2 Annuity andRisk Perspectives p9-19年金計(jì)算.doc

Unit 2 Annuity and Risk PerspectivesYou might see a new finance outlet from annuity and risk perspectives. Annuity is the Archimedes Point in financial transaction, while risk measurement is the most effective way in evaluating a future trend.Dr. XingThe Sum of the First n Items of flat Ratio SeriesRemember that we have learned flat ratio in the first year of high school? (a) (b)(a)-(b), soon get (c)If qg, applying the formula of the sum of fixed ratio series, we getIn the above deduction, it is assumed that Example Calculate the present value of a growing annuity. The dividend next year be $1.30, and will growing at the rate of 5 percent, and the discount rate is 10 percent, and supposed the above indexes will not change in future.Weighted Average of ReturnThe expected return, denoted as E(r), is the weighted average of returns in all possible scenarios, with the weights equal to the probability of that particular scenario. Example Based on the following data, find the expected rate of return on the stock index fund.State of economyscenariosProbabilityHPRBoom 10.2544%Normal growth20.5014%Recession 30.25-16%Risk MeasurementTo summarize risk with a single number, the variance, , is defined as the expected value of the squared deviation from the mean.Example based on the following data, State of economyProbabilityHPRBoom 0.1520%Recession0.15-20%Normal growth 0.25-10%(1) Calculate the expected return(2) Calculate the standard deviation Solution (1) (2) Example Here are two risk assets composed of one stock and one bond, and under three economic conditions.Economic conditionProbabilityE(r) of StockE(r) of bondRecession33.3%-7%17%normal33.3%12%7%Boom33.3%28%-3%(1) Calculate the E(r) and of stock.(2) Calculate the E(r) and of bond.(3) Suppose that the portfolio is composed of 50% of stock and 50% of bond, calculate the E(r) and of the portfolio.Solution (1) (2) (3) First , calculate the returns of the portfolio at different economic conditions.Then It is clear that this portfolio will greatly decrease the risk of investment. Covariance and Portfolio Suppose there are two assets X and Y. X has expected return E(rX) and Y has expected return E(rY). X and Y compose as a portfolio with X weight a and Ys weight 1-a. accordingly, the portfolios expected return and standard deviation can be calculated. (since )Example Suppose that two securities have the following expected returns, and standard deviations, and weights: , , , , Calculate the standard deviation of the portfolio when (1) ; (2) ; (3) ; (4) ; (5) .Solution (1) When , (2) When , (3) When ,(4) When ,(5) When ,Essential Problems2.1-2.6 are single choice questions: there are statements and articles; among A, B, C, D, only one choice is correct.2.1 Assume that you purchase a 6-year, 8 percent savings certificate for $1,000. If interest is compounded annually, what will be the value of the certificate when it matures?a. $630.17 b. $1,469.33 c. $1.677.10 d. $1,586.87 e. $1,766.33(d. ) 2.2 A friend promises to pay you $600 two years from now if you loan him $500 today. What annual interest rate is your friend offering?a. 7.5% b. 8.5% c. 9.5% d. 10.5% e. 12.5%(c. 600 = 500 (1+i)2 )2.3 You are offered an investment opportunity, which will double your investment in 5 years with annual compounding interest rate. What is the annual rate?a. 40.00% b. 100.00% c. 14.87% d. 20.00% e. 18.74%(c. i= 14.87%)2.4 You decide to begin saving towards the purchase of a new car in 5 years. If you put $1,000 at the end of each of the next 5 years in savings account with 6 percent compounded annually, how much will you accumulate after 5 years?a. $6,691.13 b. $5,637.09 c. $1,338.23 d. $5,975.32 e. $5,731.94(b.)2.5 Refer to problem 4. What would be the ending amount if the payments were made at the beginning of each year? a. $6,691.13 b. $5,637.09 c. $1,338.23 d. $5,975.32 e. $5,731.94(d. )2.6 Refer to problem 4. What would be the ending amount if $500 payments were made at the end of each 6-month period for 5 years and the account paid 6 percent compounded semiannually?a. $6,691.13 b. $5,637.09 c. $1,338.23 d. $5,975.32 e. $5,731.94(e. )2.7 A group of businessmen and women from the town of Mathews are considering filing an application with the state banking commission to charter a new bank. Due to a lack of current