Principle of finance questions

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Solving two questions in the attached word document

please include the formula's used to solve the two questions.

check attached power point slides to use the formula's

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Q1. Sally plans to send her daughter to DCU in 18 years’ time. She has decided to invest a sum of money each year on which will accumulate to €100,000 when her daughter is entering college. If Sally can obtain a return of 13% per annum, how much must she save each year?

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Q1. Sally plans to send her daughter to DCU in 18 years’ time. She has decided to invest a sum of money each year on which will accumulate to €100,000 when her daughter is entering college. If Sally can obtain a return of 13% per annum, how much must she save each year? Q2. Suppose that you take out a $250,000 house mortgage from your local savings bank. The bank requires you to repay the mortgage in equal annual installments over the next 30 years. How much will you have to pay per annum if the interest rate is 12% a year? Chapter 4 The Time Value of Money (Part 2) Learning Objectives 1. Compute the future value of multiple cash flows. 2. Determine the future value of an annuity. 3. Determine the present value of an annuity. Copyright ©2016 Pearson Education, Ltd. All rights reserved. 4-2 4.1 Future Value of Multiple Payment Streams • With unequal periodic cash flows, treat each of the cash flows as a lump sum and calculate its future value over the relevant number of periods. • Sum up the individual future values to get the future value of the multiple payment streams. Copyright ©2016 Pearson Education, Ltd. All rights reserved. 4-3 Figure 4.1 The time line of a nest egg Copyright ©2016 Pearson Education, Ltd. All rights reserved. 4-4 4.1 Future Value of Multiple Payment Streams (continued) Example 1: Future Value of an Uneven Cash Flow Stream: Jim deposits $3,000 today into an account that pays 10% per year, and follows it up with 3 more deposits at the end of each of the next three years. Each subsequent deposit is $2,000 higher than the previous one. How much money will Jim have accumulated in his account by the end of three years? Copyright ©2016 Pearson Education, Ltd. All rights reserved. 4-5 4.1 Future Value of Multiple Payment Streams (Example 1 Answer) FV = PV x (1+r)n FV FV FV FV of of of of Cash Cash Cash Cash Flow Flow Flow Flow at at at at T0 T1 T2 T3 = = = = $3,000 $5,000 $7,000 $9,000 Copyright ©2016 Pearson Education, Ltd. All rights reserved. x x x x (1.10)3 (1.10)2 (1.10)1 (1.10)0 = = = = $3,000 $5,000 $7,000 $9,000 x x x x 1.331 = $3,993.00 1.210 = $6,050.00 1.100 = $7,700.00 1.000 = $9,000.00 Total = $26,743.00 4-6 4.1 Future Value of Multiple Payment Streams (Example 1 Answer) ALTERNATIVE METHOD: Using the Cash Flow (CF) key of the calculator, enter the respective cash flows. CF0=-$3000;CF1=-$5000;CF2=-$7000; CF3=-$9000; Next calculate the NPV using I=10%; NPV=$20,092.41; Finally, using PV=-$20,092.41; n=3; i=10%;PMT=0; CPT FV=$26,743.00 Copyright ©2016 Pearson Education, Ltd. All rights reserved. 4-7 4.2 Future Value of an Annuity Stream • Annuities are equal, periodic outflows/inflows., e.g. rent, lease, mortgage, car loan, and retirement annuity payments. • An annuity stream can begin at the start of each period (annuity due) as is true of rent and insurance payments or at the end of each period, (ordinary annuity) as in the case of mortgage and loan payments. • The formula for calculating the future value of an annuity stream is as follows: FV = PMT * (1+r)n -1 r • where PMT is the term used for the equal periodic cash flow, r is the rate of interest, and n is the number of periods involved. Copyright ©2016 Pearson Education, Ltd. All rights reserved. 