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Financial Management Lecture # 02 Time Value of Money

What Is The Time Value Of Money? The principle of time value of money – the notion that a given sum of money is more valuable the sooner it is received, due to its capacity to earn interest – is the foundation for numerous applications in investment finance. The Five Components Of Interest Rates Real Risk-Free Rate – This assumes no risk or uncertainty, simply reflecting differences in timing: the preference to spend now/pay back later versus lend now/collect later. Expected Inflation - The market expects aggregate prices to rise, and the currency's purchasing power is reduced by a rate known as the inflation rate. Inflation makes real dollars less valuable in the future and is factored into determining the nominal interest rate (from the economics material: nominal rate = real rate + inflation rate). Default-Risk Premium - What is the chance that the borrower won't make payments on time, or will be unable to pay what is owed? This component will be high or low depending on the creditworthiness of the person or entity involved. Liquidity Premium- Some investments are highly liquid, meaning they are easily exchanged for cash (U.S. Treasury debt, for example). Other securities are less liquid, and there may be a certain loss expected if it's an issue that trades infrequently. Holding other factors equal, a less liquid security must compensate the holder by offering a higher interest rate. Maturity Premium - All else being equal, a bond obligation will be more sensitive to interest rate fluctuations the longer to maturity it is.

Nominal Interest Rate: Rate at which money invested grows. Real interest rate = nominal interest rate – inflation rate Real Interest Rate: Rate at which the purchasing power of an investment increases. The real rate of interest is calculated by 1 + real interest rate = 1 + nominal interest rate ____________________ 1 + inflation rate Effective Annual Interest Rate: Interest rate that is annualized using compound interest. The effective annual yield represents the actual rate of return, reflecting all of the compounding periods during the year. The effective annual yield (or EAR) can be computed given the stated rate and the frequency of compounding. Effective annual rate (EAR) = (1 + Periodic interest rate)m – 1 Where: m = number of compounding periods in one year, and periodic interest rate = (stated interest rate) / m Example: Effective Annual Rate Suppose we are given a stated interest rate of 9%, compounded monthly, here is what we get for EAR: Annual Percentage Rate (APR): Interest rate that is annualized using simple interest rate. The effective annual rate is the rate at which invested funds will grow over the course of a year. It equals the rate of interest per period compounded for the number of periods in a year. Keep in mind that the effective annual rate will always be higher than the stated rate if there is more than one compounding period (m > 1 in our formula), and the more frequent the compounding, the higher the EAR.

Future Value: Definition: The value to which a beginning lump sum or Present Value (PV) will grow in a certain number of periods, n, at a specified rate of interest, i. Formula: FV = PV (1 + i)n Where: i = the stated rate of interest n = number of years (1 + i)n = the future value interest factor Compound Interest: Interest earned on interest. Simple Interest: Interest earned only on the original investment; no interest is earned on interest. Example: In six years, Frank will be eligible for membership in the elite Flat lounger Club. A lifetime membership will cost him $14,000. Frank currently has $11,000 in a savings account that pays an annual interest rate of 4.2 percent. In six years, will he have enough money in the account to pay his membership fees? When compounding occurs more than once a year Example: Jane has inherited $4,500 dollars, and she has decided to deposit it in her savings account for six months before she decides how to spend/invest it. If Jane’s savings account pays 3 percent compounded monthly, how much money will she have in six months? Compound growth means that value increases each period by the factor (1 + growth rate). The value after t periods will equal the initial value times (1 + growth rate)t. When money is invested at compound interest, the growth rate is the interest rate.

Present Value: A dollar today is worth more than a dollar tomorrow. The time line in the future and look back toward time 0 to see what was the beginning amount. Formula: PV= FV / ( 1+ r )n

Discount Rate: Interest rate used to compute present values of future cash flows. Example: If I promised to give you one million dollars 50 years from now, what would it be worth today if the discount rate is 15 percent compounded annually? When compounding occurs more than once a year Example : In our previous example, I asked what a million dollars would be worth 50 years from now at 15 percent compounded annually. What would it be worth if the discount rate were 15 percent compounded semiannually? Example: Will Williams grandmother always gives him $200 on his birthday which is nine months away. Will needs the money now in order to buy his finance text. Fred Fredrickson has agreed to lend Will some money at 18 percent compounded quarterly. If Will plans to pay back Fred in nine months with his birthday money, how much can he borrow from Fred now? Example: If you borrow $10,000 today and pay back $12,167 at end of 5 years what rate of interest did you pay on the loan? Example: Suppose you invest $2,000 at 4.5% and want your investment to grow to $4,205. How long will it take? Recall from your math classes that ln(xn)= n ln(x) Example: Which rate would you prefer as a borrower? 10 percent compounded semiannually or 9.8 percent compounded daily (365 days in a year)

Annuity: Equally spaced level stream of cash flows. Perpetuity: Stream of level cash payments that never ends. Present Value of a Perpetuity A perpetuity starts as an ordinary annuity (first cash flow is one period from today) but has no end and continues indefinitely with level, sequential payments PV of a perpetuity = annuity payment A interest rate r Example: perpetuity paying $1,000 annually at an interest rate of 8% would be worth?

