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**Chapter 5: Time Value of Money – Advanced Topics**

FINC3131 Business Finance Chapter 5: Time Value of Money – Advanced Topics

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Learning Objectives Use a financial calculator to solve TVM problems involving multiple periods and multiple cash flows. Solve TVM problems when the period of compounding is less than a year. Tell the difference between an ordinary annuity and an annuity due. Solve TVM problems involving an annuity due. Prepare an amortization schedule

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**Preparing BAII Plus for use**

Press ‘2nd’ and [Format]. The screen will display the number of decimal places that the calculator will display. If it is not eight, press ‘8’ and then press ‘Enter’. Press ‘2nd’ and then press [P/Y]. If the display does not show one, press ‘1’ and then ‘Enter’. Press ‘2nd’ and [BGN]. If the display is not END, that is, if it says BGN, press ‘2nd’ and then [SET], the display will read END.

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**The Formula for Future Value**

Number of periods Rate of return or discount rate or interest rate or growth per period Present Value

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**The Formula for Present Value**

From before, we know that Solving for PV, we get Unless otherwise stated, r stated on an annual basis. Again, now we deal with PV problems where n > 2

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**Special keys for TVM problems**

N: Number of periods (e.g., years) I/Y: Interest rate/ discounting rate per period 3. PV: Present value PMT: Periodic fixed cash flow 5. FV: Future value 6. CPT: Compute

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**What is the future value of an initial $100 after 3 years, if I/YR = 10%?**

Finding the FV of a cash flow or series of cash flows is called compounding. FV can be solved by using the step-by-step, financial calculator, and spreadsheet methods. FV = ? 1 2 3 10% 100

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**The step-by-step and formula methods**

After 1 year: FV1 = PV (1 + I) = $100 (1.10) = $110.00 After 2 years: FV2 = PV (1 + I)2 = $100 (1.10) =$121.00 After 3 years: FV3 = PV (1 + I)3 = $100 (1.10) =$133.10 After N years (general case): FVN = PV (1 + I)N

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**The calculator method Solves the general FV equation.**

Requires 4 inputs into calculator, and will solve for the fifth. 3 10 -100 INPUTS CPT N I/YR PV PMT FV OUTPUT 133.10

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Multi-period, Find PV Find the present value of $6,000 that occurs at t = 6. The discount rate is 14 percent. Use PV = FV/(1+r)6 FV=6000, N = 6, I/Y = 14, PMT = 0. Press CPT and then PV

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Multi-period, find FV Suppose you deposit $150 in an account today and the interest rate is 6 percent p.a.. How much will you have in the account after 33 years? Use FV = PV x (1+r)33 Press PV=-150, N=33,I/Y=6, PMT=0 Press CPT then FV

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Multi-period, find r You deposited $15,000 in an account 22 years ago and now the account has $50,000 in it. What was the annual rate of return on this investment? PV = , N = 22, PMT = 0, FV = 50000, I/Y = ?

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Multi-period, find n You currently have $38,000 in an account that has been paying 5.75 percent p.a.. You remember that you had opened this account quite some years ago with an initial deposit of $19,000. You forget when the initial deposit was made. How many years ago did you make the initial deposit? PV = , PMT = 0, FV = 38000, I/Y = 5.75, N = ?

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Perpetuity 1 Perpetuity: a stream of equal cash flows ( C ) that occur at the end of each period and go on forever. PV of perpetuity = C is the cash flow at the end of each period r is the discount rate

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Perpetuity 2 So what? We use the idea of a perpetuity to determine the value of A preferred stock A perpetual debt

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Perpetuity questions Suppose the value of a perpetuity is $38,900 and the discount rate is 12 percent p.a.. What must be the annual cash flow from this perpetuity? Use C = PV x r. Verify that C = $4,668. An asset that generates $890 per year forever is priced at $6,000. What is the required rate of return? Use r = C/PV. Verify that r = percent

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Annuity Annuity (ordinary annuity): a cash flow stream where a fixed amount is received at the end of every period for a fixed number of periods. Example: You borrow a loan for $12,000 and pay 5% interest at the end of every year. In many TVM problems, the cash flow stream is An annuity combined with a single cash flow (often at the beginning or the end) A combination of two or more annuities.