4-8 4.2 Future Value of an Annuity Stream (continued) Example 2: Future Value of an Ordinary Annuity Stream Jill has been faithfully depositing $2,000 at the end of each year since the past 10 years into an account that pays 8% per year. How much money will she have accumulated in the account? Copyright ©2016 Pearson Education, Ltd. All rights reserved. 4-9 4.2 Future Value of an Annuity Stream (continued) Example 2 Answer Future Value of Payment One = $2,000 x 1.089 = $3,998.01 Future Value of Payment Two = $2,000 x 1.088 = $3,701.86 Future Value of Payment Three = $2,000 x 1.087 = $3,427.65 Future Value of Payment Four = $2,000 x 1.086 = $3,173.75 Future Value of Payment Five = $2,000 x 1.085 = $2,938.66 Future Value of Payment Six = $2,000 x 1.084 = $2,720.98 Future Value of Payment Seven = $2,000 x 1.083 = $2,519.42 Future Value of Payment Eight = $2,000 x 1.082 = $2,332.80 Future Value of Payment Nine = $2,000 x 1.081 = $2,160.00 Future Value of Payment Ten = $2,000 x 1.080 = $2,000.00 Total Value of Account at the end of 10 years $28,973.13 Copyright ©2016 Pearson Education, Ltd. All rights reserved. 4-10 4.2 Future Value of an Annuity Stream (continued) Example 2 (Answer) FORMULA METHOD FV = PMT * (1+r)n -1 r where, PMT = $2,000; r = 8%; and n=10. FVIFA ➔[((1.08)10 - 1)/.08] = 14.486562, FV = $2000*14.486562 ➔ $28,973.13 USING A FINANCIAL CALCULATOR N= 10; PMT = -2,000; I = 8; PV=0; CPT FV = 28,973.13 Copyright ©2016 Pearson Education, Ltd. All rights reserved. 4-11 4.2 Future Value of an Annuity Stream (continued) USING AN EXCEL SPREADSHEET Enter =FV(8%, 10, -2000, 0, 0); Output = $28,973.13 Rate, Nper, Pmt, PV,Type Type is 0 for ordinary annuities and 1 for annuities due USING FVIFA TABLE (A-3) Find the FVIFA in the 8% column and the 10 period row; FVIFA = 14.486 FV = 2000*14.4865 = $28.973.13 Copyright ©2016 Pearson Education, Ltd. All rights reserved. 4-12 FIGURE 4.3 Interest and principal growth with different interest rates for $100-annual payments. Copyright ©2016 Pearson Education, Ltd. All rights reserved. 4-13 4.3 Present Value of an Annuity To calculate the value of a series of equal periodic cash flows at the current point in time, we can use the following simplified formula:   1 1 −  n ( ) 1 + r  PV = PMT   r    The last portion of the equation, is the Present Value Interest Factor of an Annuity (PVIFA). Practical applications include figuring out the nest egg needed prior to retirement or lump sum needed for college expenses. Copyright ©2016 Pearson Education, Ltd. All rights reserved. 4-14 FIGURE 4.4 Time line of present value of annuity stream. Copyright ©2016 Pearson Education, Ltd. All rights reserved. 4-15 4.3 Present Value of an Annuity (continued) Example 3: Present Value of an Annuity. John wants to make sure that he has saved up enough money prior to the year in which his daughter begins college. Based on current estimates, he figures that college expenses will amount to $40,000 per year for 4 years (ignoring any inflation or tuition increases during the 4 years of college). How much money will John need to have accumulated in an account that earns 7% per year, just prior to the year that his daughter starts college? Copyright ©2016 Pearson Education, Ltd. All rights reserved. 4-16 4.3 Present Value of an Annuity (continued) Example 3 Answer Using the following equation:   1   1−  n   (1+ r )   PV = PMT  r 1. Calculate the PVIFA value for n=4 and r=7%➔3.387211. 2. Then, multiply the annuity payment by this factor to get the PV, PV = $40,000 x 3.387211 = $135,488.45 Copyright ©2016 Pearson Education, Ltd. All rights reserved. 4-17 4.3 Present Value of an Annuity (continued) Example 3 Answer—continued FINANCIAL CALCULATOR METHOD: Set the calculator for an ordinary annuity (END mode) and then enter: N= 4; PMT = 40,000; I = 7; FV=0; CPT PV = 135,488.45 SPREADSHEET METHOD: Enter =PV(7%, 4, 40,000, 0, 0); Output = $135,488.45 Rate, Nper, Pmt, FV, Type Copyright ©2016 Pearson Education, Ltd. All rights reserved. 4-18 4.3 Present Value of an Annuity (continued) Example 3 Answer—continued PVIFA TABLE (APPENDIX A-4) METHOD For r =7% and n = 4; PVIFA =3.3872 PVA = PMT*PVIFA = 40,000*3.3872 = $135,488 (Notice the slight rounding error!) Copyright ©2016 Pearson Education, Ltd. All rights reserved. 