如何計算內(nèi)部收益率IRR

1、Chapter 7Internal Rate of Return7-1Tony invested SI5.000 in a high yield account. At the end of 30 years he closed the account and received $539.250. Compute the effective interest rate he received on the account.SolutionRecall that F = P(1 + /)539.250 = 15.000( 1 + i)30 => 539.250/15.000 = (1 +

2、r)3035.95 = (1 +r)3° 35.95 = 1 +11.1268= l+i 1268 = /i = 12.68%7-2The heal loss (hrough (he exterior walls of a processing plant is esiimaied io cosi (he owner S3.000 next year A salesman from Superfiber. Inc. claims he can reduce the heat loss by 80 % with the installation of $ 15.000 of Super

3、fiber now. If the cost of heat loss rises by $200 per year, after next year (gradient) and the owner plans to keep the building ten more years, what is his rate of retunx neglecting depreciation and taxes?SolutionNPW = 0 at the rate of returnTry 12%NPW = -15.000 + &3000)(P/A. 12%, 10) + 8(2OO)(P

4、/G, 12%, 10)= $1,800.64Try 15%NPW = -S237.76By interpolation i = 14.7%105Chapter 7 Internal Rate of Return#70Does the following project have a positive or negative rate of return? Show how this is known lo be trueInvestment cost Net benefits Salvage Useful lifeS2.5OO$300 in Year L increasine by $200

5、/year $504 yearsSolutionYearBenefits1300Total Benefits obtained are less2500than the investment, so the HrcturnM3700on the investment is negative4900550Total:=2.450 < Cost74At what interest rate would $1000 at the end of 1995 be equivalent to $2000 at the end of 2002?Solution(1 +i)7 = 2;i = (2),z

6、7 1 =0.1041 or 10.41%Chapter 7 Internal Rate of Return#Chapter 7 Internal Rate of Return#75A painting, purchased one month ago for $1000. has just been sold for SI700 What nominal annual rate of return did the owner receive on his investment?Solutioni = 7.000/L000 = 70% IRR = 70x 12 = 840%7$Find the

7、 rale of return for a $10,000 investment that will pay $1000/ycar for 20 years.Solution10.000 = l000(P/A i% 20)(P/A. i%. 20)= 10Chapter 7 Internal Rate of Return#Chapter 7 Internal Rate of Return#From tables: 7% < i < 8% interpolate i = 7.77%Chapter 7 Internal Rate of Return109fori= 1% for i =

8、 1V4%!%<i< I%interpolate7-9A young engineer has a mortgage loan at a 15% interest rate, which he got some time ago. for a total of $52,000. He has to pay 120 more monthly payments of $620.72 each. As interest rates are going down he inquires about the conditions under which he could refinance

9、the loan. If the bank charges a new loan fee of 4% of the amount to be financed, and if the bank and the engineer agree on paying this fee by borrowing the additional 4% under the same terms as the new loan, what percentage rate would make the new loan attractive, if the conditions require him to re

10、pay it in the same 120 payments?SolutionThe amount to be refinanced:a) PW of 120 monthly payments left P = A(P/A, i, n)=620.72( P/A. 15%/I2, 120)=S3&474.09b) New loan fee (4 %)38.474.09 x 0.04 = $1,53&96=> Total amount to refinance = 38.474.09 + 1.538.96 = 40.013.05The new monthly payment

11、s are: Anew = 40.013.05 (A/P i. 120)The current payments are: Aold = 620.72We waiit Anew < AqloSubstituting =>40.013.05(A/P, i. 120) < 620.72(A/P. i. 120) < 620.72/40.013.05 = 0.0155(A/P. % 120) = 0.0143(A/P. 1%. 120) = 0.0161i= 1.1667%This corresponds to a nominal annual percentage rate

12、 of 12 x 0.011667 = 14 % Therefore, she has to wait until interest rates are less than 14%7-10Your company has been presented with an opportunity to invest in a prqjccl The facts on the project are presented below:Investment required Annual Gross income Annual Operating costs:$60000.00020.000.000Lab

13、orMaterials, licenses insurance etc Fuel and other costsMaintenance costsSalvage value after 10 years2500.0001.000.0001.500.000500.0000The project is expected to operate as shown for ten years【f your management expects to make 25% on its investments before taxes, would you recommend this project?Sol

14、utionNet income = 20M 一 (2.5M + 1M+ 1.5M + .5M)= $14.500.000NPW = 0 al the rate of return0 = - 60,000.000 + 14.500.000( P/A i. 10)(P/A. i, 10) = 60/14.5= 4.13820%P/A = 4.19225%P/A = 3.57l20% < i < 25% :. interpolatei = 20.43%IRR < 25 % . Reject Project7-11Consider the following investment i

