shí)，可以先不區(qū)分現(xiàn)金流方向，如若計(jì)算器出現(xiàn)ERROR的提示或者計(jì)算結(jié)果與可選項(xiàng)無(wú)一相同，應(yīng)首先想到是否因現(xiàn)金流方向錯(cuò)誤導(dǎo)致。從投資賬戶的角度考慮，凡是現(xiàn)金流入投資賬戶，則記為+；流出投資賬戶，則記為-。
　　        知識(shí)點(diǎn)對(duì)應(yīng)的試題如下：
　　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，F(xiàn)V=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%.
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