4-19 Chapter 3 The Time Value of Money (Part 1) Learning Objectives 1. Calculate future values and understand compounding. 2. Calculate present values and understand discounting. 3. Calculate implied interest rates and waiting time from the time value of money equation. 4. Apply the time value of money equation using formula, and spreadsheet. Copyright ©2016 Pearson Education, Ltd. All rights reserved. 3-2 3.1 Future Value and Compounding Interest • The value of money at the end of the stated period is called the future or compound value of that sum of money. – Determine the attractiveness of alternative investments – Figure out the effect of inflation on the future cost of assets, such as a car or a house. Copyright ©2016 Pearson Education, Ltd. All rights reserved. 3-3 3.1 (A) The Single-Period Scenario FV = PV + PV x interest rate, or FV = PV(1+interest rate) (in decimals) Example 1: Let’s say John deposits $200 for a year in an account that pays 6% per year. At the end of the year, he will have: FV = $200 + ($200 x .06) = $212 = $200(1.06) = $212 Copyright ©2016 Pearson Education, Ltd. All rights reserved. 3-4 3.1 (B) The Multiple-Period Scenario FV = PV x (1+r)n Example 2: If John closes out his account after 3 years, how much money will he have accumulated? How much of that is the interest-on-interest component? What about after 10 years? FV3 = $200(1.06)3 = $200*1.191016 = $238.20, where, 6% interest per year for 3 years = $200 x.06 x 3=$36 Interest on interest = $238.20 - $200 - $36 =$2.20 FV10 = $200(1.06)10 = $200 x 1.790847 = $358.17 where, 6% interest per year for 10 years = $200 x .06 x 10 = $120 Interest on interest = $358.17 - $200 - $120 = $38.17 Copyright ©2016 Pearson Education, Ltd. All rights reserved. 3-5 3.1 (C) Methods of Solving Future Value Problems • Method 1: The formula method – Time-consuming, tedious • Method 2: The financial calculator approach – Quick and easy • Method 3: The spreadsheet method – Most versatile • Method 4: The use of Time Value tables: – Easy and convenient but most limiting in scope Copyright ©2016 Pearson Education, Ltd. All rights reserved. 3-6 3.1 (C) Methods of Solving Future Value Problems (continued) Example 3: Compounding of Interest Let’s say you want to know how much money you will have accumulated in your bank account after 4 years, if you deposit all $5,000 of your high-school graduation gifts into an account that pays a fixed interest rate of 5% per year. You leave the money untouched for all four of your college years. Copyright ©2016 Pearson Education, Ltd. All rights reserved. 3-7 3.1 (C) Methods of Solving Future Value Problems (continued) Example 3: Answer Formula Method: FV = PV x (1+r)n➔$5,000(1.05)4=$6,077.53 Calculator method: PV =-5,000; N=4; I/Y=5; PMT=0; CPT FV=$6077.53 Spreadsheet method: Rate = .05; Nper = 4; Pmt=0; PV=-5,000; Type =0; FV=6077.53 Time value table method: FV = PV(FVIF, 5%, 4) = 5000*(1.215506)=6077.53, where (FVIF, 5%,4) = Future value interest factor listed under the 5% column and in the 4-year row of the Future Value of $1 table. Copyright ©2016 Pearson Education, Ltd. All rights reserved. 3-8 3.1 (C) Methods of Solving Future Value Problems (continued) Example 4: Future Cost due to Inflation Let’s say that you have seen your dream house, which is currently listed at $300,000, but unfortunately, you are not in a position to buy it right away and will have to wait at least another 5 years before you will be able to afford it. If house values are appreciating at the average annual rate of inflation of 5%, how much will a similar house cost after 5 years? Copyright ©2016 Pearson Education, Ltd. All rights reserved. 3-9 3.1 (C) Methods of Solving Future Value Problems (continued) Example 4 (Answer) PV = current cost of the house = $300,000; n = 5 years; r = average annual inflation rate = 5%. Solving for FV, we have FV = $300,000*(1.05)(1.05)(1.05)(1.05)(1.05) = $300,000*(1.276282) = $382,884.5 So the house will cost $382,884.5 after 5 years Copyright ©2016 Pearson Education, Ltd. All rights reserved. 