Present value of t-year annuity = payment annuity factor Future Value Annuity Factor = (1 + r)n - 1 r Present Value Annuity Factor = 1 1 (1 + r)n r Annuity Due: Level stream of cash flow starting immediately. Example: FV and PV of ordinary annuity and annuity due An individual deposits $10,000 at the beginning of each of the next 10 years, starting today, into an account paying 9% interest compounded annually. The amount of money in the account of the end of 10 years will be closest to:

Answer: The problem gives the annuity amount A = $10,000, the interest rate r = 0.09, and time periods N = 10. Time units are all annual (compounded annually) so there is no need to convert the units on either r or N. However, the starting today introduces a wrinkle. The annuity being described is an annuity due, not an ordinary annuity, so to use the FV annuity factor, we will need to change our perspective to fit the definition of an ordinary annuity. The definition of an ordinary annuity is a cash flow stream beginning in one period, so the annuity being described in the problem is an ordinary annuity starting last year, with 10 cash flows from t0 to t9. Using the FV annuity factor formula, we have the following: FV annuity factor = ((1 + r)n – 1)/r = (1.09)10 – 1)/0.09 = (1.3673636)/0.09 = 15.19293

Multiplying this amount by the annuity amount of $10,000, we have the future value at time period 9. FV = ($10,000)*(15.19293) = $151,929. To finish the problem, we need the value at t10. To calculate, we use the future value of a lump sum, FV = PV*(1 + r)N, with N = 1, PV = the annuity value after 9 periods, r = 9.FV = PV*(1 + r)N = ($151,929)*(1.09) = $165,603.

Time period 1

2

3

4

5

Cash Flow

1500

2000

500

3000

1000

Example: Assume that we are to receive a sequence of uneven cash flows from an annuity and we're asked for the present value of the annuity at a discount rate of 8%. Scratch out a table similar to the one below, with periods in the first column, cash flows in the second, formulas in the third column and computations in the fourth.

Example: Monthly Mortgage Payments Assuming a 30-year loan with monthly compounding (so N = 30*12 = 360 months), and a rate of 6% (so r = .06/12 = 0.005), we first calculate the PV annuity factor: PV annuity factor = (1 – (1/(1 + r)N)/r = (1 – (1/(1.005)360)/0.005 = 166.7916 With a loan of $250,000, the monthly payment in this example would be $250,000/166.7916, or $1,498.88 a month. Example: An investor wants to have $1 million when she retires in 20 years. If she can earn a 10% annual return, compounded annually, on her investments, the lump-sum amount she would need to invest today to reach her goal is closest to:

EXAMPLE Home Mortgages Sometimes you may need to find the series of cash payments that would provide a given value today. For example, home purchasers typically borrow the bulk of the house price from a lender. The most common loan arrangement is a 30-year loan that is repaid in equal monthly installments. Suppose that a house costs $125,000, and that the buyer puts down 20 percent of the purchase price, or $25,000, in cash, borrowing the remaining $100,000 from a mortgage lender such as the local savings bank. What is the appropriate monthly mortgage payment? Self-Test : What will be the monthly payment if you take out a $100,000 fifteen-year mortgage at an interest rate of 1 percent per month? How much of the first payment is interest and how much is amortization?

Saving for Retirement In only 50 more years, you will retire. (That’s right—by the time you retire, the retirement age will be around 70 years. Longevity is not an unmixed blessing.) Have you started saving yet? Suppose you believe you will need to accumulate $500,000 by your retirement date in order to support your desired standard of living. How much must you save each year between now and your retirement to meet that future goal? Let’s say that the interest rate is 10 percent per year. You need to find how large the annuity in the following figure must be to provide a future value of $500,000:

We know that if you were to save $1 each year your funds would accumulate to Future value of annuity of $1 a year = (1 + r)t – 1 = (1.10)50 – 1 _________ __________ = $1,163.91 r 0.10 Therefore, if we save an amount of $C each year, we will accumulate $C × 1,163.91. We need to choose C to ensure that $C × 1,163.91 = $500,000. Thus C = $500,000/1,163.91 = $429.59. This appears to be surprisingly good news. Saving $429.59 a year does not seem to be an extremely demanding savings program. Don’t celebrate yet, however. The news will get worse when we consider the impact of inflation.