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Annuity, find PV You are considering buying a rental property. The yearly rent from this property is $18,000 to be paid at year-end. You expect that the property will generate this rent for the next twenty years after which you will be able to sell it for $250,000. If your required rate of return is 12 percent p.a., what is the maximum amount that you would pay for this property? PMT=18000, FV=250,000, I/Y=12, N=20, PV=?

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**Time Line Analysis 18,000 lasts for 20 years.**

PV 250000 18000 18000 18000 18000 1 2 19 20 18,000 lasts for 20 years. 250,000 occurs one minute right after the last 18,000. PV actually is the price your are willing to pay.

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Annuity, find FV You open an account today with $20,000 and at the end of each of the next 15 years, you deposit $2,500 in it. At the end of 15th years, what will be the balance in the account if the interest rate is 7 percent p.a.? PV=-20000, PMT=-2500, N=15, I/Y=7, FV=?

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Annuity, find I/Y You lend your friend $100,000. He will pay you $12,000 per year for the ten years and a balloon payment at t = 10 of $50,000. What is the interest rate that you are charging your friend? PV=-100,000, FV=50,000, PMT=12,000, N = 10, I/Y=?

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Annuity, find PMT Next year, you will start to make deposits of $3,000 per year for 35 years in your Individual Retirement Account. With the money accumulated at t=35, you will then buy a retirement annuity of 20 years with equal yearly payments from a life insurance company (payments from t=36 to t=55). If the annual rate of return over the entire period is 8%, what will be the annual payment of the annuity?

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continued There are two annuity. One is from t=0 to 35, the other is from 35 to 55. To find the PMT in the second annuity, we need to input PV=, N=20, FV=0, I/Y=8. However, PV (the price) is not known. But the PV in the second annuity equals the FV in the first annuity because you use all the money accumulated up to t=35 to buy the second annuity. To find FV in the first annuity, you need to input PV=0, N=35, PMT=-3000, I/Y=8. -3000 -3000 FV -3000 -3000 34 35 1 2 PMT=? PV 35 36 55

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Uneven Cash Flows 100 50 75 1 2 3 I% -50

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Uneven cash flows 1 Your account pays interest at a rate of 5 percent p.a.. You deposit $8,000 in it today. You must have exactly $3,000 in the account after two years. How much should you withdraw at the end of the first year to ensure this?

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**What is the PV of this uneven cash flow stream?**

100 1 300 2 3 10% -50 4

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**What is the PV of this uneven cash flow stream?**

100 1 300 2 3 10% -50 4 90.91 247.93 225.39 -34.15 = PV

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**Solving for PV: Uneven cash flow stream**

Input cash flows in the calculator’s “CF” register: Press CF key CF0 = 0, ENTER, C01 = 100, ENTER, F01=1, ENTER, C02 = 300, ENTER, F02=2, ENTER, C03= 50, +/- key, ENTER, F03=1, ENTER, Press NPV key I = 10, ENTER, press CPT key to get NPV = (Here NPV = PV.)

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**Use calculator Remember to push ENTER and keys after each step !**

ENTER is not = ! F01, F02,… refers to the frequency of the cash flows. Since only one 100, so F01=1, but F02=2 because there are two consecutive 300.

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Uneven cash flows 2 An asset promises to produce the following series of cash flows. At the end of each of the first three years, $5,000. At the end of each of the following four years, $7,000. And, at the end of each of following five years, $9,000. If your required rate of return is 10 percent, how much is this asset worth to you? Find PV of this series of cash flows. PV = $46,612.68

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Uneven cash flows 3 You will need to pay for your son’s private school tuition (first grade through 12th grade) a sum of $8,000 per year for Years 1 through 5, $10,000 per year for Years 6 through 8, and $12,500 per year for Years 9 through 12. Assume that all payments are made at the beginning of the year, that is, tuition for Year 1 is paid now (i.e., at t = 0), tuition for Year 2 is paid one year from now, and so on. In addition to the tuition payments you expect to incur graduation expenses of $2,500 at the end of Year 12. If a bank account can provide a certain 10 percent p.a. rate of return, how much money do you need to deposit today to be able to pay for the above expenses?