3-10 3.1 (C) Methods of Solving Future Value Problems (continued) Calculator method: PV =-300,000; N=5; I/Y=5; PMT=0; CPT FV=$382,884.5 Spreadsheet method: Rate = .05; Nper = 5; Pmt=0; PV=-$300,000; Type =0; FV=$382,884.5 Time value table method: FV = PV(FVIF, 5%, 5) = 300,000*(1.27628)=$382,884.5; where (FVIF, 5%,5) = Future value interest factor listed under the 5% column and in the 5-year row of the future value of $1 table=1.276 Copyright ©2016 Pearson Education, Ltd. All rights reserved. 3-11 3.2 Present Value and Discounting • Involves discounting the interest that would have been earned over a given period at a given rate of interest. • It is therefore the exact opposite or inverse of calculating the future value of a sum of money. • Such calculations are useful for determining today’s price or the value today of an asset or cash flow that will be received in the future. • The formula used for determining PV is as follows: PV = FV x 1/ (1+r)n Copyright ©2016 Pearson Education, Ltd. All rights reserved. 3-12 3.2 (A) The Single-Period Scenario When calculating the present or discounted value of a future lump sum to be received one period from today, we are basically deducting the interest that would have been earned on a sum of money from its future value at the given rate of interest. i.e. PV = FV/(1+r)➔ since n = 1 So, if FV = 100; r = 10%; and n =1; ➔PV = 100/1.1=90.91 Copyright ©2016 Pearson Education, Ltd. All rights reserved. 3-13 3.2 (B) The Multiple-Period Scenario When multiple periods are involved… The formula used for determining PV is as follows: PV = FV x 1/(1+r)n where the term in brackets is the present value interest factor for the relevant rate of interest and number of periods involved, and is the reciprocal of the future value interest factor (FVIF) Copyright ©2016 Pearson Education, Ltd. All rights reserved. 3-14 3.2 Present Value and Discounting (continued) Example 5: Discounting Interest Let’s say you just won a jackpot of $50,000 at the casino and would like to save a portion of it so as to have $40,000 to put down on a house after 5 years. Your bank pays a 6% rate of interest. How much money will you have to set aside from the jackpot winnings? Copyright ©2016 Pearson Education, Ltd. All rights reserved. 3-15 3.2 Present Value and Discounting (continued) Example 5 (Answer) FV = amount needed = $40,000 N = 5 years; Interest rate = 6%; • PV = FV x 1/ (1+r)n • PV = $40,000 x 1/(1.06)5 • PV = $40,000 x 0.747258 • PV = $29,890.33➔ Amount needed to set aside today Copyright ©2016 Pearson Education, Ltd. All rights reserved. 3-16 3.2 Present Value and Discounting (continued) Calculator method: FV 40,000; N=5; I/Y =6%; PMT=0; CPT PV=-$29,890.33 Spreadsheet method: Rate = .06; Nper = 5; Pmt=0; Fv=$40,000; Type =0; Pv=-$29,890.33 Time value table method: PV = FV(PVIF, 6%, 5) = 40,000*(0.7473)=$29,892 where (PVIF, 6%,5) = Present value interest factor listed under the 6% column and in the 5-year row of the Present Value of $1 table=0.7473 Copyright ©2016 Pearson Education, Ltd. All rights reserved. 3-17 3.2 (C) Using Time Lines • When solving time value of money problems, especially the ones involving multiple periods and complex combinations (which will be discussed later) it is always a good idea to draw a time line and label the cash flows, interest rates and number of periods involved. Copyright ©2016 Pearson Education, Ltd. All rights reserved. 3-18 3.2 (C) Using Time Lines (continued) FIGURE 3.1 Time lines of growth rates (top) and discount rates (bottom) illustrate present value and future value. Copyright ©2016 Pearson Education, Ltd. All rights reserved. 3-19 3.4 Applications of the Time Value of Money Equation • Calculating the amount of saving required for retirement • Determining future value of an asset • Calculating the cost of a loan • Calculating growth rates of cash flows • Calculating number of periods required to reach a financial goal. Copyright ©2016 Pearson Education, Ltd. All rights reserved. 