banking facilities within a 10-mile radius of the community, the organizing group estimates that the initial banking facility would cost about $3,200,000 to build along with another $700,000 in other organizing expenses and would last for about 20 years. Total revenues are projected to be $510,000 the first year, while total operating expenses are projected to reach $180,000 in year 1. Revenues are expected to increase 6 percent annually after the first year, while expenses will grow an estimated 5 percent annually after year 1. If the organizers require a minimum of a 10% annual rate of return on their investment of capital in the proposed new bank, are they likely to proceed with their charter application given the above estimates?SolutionGrwoth Rate of Revenue=6%Grwoth Rate of Expense=5%Initial investment=3,200,000+700,000 = $3,900,000Expected return=10%2.7 dada processingno. Revenue ExpenseNet Profitdiscount rateNPV of Op.1510000 180000 330000 1.1000 300000 2540600 189000 351600 1.2100 290579 3573036 198450 374586 1.3310 281432 4607418 208373 399046 1.4641 272554 5643863 218791 425072 1.6105 263936 6682495 229731 452764 1.7716 255574 7723445 241217 482228 1.9487 247459 8766851 253278 513573 2.1436 239586 9812863 265942 546921 2.3579 231948 10861634 279239 582395 2.5937 224539 11913332 293201 620131 2.8531 217352 12968132 307861 660271 3.1384 210383 131026220 323254 702966 3.4523 203624 141087793 339417 748377 3.7975 197071 151153061 356388 796673 4.1772 190717 161222245 374207 848038 4.5950 184558 171295579 392917 902662 5.0545 178587 181373314 412563 960751 5.5599 172799 191455713 433191 1022522 6.1159 167190 201543056 454851 1088205 6.7275 161755 4491642 NPV of cash flows=4,491,642NPV of project=4,491,642-1,900,000=591,642Given the above information, the organizers are likely to proceed given that the net present value of this investment is positive. The return they are going to earn is greater than the 10% they need to earn. 2.8 Andover Savings Bank is considering the establishment of a new branch office at the corner of Lafayette and Connecticut Avenues. The savings associations Economics Department projects annual operating revenues of $1.75 million from services sold to generate fee income and annual branching operating expenses of $880,000. The cost of procuring the property is $2.5 million and branch construction will total an estimated $2.32 million; the facility is expected to last 16 years. If the savings bank has a minimum acceptable rate of return on its invested capital of 12 percent, will Andover likely proceed with this branch office project?Solution Initial Investment $4,820,000 Required Rate of Return 0.12It is needed to calculate NPV of future cash flow. Net yearly cash flow=1750000-880000=870000NPV of Investment=6,067,368-4,820,000=1,247,368 Andover is likely to proceed with this project because the net present value is positive. This means that the interest rate that Andover will earn on this project is higher than the 12% they need to earn.2.9 Jackson Bank of Commerce estimates that building a new branch office in the newly developed Guidar residential township will yield an annual expected return of 13 percent. The expected annual return from the banks existing facilities and other assets is 10 percent. The branch will represent just 10 percent of Jacksons total assets. Will the proposed branch increase Sullivans overall rate of return? And how much is its variance of overall return (the new branch included)? Solution The estimated total rate of return would be:E (R) = 0.10 (13%) + 0.90 (10%) = 10.3% Accordingly, the variance of overall return is 2.10 The following statistics and estimates were compiled by First Savings Bank of Talbot regarding a proposed new branch office and the bank itself: Branch Office Expected Return = 16% Standard Deviation of Return = 7% Banks original expected return = 10% Standard deviation of banks return = 3% Branch Asset Value as a Percent of Total Bank Assets = 15% Correlation of Cash Flows = + 0.27 What will happen to the Talbots total expected return and overall risk if the proposed new branch is adopted? Solution The banks total expected return is: E (R) = 0.15 (16%) + 0.85 (10%) = 10.9% The banks risk exposure is: =0.0009051s =The proposed project raises the savings banks expected return slightly and does not affect the risk of the