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What Is The Time Value Of Money? The principle of time value of money – the notion that a given sum of money is more valuable the sooner it is received, due to its capacity to earn interest – is the foundation for numerous applications in investment finance. The Five Components Of Interest Rates Real Risk-Free Rate – This assumes no risk or uncertainty, simply reflecting differences in timing: the preference to spend now/pay back later versus lend now/collect later. Expected Inflation - The market expects aggregate prices to rise, and the currency's purchasing power is reduced by a rate known as the inflation rate. Inflation makes real dollars less valuable in the future and is factored into determining the nominal interest rate (from the economics material: nominal rate = real rate + inflation rate). Default-Risk Premium - What is the chance that the borrower won't make payments on time, or will be unable to pay what is owed? This component will be high or low depending on the creditworthiness of the person or entity involved. Liquidity Premium- Some investments are highly liquid, meaning they are easily exchanged for cash (U.S. Treasury debt, for example). Other securities are less liquid, and there may be a certain loss expected if it's an issue that trades infrequently. Holding other factors equal, a less liquid security must compensate the holder by offering a higher interest rate. Maturity Premium - All else being equal, a bond obligation will be more sensitive to interest rate fluctuations the longer to maturity it is.

Nominal Interest Rate: Rate at which money invested grows. Real interest rate = nominal interest rate – inflation rate Real Interest Rate: Rate at which the purchasing power of an investment increases. The real rate of interest is calculated by 1 + real interest rate = 1 + nominal interest rate ____________________ 1 + inflation rate Effective Annual Interest Rate: Interest rate that is annualized using compound interest. The effective annual yield represents the actual rate of return, reflecting all of the compounding periods during the year. The effective annual yield (or EAR) can be computed given the stated rate and the frequency of compounding. Effective annual rate (EAR) = (1 + Periodic interest rate)m – 1 Where: m = number of compounding periods in one year, and periodic interest rate = (stated interest rate) / m Example: Effective Annual Rate Suppose we are given a stated interest rate of 9%, compounded monthly, here is what we get for EAR: Annual Percentage Rate (APR): Interest rate that is annualized using simple interest rate. The effective annual rate is the rate at which invested funds will grow over the course of a year. It equals the rate of interest per period compounded for the number of periods in a year. Keep in mind that the effective annual rate will always be higher than the stated rate if there is more than one compounding period (m > 1 in our formula), and the more frequent the compounding, the higher the EAR.

Future Value: Definition: The value to which a beginning lump sum or Present Value (PV) will grow in a certain number of periods, n, at a specified rate of interest, i. Formula: FV = PV (1 + i)n Where: i = the stated rate of interest n = number of years (1 + i)n = the future value interest factor Compound Interest: Interest earned on interest. Simple Interest: Interest earned only on the original investment; no interest is earned on interest. Example: In six years, Frank will be eligible for membership in the elite Flat lounger Club. A lifetime membership will cost him $14,000. Frank currently has $11,000 in a savings account that pays an annual interest rate of 4.2 percent. In six years, will he have enough money in the account to pay his membership fees? When compounding occurs more than once a year Example: Jane has inherited $4,500 dollars, and she has decided to deposit it in her savings account for six months before she decides how to spend/invest it. If Jane’s savings account pays 3 percent compounded monthly, how much money will she have in six months? Compound growth means that value increases each period by the factor (1 + growth rate). The value after t periods will equal the initial value times (1 + growth rate)t. When money is invested at compound interest, the growth rate is the interest rate.

Present Value: A dollar today is worth more than a dollar tomorrow. The time line in the future and look back toward time 0 to see what was the beginning amount. Formula: PV= FV / ( 1+ r )n

Discount Rate: Interest rate used to compute present values of future cash flows. Example: If I promised to give you one million dollars 50 years from now, what would it be worth today if the discount rate is 15 percent compounded annually? When compounding occurs more than once a year Example : In our previous example, I asked what a million dollars would be worth 50 years from now at 15 percent compounded annually. What would it be worth if the discount rate were 15 percent compounded semiannually? Example: Will Williams grandmother always gives him $200 on his birthday which is nine months away. Will needs the money now in order to buy his finance text. Fred Fredrickson has agreed to lend Will some money at 18 percent compounded quarterly. If Will plans to pay back Fred in nine months with his birthday money, how much can he borrow from Fred now? Example: If you borrow $10,000 today and pay back $12,167 at end of 5 years what rate of interest did you pay on the loan? Example: Suppose you invest $2,000 at 4.5% and want your investment to grow to $4,205. How long will it take? Recall from your math classes that ln(xn)= n ln(x) Example: Which rate would you prefer as a borrower? 10 percent compounded semiannually or 9.8 percent compounded daily (365 days in a year)

Annuity: Equally spaced level stream of cash flows. Perpetuity: Stream of level cash payments that never ends. Present Value of a Perpetuity A perpetuity starts as an ordinary annuity (first cash flow is one period from today) but has no end and continues indefinitely with level, sequential payments PV of a perpetuity = annuity payment A interest rate r Example: perpetuity paying $1,000 annually at an interest rate of 8% would be worth?