15、n a piece of landPurchase priceAnnual maintenance:Expected sale price after 5 years:$10.000100$20.000Determine:(a) A trial value for i(b) The rate of return (to 1/100 percent)(c) What is the lowest sale price the investor should accept if she wishes to earn a return of 10% after keeping the land for

16、 10 years?Solution(a) (F/P. i % 5) = 20.000/10.000=2i= 15%(b) NPW = -10.000 - IOO(P/A, i%,5) + 20.000(P/F. i%.5) = 0Try i= 15%: = -391.2Try i = 12%: = +987.515% < i < 12% :. interpolatei= 14.15%(c) NFW = 0 = -10.000(F/P. 10%. 10) 100 (F/A 10%, 10) + Sale PriceSale Price = $27.5347-12Calculate

17、the rale of return of the following cash flow with accuracy to the nearest l/10 percent.A = $1.0001 102L45'$3100SolutionNPW = 0 al the rate of return0 =-3.100 + L00(XP/A i,5)(P/A. i.5) = 3.1(P/A. 20%. 5) = 2.991(P/A. 18%, 5) = 3.12718% < i < 20% J. interpolatei= 18.4%Chapter 7 Internal Rat

18、e of Return#7-13An investment that cost $1000 is sold five years later for $1261 What is the nominal rate of return on the investment (interest rate)?SolutionF = P(F/Pi%,5)1.261 kOOO (F/P, i %.5)(F/P.i%.5)= L261/LOOO=1.2610From tables:(F/P4% 5)= 1.246(F/P. 5%.5)= 1.2764l/2% < i < 5% ：. interpo

19、latei = 4.75%7-14Elizabeth made an initial investment of $5000 in a trading account with a stock brokerage house. After a period of 17 months the value of the account had increased to S8400. Whal is the nominal annual interest rate realized on the initial investment if it is assumed there were no ad

20、ditions or withdrawals from the account?SolutionF = P(F/P i. 17)F/P = 8.400/5.000 =1.68(1 +i),7= 1.681 + i = (1.68),/,7 =1.031Annual interest rate = 3x 12 = 37.2%7-15You borrowed $25 from a friend and after five months repaid $27 What interest rate did you pay (annual basis)?Chapter 7 Internal Rate

21、of Return111SolutionF = P(1 +i)n 27 = 25(1 +i)5 (27/25嚴=i +i=1.0155i 1.55% per month or 18.6% per year7-16You have a choice of $3.000 now. or $250 now with $ 150 a month for two years. What interest rate will make these choices comparable?Solution3.000 = 250 + 150 (P/A, i. 24)P/A= 18.33 2%P/A= 18.91

22、4 2%P/A= 17.8852% < i < 2% A interpolatei = = 2.28% or 27.36% per year7-17Sain and Lewis Investment Management Inc., is considering the purchase of a number of bonds to be issued by Southeast Airlines. The bonds have a face viilue of $10.000 and a face rate of 7.5% payable annually. The bonds

23、will mature ten years after they are issued The issue price is expected to be $&75O Determine the yield to maturity (1RR) for the bonds If SLIM Inc. requires at least a 10% return on all investments should they invest in the bonds?10%<lRR< 12%SolutionYear10%12%0First Cost-8.750.00&750.

24、00MOInterest 750(P/A. i%. 10)+4.608.75+4,813.5010Maturity 10.000(P/F i% n)+3.855.00+4.224.00NPW = $286.25S287.5Ointerpolatei = 9.5% => Do not invest.7-18A 9.25% coupon bond issued by Gurley Gears LLC is purchased January L 2002 and matures December 3L 2010. The purchase price is Si079 and interes

25、t is paid semi-annually If lhe face value of the bond is $1.000 determine the effective internal rate of returnSolutionn 9 x 2 18 Viyear periodsl/2 Year4%0First Cost LO79.OO1-18Interest 46.25(P/A, % 18)+585.4818Maturity 1000(P/Ft i%. 18)+493.60NPW =$.08IRR = (1 + .04)2 - 1 = 8.16%7-19Joe's Billi

26、ards Inc stock can be purchased for SI4.26 per share If dividends are paid each quarter at a rate of $0.16 per share dclcmiinc the effective / if after 4 years the stock is sold for $21.36 per share.% Year3%y/20Cost 14.26-14.261-16Dividends .16( P/A. i%, 16)+2.01+ 1.9316Sale 21.36( P/F, % 16)+ 13.31

27、+ 12.32NPW =$1.06-$.01i = V/i per quarter effective / = (1 + .035)4 - l=14.75%7-20A bond with a face value of $1.500 can be purchased for $800. The bond will mature five years from now and the bond dividend rate is 12% If dividends arc paid every three months, what effective interest rate would an investor receive if she purchases the bond?Chapter 7 Internal Rate of Return113Solut