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2-1 Time Value of Money (Examples) The sample problems are not in order of topic or difficulty. 2-2 Part 1: Example 1 - PV If you invest $15,000 for ten years, you receive $30,000. What is your annual return? 2-3 Example 2 - The Multi-Period Case Assume that the average college tuition costs $20,000 dollars per annum (paid at the end of the year). For a freshman just starting college, what is the present value of the cost of a four year degree when the interest rate is 10%? If the tuition is paid at the beginning of the year, what is the PV? Ex. 3 - Computing Payments with APRs 2-4 Suppose you want to buy a new computer system and the store is willing to sell it to allow you to make monthly payments. The entire computer system costs $3500. The loan period is for 2 years and the interest rate is 16.9% (APR) with monthly compounding. What is your monthly payment? An annuity problem... Monthly rate = Number of months = 3500 = C= 2-5 Ex. 4 - Future Values with Monthly Compounding Suppose you deposit $50 a month into an account that has an APR of 9%, based on monthly compounding. How much will you have in the account in 35 years? Monthly rate = Number of months = FV = Ex. 5 - Present Value with Daily Compounding 2-6 You need $15,000 in 3 years for a new car. If you can deposit money into an account that pays an effective annual rate of 5.63% based on daily compounding, how much would you need to deposit? Daily rate = Number of days = FV = 2-7 Example 6 - Pure Discount Loans A T-bill promises to repay $10,000 in 6 months. If the bill sells for $9,600 in the market, what is the annual rate of return (simple and compound)? HPR = Simple annual return= Compound annual return= 2-8 Example 7 - Investment Your broker calls you and tells you that he has this great investment opportunity. If you invest $100 today, you will receive two cash flows: $40 next year and $75 in two years. If you require a 15% return on investments of this risk, should you take the investment? 2-9 Ex. 8 - Saving For Retirement You are offered the opportunity to put some money away for retirement now. You will receive ten annual payments of $25,000 each beginning in 30 years. How much would you be willing to invest today if your money can earn an interest rate of 12% per annum over the years. 2-10 Ex. 8 - Saving For Retirement Timeline 0 1 2 … ? 0 0 … 29 0 30 31 32 25K 25K 25K Notice that the year 0 cash flow = CF0 = ? The cash flows years 1 – 29 are 0 The cash flows years 30 – 39 are 25,000 ... 39 25K 25K 2-11 Example 9 - Buying a House Fixed rate mortgage loans are basically annuities. They promise a specified stream of cash payments to a lender. Suppose, you want to take out a 30-year mortgage loan for 500,000$ at an interest rate of 7.5% per annum. A) What would be your monthly payment? B) How much would you owe to the bank after third monthly payment? Show the amortization schedule for the first three months. C) How much would you owe to the bank after the 60th payment? Definitions – EAR – Birleşik Faiz Effective Annual Rate (EAR) This is the amount of interest you would earn in one year assuming that you rollover the loan and reinvest all interest payments as often as is allowed by the terms of the loan, that is, the loan is compounded as often as possible during the year. 2-12 2-13 Definitions – APR – Basit Faiz Annual Percentage Rate (APR) This is the amount of interest you would earn in one year assuming that you rollover the loan but do not reinvest any interest payment paid during the year, that is, the loan is not compounded. 2-14 Ex. 10 - Computing EARs Suppose you can earn 1% per month on $1 invested today. What is the APR? How much are you effectively earning? Suppose you put it in another account, you earn 3% per quarter. What is the APR? How much are you effectively earning? Which account would you prefer? Monthly or quarterly compounding? 2-15 Ex. 11 – EAR’s You are looking at two savings accounts. One pays 5.25%, with daily compounding. The other pays 5.3% with semiannual compounding. Which account should you use? First account: EAR = Second account: EAR = ...................... = Which account? 2-16 Ex. 11 – EAR’s Let’s verify the choice. Suppose you invest $100 in each account. How much will you have in each account in one year? First Account: Second Account: 2-17 Example 11/A – EAR’s Your money is promised a return of 100% over 15 years. Is this a lot or a little? How does this compare to 1% over 3 months? 2-18 Example 12 - APR Suppose you want to earn an effective rate of 12% and you are looking at an account that compounds on a monthly basis. What APR must the account pay? 2-19 Example 14 - APR If a 3 month bond has a 8% APR, how much interest will I earn over the life of the bond? . 2-20 Example 15 - EAR If a 3 month bond has a 8% EAR, how much interest will I earn over the life of the bond? Since the EAR quote does include interest on interest and since a 3 month bond can be reinvested 4 times during the year, Why is this rate lower? 