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**Special topics Compounding period is less than 1 year Annuity due**

Loan amortization

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**Compounding period is less than 1 year**

Saying that compounding period is less than 1 year is equivalent to saying frequency of compounding is more than once per year

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**Six-months / semiannual**

Common examples Compounding period Compounding frequency Six-months / semiannual 2 Quarter 4 Month 12 Day 365

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Example (1) Suppose that your bank “states” that the interest on your account is eight percent p.a.. However, interest is paid semi-annually, that is every six months or twice a year. The 8% is called the stated interest rate. (also called the nominal interest rate) But, the bank will pay you 4% interest every 6 months.

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Example (2) Ok, so we know how much interest is paid every 6 months. Over a year, what is the percentage interest I actually earn? In other words, I want to know the effective annual interest rate (or effective interest rate, or annual percentage yield)

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**Example (3) Suppose you deposit $100 into the account today.**

Account balance at end of 6 months: 100 x 1.04 = 104 Account balance at end of 1 year: 104 x 1.04 =108.16 Effective interest rate = ( – 100)/100 = or 8.16%

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**When frequency of compounding is more than once a year**

‘n’ = number of years ‘m’ = frequency of compounding per year ‘r’ = stated interest rate

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**Can the effective rate ever be equal to the nominal rate?**

Yes, but only if annual compounding is used, i.e., if m = 1. If m > 1, effective rate will always be greater than the nominal rate.

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**Can’t remember those formula?**

Formula are hard to remember. The examples in the following slides are easy to remember. is the rate for each period, that is, periodic rate.

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Examples If you deposit $100 today for 3 years. The stated annual interest rate is 12% a year. How much can you withdraw after 3 years? What is the effective rate? * The answer depends on how often the interest is compounded or paid. There are 4 possible answers.

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Possible answers 1. The interest is paid once 1 year: 2. The interest is paid every 6 months: 3. The interest is paid every 3 months: 4. The interest is paid every month:

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**Effective rate example**

You have decided to buy a car priced at $45,000. The dealer offers to finance the entire amount and requires 60 monthly payments of $950 per month. What are the yearly stated and effective interest rates for this financing? Answer: stated = % p.a. effective = % p.a.

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**Car buying with down payment**

You are considering buying a new car. The sticker price is $15,000, and you have $3000 for down payment. You obtain a 5-year car loan at a nominal annual interest rate of 12%. What is your monthly loan payment?

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**Annuity with monthly compounding**

Compute the future value at the end of year 25 of a $100 deposited every month (with the first deposit made one month from today) into an account that pays 9 percent p.a.

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**Annuity with semiannual compounding**

You would like to accumulate $16,500 over the next 8 years. How much must you deposit every six months, starting six months from now, given a 4 percent annual rate with semiannual compounding?

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Effective rate Your bank’s stated interest rate on a three month certificate of deposit is 4.68 percent p.a. and the interest is paid quarterly. What is the effective interest rate?

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Find period The stated interest rate for a bank account is 7 percent and interest is paid semi-annually. How many years will it take you to double your money in this account?

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**More frequent compounding, more $**

All else constant, for a given nominal interest rate, an increase in the number of compounding periods per year will cause the future value of some current sum of money to: Increase Decrease Remain the same May increase, decrease or remain the same depending on the number of years until the money is to be received. Will increase if compounding occurs more often than 12 times per year and will decrease if compounding occurs less than 12 times per year.

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**Annuity Due 1 Up till now, we deal with ordinary annuities.**

For an ordinary annuity, payment occurs at the end of each period. For an annuity due, payment occurs at the beginning of each period.

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**Consider an annuity that pays $300 per year for three years.**

If ordinary annuity, time line is: If annuity due, time line is: $300 $300 $300 T = 1 T = 3 T = 0 T = 2 $300 $300 $300 T = 3 T = 0 T = 1 T = 2

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**Is there a relationship between ordinary annuity and annuity due?**

Yes ! PV of annuity due = (PV of ordinary annuity) x (1 + r) FV of annuity due = (FV of ordinary annuity) x (1 + r) ‘ordinary annuity’ and ‘regular annuity’ mean the same thing.

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Example You have a rental property that you want to rent for 10 years. Prospective tenant A promises to pay you a rent of $12,000 per year with the payments made at the end of each year. Prospective tenant B promises to pay $12,000 per year with payments made at the beginning of each year. Which is a better deal for you if the appropriate discount rate is 10 percent? Set PMT = 12,000, N = 10, I/Y = 10, FV=0 To answer question, focus on dollar amount of each PV.