3-20 Example 3.3 Saving for retirement Copyright ©2016 Pearson Education, Ltd. All rights reserved. 3-21 Example 3.3 Saving for retirement (continued) Copyright ©2016 Pearson Education, Ltd. All rights reserved. 3-22 Example 3.3 Saving for retirement (continued) Copyright ©2016 Pearson Education, Ltd. All rights reserved. 3-23 Example 3.4 Let’s make a deal (future value) Copyright ©2016 Pearson Education, Ltd. All rights reserved. 3-24 Example 3.4 Let’s make a deal (continued) Copyright ©2016 Pearson Education, Ltd. All rights reserved. 3-25 Example 3.4 Let’s make a deal (continued) Copyright ©2016 Pearson Education, Ltd. All rights reserved. 3-26 Example 3.5 What’s the cost of that loan? Copyright ©2016 Pearson Education, Ltd. All rights reserved. 3-27 Example 3.5 What’s the cost of that loan? (continued) Copyright ©2016 Pearson Education, Ltd. All rights reserved. 3-28 Example 3.5 What’s the cost of that loan? (continued) Copyright ©2016 Pearson Education, Ltd. All rights reserved. 3-29 Example 3.6 Boomtown, USA (growth rate) Copyright ©2016 Pearson Education, Ltd. All rights reserved. 3-30 Example 3.6 Boomtown, USA (continued) Copyright ©2016 Pearson Education, Ltd. All rights reserved. 3-31 Example 3.6 Boomtown, USA (continued) Copyright ©2016 Pearson Education, Ltd. All rights reserved. 3-32 Example 3.7 When will I be rich? Copyright ©2016 Pearson Education, Ltd. All rights reserved. 3-33 Example 3.7 When will I be rich? (continued) Copyright ©2016 Pearson Education, Ltd. All rights reserved. 3-34 Example 3.7 When will I be rich? (continued) Copyright ©2016 Pearson Education, Ltd. All rights reserved. 3-35 Additional Problems with Answers Problem 1 Joanna’s Dad is looking to deposit a sum of money immediately into an account that pays an annual interest rate of 9% so that her first-year college tuition costs are provided for. Currently, the average college tuition cost is $15,000 and is expected to increase by 4% (the average annual inflation rate). Joanna just turned 5, and is expected to start college when she turns 18. How much money will Joanna’s Dad have to deposit into the account? Copyright ©2016 Pearson Education, Ltd. All rights reserved. 3-36 Additional Problems with Answers Problem 1 (Answer) Step 1. Calculate the average annual college tuition cost when Joanna turns 18, i.e., the future compounded value of the current tuition cost at an annual increase of 4%. PV = -15,000; n= 13; i=4%; PMT=0; CPT FV=$24, 976.10 OR FV= $15,000 x (1.04)13 = $15,000 x 1.66507 = $24,976.10 Copyright ©2016 Pearson Education, Ltd. All rights reserved. 3-37 Additional Problems with Answers Problem 1 (Answer) (continued) Step 2. Calculate the present value of the annual tuition cost using an interest rate of 9% per year. FV = 24,976.10; n=13; i=9%; PMT = 0; CPT PV = $8,146.67 (rounded to 2 decimals) OR PV = $24,976.10 x (1/(1+0.09)13=$24,976.10 x 0.32618 = $8,146.67 So, Joanna’s Dad will have to deposit $8,146.67 into the account today so that she will have her first-year tuition costs provided for when she starts college at the age of 18. Copyright ©2016 Pearson Education, Ltd. All rights reserved. 3-38 Additional Problems with Answers Problem 2 Bank A offers to pay you a lump sum of $20,000 after 5 years if you deposit $9,500 with them today. Bank B, on the other hand, says that they will pay you a lump sum of $22,000 after 5 years if you deposit $10,700 with them today. Which offer should you accept, and why? Copyright ©2016 Pearson Education, Ltd. All rights reserved. 3-39 Additional Problems with Answers Problem 2 (Answer) To answer this question, you have to calculate the rate of return that will be earned on each investment and accept the one that has the higher rate of return. Bank A’s Offer: PV = -$9,500; n=5; FV =$20,000; PMT = 0; CPT I = 16.054% OR Rate = (FV/PV)1/n - 1 = ($20,000/$9,500)1/5 – ...
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