2-21 Ex. 16 - Finding the Number of Payments Suppose you borrow $2000 at 5% and you are going to make annual payments of $734.42. How long before you pay off the loan? Ex. 17 - Amortized Loan with Fixed Principal2-22 Payment Consider a $50,000, 10 year loan at 8% interest. The loan agreement requires the firm to pay $5,000 in principal each year plus interest for that year. Construct the amortization schedule on an excel file. 2-23 Ex. 18 - Amortized Loan with Fixed Payment Each payment covers the interest expense plus reduces principal Consider a 4 year loan with annual payments. The interest rate is 8% and the principal amount is $5000. What is the annual payment? .................................................. PMT = Construct the amortization table 2-24 Example 19 - Perpetuity You made your fortune in the dot-com boom (and got out in time!). As part of your legacy, you want to endow an annual MBA graduation party at your alma matter. You want it to be a memorable, so you budget $30,000 per year for the party. If the university earns 8% per year on its investments, and if the first party is in one year’s time, how much will you need to donate to endow the party? 2-25 Example 19 - Cont’d Suppose instead the first party was scheduled to be held 2 years from today. How would this change the amount of the donation required? 2-26 Non-standard TVM Problems Sometimes the problem we face is not stated as a typical time value of money problem and so does not exactly fit any formula. In these cases, we often have to “back into” a solution. 2-27 Example 24– Deceptive Ads (1) Motown Autos (MA) Advertisement: American Classic Cars! Finance Special! Sprite Conversion! Now Only $15,000! Just $1,000 Down, 0% interest, and 3years to pay with easy monthly payments! What is the true interest rate in this deal? You can use the additional info below: Classic Car News has an almost identical car advertised for $9,000, but it needs $3,000 of work to match the condition of the car offered by MA. 2-28 Example 25 You have $30,000 in student loans that call for monthly payments over 10 years. $15,000 is financed at seven percent APR $8,000 is financed at eight percent APR and $7,000 at 15 percent APR What is the interest rate on your portfolio of debt? Hint: don’t even think about doing this: = 15,000 × 7% + 8,000 × 8% + 7,000 × 15% 30,000 30,000 30,000 2-29 Example 27 You are considering the purchase of a prepaid tuition plan for your 8-year old daughter. She will start college in exactly 10 years, with the first tuition payment of $12,500 due at the start of the year. Sophomore year tuition will be $15,000; junior year tuition $18,000, and senior year tuition $22,000. How much money will you have to pay today to fully fund her tuition expenses? The discount rate is 14% 2-30 Example 28 You are thinking of buying a new car. You bought your current car exactly 3 years ago for $25,000 and financed it at 7% APR for 60 months. You need to estimate how much you owe on the loan to make sure that you can pay it off when you sell the old car. 2-31 Example 29 You have just landed a job and are going to start saving for a down-payment on a house. You want to save 20 percent of the purchase price and then borrow the rest from a bank. You have an investment that pays 10 percent APR. Houses that you like and can afford currently cost $100,000. Real estate has been appreciating in price at 5 percent per year and you expect this trend to continue. How much should you save every month in order to have a down payment saved five years from today? 2-32 Part 2: Some Problems for practice: Q1) On January 1 you deposit $100 in an account that pays a nominal interest rate of 11.33463%, with daily compounding (365 days). How much will you have on October 1, or after 9 months (273 days)? (Days given.) 4/9/2015 32 2-33 Q2) What’s the value at the end of Year 3 of the following CF stream if the quoted interest rate is 10%, compounded semiannually? 0 1 2 3 4 6 5 5% 100 4/9/2015 100 6-mos. periods 100 33 2-34 Q3) You are offered a note which pays $1,000 in 15 months (or 456 days) for $850. You have $850 in a bank which pays a 6.76649% nominal rate, with 365 daily compounding, which is a daily rate of 0.018538% and an EAR of 7.0%. You plan to leave the money in the bank if you don’t buy the note. The note is riskless. Should you buy it? 4/9/2015 34 2-35 Q4) Future Values Suppose you just started a new job at a current annual salary of $25,000. Average expected inflation rate is 4% for the next 40 years. If you receive annual cost-of-living raises tied to the inflation rate, what would be your ending salary? Also today’s $20,000 car will cost $96,020 under the same assumptions. 35 2-36 Q5) Future Values Suppose you had a relative deposit $10 at 5.5% interest 200 years ago. How much would the investment be worth today? What is the effect of compounding? 5F-36 2-37 Q6) Present Values You want to begin saving for your daughter’s college education and you estimate that she will need $150,000 in 17 years. If you feel confident that you can earn 8% per year, how much do you need to invest today? 37 2-38 Q7) Discount Rate Suppose you are offered an investment that will allow you to double your money in 6 years. You have $10,000 to invest. What is the implied rate of interest? 38 2-39 Q8) Number of Periods Suppose you want to buy a new house. You currently have $15000 and you figure you need to have a 10% down payment plus an additional 5% of the loan amount for closing costs. Assume the type of house you want will cost about $150,000 and you can earn 7.5% per year, how long will it be before you have enough money for the down payment and closing costs? 39