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Another example What is the present value of an annuity of $1200 per year for 10 years (with the first payment to be made today and the last payment to be made 9 years from today) given an interest rate of 5.5 percent p.a.?

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Loan Amortization Amortization is the process of separating a payment into two parts: The interest payment The repayment of principal Note: Interest payment decreases over time Principal repayment increases over time

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**Example of loan amortization 1**

You have borrowed $8,000 from a bank and have promised to repay the loan in five equal yearly payments. The first payment is at the end of the first year. The interest rate is 10 percent. Draw up the amortization schedule for this loan. Amortization schedule is just a table that shows how each payment is split into principal repayment and interest payment.

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**Example of loan amortization 2**

1) Compute periodic payment. PV=8000, N=5, I/Y=10, FV=0, PMT=? Verify that PMT = -2,110.38 Amortization for first year Interest payment = 8000 x 0.1 = 800 Principal repayment = 2, – 800 = Immediately after first payment, the principal balance is = 8000 – = 6,689.62

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**Example of loan amortization 3**

Amortization for second year Interest payment = x 0.1 = (using the new balance!) Principal repayment = 2, – = Immediately after second payment, the principal balance is = 6, – = 5,248.20 Verify the entire schedule (on following slide)

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**Verify the amortization schedule**

Year Beg. Balance Payment Interest Principal End. 8,000.00 1 2,110.38 800.00 1,310.38 6,689.62 2 668.96 1,441.42 5,248.20 3 524.82 1,585.56 3,662.64 4 366.26 1,744.12 1,918.53 5 191.85 0.00

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**Using financial calculator to generate amortization schedule 1**

Very often, amortization problems involve long periods of time, e.g., 30 year mortgage with monthly payments => 360 periods. To generate amortization schedule in such problems, it’s more efficient to use the financial calculator. Let’s reuse the last problem (Problem 7.25). First, find the monthly payment. Key in: PV=8000, N=5, I/Y=10, FV=0, PMT=? We already worked out that PMT = -2,

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**Using financial calculator to generate amortization schedule 2**

Suppose we want to work out the remaining balance immediately after the 2nd payment. Press [2ND], [AMORT] to activate the Amortization worksheet in BA II Plus. Press P1=2, [ENTER], , Press P2=2, [ENTER], , You will see BAL=5,248.20 P1 = starting point in a range of payments, the first payment of interest P2 = ending point in a range of payments, the last payment of interest

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**Using financial calculator to generate amortization schedule 2**

Press again and you see the portion of the year 2 payment going towards repaying principal, i.e., PRN = -1,441.42 Press again and you see the portion of year 2 payment going towards interest, i.e., INT = To get out of the Amortization schedule, press [2ND], Quit.

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What are P1 and P2? P1 is the first payment of the period your are interested in, while p2 is the ending payment of the period you are interested in. In the example, you are interested in the second payment only. So P1=P2=2. If you want to find the total interest paid from the 3rd, 4th, and 5th payments, then P1=3 and P2=5.

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All together now (1) Which of the following statements is most correct? A 5-year $100 annuity due will have a higher future value than a 5-year $100 ordinary annuity. A 15-year mortgage will have smaller monthly payments than a 30-year mortgage of the same amount and same interest rate. All else being constant, for a given nominal interest rate, an increase in the number of compounding periods per year will cause the future value of some current sum of money to increase. Statements A and C are correct. All of the statements above are correct.

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All together now (2) Which of the following statements is most correct? An investment that compounds interest semiannually, and has a nominal rate of 15 percent, will have an effective rate less than 15 percent. The present value of a three-year $1000 annuity due is less than the present value of a three-year $1000 ordinary annuity. The portion of the payment of a fully amortized loan that goes toward interest declines over time. Statements A and C are correct. None of the answers above is correct.

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One more example You borrowed $300,000 mortgage loan to buy a house at the first day of The loan is for 15 years and the stated rate is 12% p.a. All mortgage loan computes interest monthly. Compute the total mortgage interest paid in 2012.

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**Summary TVM problems with multiple periods and multiple cash flows**

Solving TVM problems using financial calculator and time lines Special topics Compounding period < one year Continuous compounding Annuity due Loan amortization

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Assignment Chapter 5 Self-test ST-3 ST-4 Questions Problems

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