Present value of t-year annuity = payment annuity factor Future Value Annuity Factor = (1 + r)n - 1 r Present Value Annuity Factor = 1 1 (1 + r)n r Annuity Due: Level stream of cash flow starting immediately. Example: FV and PV of ordinary annuity and annuity due An individual deposits $10,000 at the beginning of each of the next 10 years, starting today, into an account paying 9% interest compounded annually. The amount of money in the account of the end of 10 years will be closest to:

Answer: The problem gives the annuity amount A = $10,000, the interest rate r = 0.09, and time periods N = 10. Time units are all annual (compounded annually) so there is no need to convert the units on either r or N. However, the starting today introduces a wrinkle. The annuity being described is an annuity due, not an ordinary annuity, so to use the FV annuity factor, we will need to change our perspective to fit the definition of an ordinary annuity. The definition of an ordinary annuity is a cash flow stream beginning in one period, so the annuity being described in the problem is an ordinary annuity starting last year, with 10 cash flows from t0 to t9. Using the FV annuity factor formula, we have the following: FV annuity factor = ((1 + r)n – 1)/r = (1.09)10 – 1)/0.09 = (1.3673636)/0.09 = 15.19293

Multiplying this amount by the annuity amount of $10,000, we have the future value at time period 9. FV = ($10,000)*(15.19293) = $151,929. To finish the problem, we need the value at t10. To calculate, we use the future value of a lump sum, FV = PV*(1 + r)N, with N = 1, PV = the annuity value after 9 periods, r = 9.FV = PV*(1 + r)N = ($151,929)*(1.09) = $165,603.

Time period 1

2

3

4

5

Cash Flow

1500

2000

500

3000

1000

Example: Assume that we are to receive a sequence of uneven cash flows from an annuity and we're asked for the present value of the annuity at a discount rate of 8%. Scratch out a table similar to the one below, with periods in the first column, cash flows in the second, formulas in the third column and computations in the fourth.

Example: Monthly Mortgage Payments Assuming a 30-year loan with monthly compounding (so N = 30*12 = 360 months), and a rate of 6% (so r = .06/12 = 0.005), we first calculate the PV annuity factor: PV annuity factor = (1 – (1/(1 + r)N)/r = (1 – (1/(1.005)360)/0.005 = 166.7916 With a loan of $250,000, the monthly payment in this example would be $250,000/166.7916, or $1,498.88 a month. Example: An investor wants to have $1 million when she retires in 20 years. If she can earn a 10% annual return, compounded annually, on her investments, the lump-sum amount she would need to invest today to reach her goal is closest to:

EXAMPLE Home Mortgages Sometimes you may need to find the series of cash payments that would provide a given value today. For example, home purchasers typically borrow the bulk of the house price from a lender. The most common loan arrangement is a 30-year loan that is repaid in equal monthly installments. Suppose that a house costs $125,000, and that the buyer puts down 20 percent of the purchase price, or $25,000, in cash, borrowing the remaining $100,000 from a mortgage lender such as the local savings bank. What is the appropriate monthly mortgage payment? Self-Test : What will be the monthly payment if you take out a $100,000 fifteen-year mortgage at an interest rate of 1 percent per month? How much of the first payment is interest and how much is amortization?

Saving for Retirement In only 50 more years, you will retire. (That’s right—by the time you retire, the retirement age will be around 70 years. Longevity is not an unmixed blessing.) Have you started saving yet? Suppose you believe you will need to accumulate $500,000 by your retirement date in order to support your desired standard of living. How much must you save each year between now and your retirement to meet that future goal? Let’s say that the interest rate is 10 percent per year. You need to find how large the annuity in the following figure must be to provide a future value of $500,000:

We know that if you were to save $1 each year your funds would accumulate to Future value of annuity of $1 a year = (1 + r)t – 1 = (1.10)50 – 1 _________ __________ = $1,163.91 r 0.10 Therefore, if we save an amount of $C each year, we will accumulate $C × 1,163.91. We need to choose C to ensure that $C × 1,163.91 = $500,000. Thus C = $500,000/1,163.91 = $429.59. This appears to be surprisingly good news. Saving $429.59 a year does not seem to be an extremely demanding savings program. Don’t celebrate yet, however. The news will get worse when we consider the impact of inflation.

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