The time value of money means that money accumulates value over time. This concept is based on the principle of compounding, where interest is earned on both the initial principal and the accumulated interest.

Key action words for learning outcomes include interpret, explain, and demonstrate. Understanding different types of interest rates is necessary for financial calculations, and using a financial calculator is essential for solving time value of money problems.

Receiving money today is generally preferred over receiving the same amount in the future due to potential investment opportunities. Inflation can also affect the value of money over time, making present money more valuable.

For example, if an investor has $100todayandthesavingsrateis10$100 will grow to $110 in one year.

• ★The time value of money means that money accumulates value over time.0:04
• ★Compounding refers to earning interest on both the principal and accumulated interest.0:30
• ★Receiving money today is preferred over receiving the same amount in the future.2:30
• ★$100$ today grows to $110$ in one year at a $10\%$ interest rate.3:00

Example: If an investor has $100today,itwillgrowto$110 in one year at a 10% interest rate.3:00

Consider investing $100atasavingsrateof4.3467$100$by$0.043467$toget$4.3467$ininterest,thenaddthistotheinitialinvestment,resultingin$104.3467$.

The 'new short way math' combines these steps. We use the time value factor, defined as $(1+r{)}^n$, where $r$ is the interest rate and $n$ is the number of periods. Thus, we calculate the future value as $100\times (1+0.043467{)}^1$.

If we want to calculate the future value two years from now, we must account for compounding. The future value is not simply $104.3467+4.3467$ for the second year. Instead, we calculate it as $100\times (1+0.043467{)}^2$.

• ★The time value factor is $(1+r{)}^n$.4:08
• ★The 'old long way math' requires separate multiplication and addition.4:10
• ★The 'new short way math' combines calculations into one formula.4:14
• ★Compounding interest affects future value calculations.4:30

Example: Using the new method, the future value after one year is $100\times (1+0.043467{)}^1=104.3467$.4:20

Compounding adds an extra dollar on top of interest and principal. For an initial amount of $100$ at an interest rate of 10\text{%} over two periods, the future value is $121$. This is calculated using the formula $(1+0.10{)}^2$.

To compute future values, a financial calculator can be used. Ensure the calculator is set to two decimal places for precision. Adjust the payments per year to one for annual compounding.

When using the financial calculator, input the present value, interest rate, and number of periods. The order of inputs does not matter, but the output depends on which variable is being solved.

• ★The future value with compounding is $121$ instead of $120$.7:26
• ★Use the formula $(1+r{)}^n$ for future value calculations.7:26
• ★Set payments per year to $1$ for annual compounding in the calculator.7:26
• ★The future value is displayed as a negative number in the calculator.7:26

Example: Using a financial calculator, input $100$ as present value, 10\text{%} as interest rate, and $1$ as the number of periods to compute the future value of $110$.7:26

The time value of money applies equally to both sides of a financial contract. We can compute future value using present value and interest rate. For example, if Bob saves $500 today at a 7% interest rate, we can calculate the future value after five years.

Using a financial calculator, we input $500asthepresentvalue,$5$asthenumberofperiods($n$),and$7$astheinterestrate($i$).Wethencomputethefuturevalue,whichresultsin$701.

In another scenario, if Betty needs $1,000in11yearsata9$11$as$n$,$9$as$i$,and$1,000$asthefuturevalue.Wecomputethepresentvalue,whichgivesus$387.

• ★Future value can be computed from present value and interest rate.12:25
• ★Input order in financial calculators does not affect the outcome.12:25
• ★Example: $500savedtodayat7\%$ grows to $701$ in five years.12:25
• ★To achieve $1,000$ in 11 years at $9\%$, save $387$ today.12:25

Example: Bob saves $500today;futurevalueafterfiveyearsat7$701.12:26

When calculating interest rates, present value should be entered as negative. For example, if Bill has $49todayandneeds$92 in six years, we enter $PV=-49$, $FV=92$, $n=6$, and $PMT=0$ to compute the interest rate, which results in $i=11.07\%$. If both present value and future value are positive, the calculator will indicate no solution exists.

An annuity is defined as a series of equal payments made at regular intervals. Bonnie promises to pay $200peryearforeightyears,whichisanordinaryannuity.Weset$PV = 0$,$n = 8$,$i = 11$,and$PMT = 200$tofindthefuturevalue,resultingin$FV = 23.71$.

In another example, Bruce has $9000todayandpromisestopayhissonseveryyearforsevenyears.Weinput$PV = -9000$,$n = 7$,$i = 3$,and$FV = 0$tosolveforthepayment,yielding$PMT = 1444.55$. This illustrates how to enter four variables and solve for the fifth.

The required rate of return is interpreted as the minimum return expected from an investment. This is crucial when entering a contract, as it defines the expected cash flows.

• ★Present value should be negative when calculating interest rates.15:48
• ★An annuity consists of equal payments made at regular intervals.15:56
• ★Bonnie's payment of $200 per year for eight years is an ordinary annuity.16:04
• ★Bruce's present value of $9000 allows calculation of annual payments.16:13
• ★The required rate of return is the minimum expected return from an investment.17:00

Example: Bill needs $92insixyearsstartingfrom$49 today, leading to an interest rate of $11.07\%$.15:50

The required rate of return is the minimum return an investor must earn to justify an investment. For example, if a lender requires 10 percent on a $100 loan, the borrower must earn at least 10 percent to satisfy the contract.

The discount rate is a key concept in the time value of money. It is used to calculate the present value of future cash flows. This rate reflects the opportunity cost of capital.

Opportunity cost is the value of the best alternative that is foregone when making an investment decision. If a more profitable opportunity arises after lending money, the investor may regret the decision.

Interest rates can be broken down into a real risk-free rate and various risk premiums. The real risk-free rate represents the return on an investment with no risk of financial loss.

• ★The required rate of return is the minimum return an investor must earn.20:47
• ★The discount rate is used to calculate present value of future cash flows.20:55
• ★Opportunity cost is the value of the best alternative foregone.21:05
• ★Interest rates consist of a real risk-free rate and risk premiums.21:15

Example: If a lender requires a 10% return on a $100loan,theborrowermustinvestittoearnatleast$10 to satisfy the lender.20:50

Collateral, like the Pink Panther diamond, makes a loan risk-free. If I lend you $50,000 and you provide this diamond as collateral, I am guaranteed to get my capital and interest back.

The real interest rate reflects how much better off you want to be over time, accounting for inflation. If a loaf of bread costs $1todayand$1.10 next year, the inflation rate is 10%.

To calculate the nominal interest rate, add the real interest rate to the inflation rate. For example, if the real rate is 2% and inflation is 10%, the nominal rate is 2\text{%} + 10\text{%} = 12\text{%}.

Additional premiums are added to the nominal interest rate. A default risk premium accounts for the borrower's creditworthiness, while a liquidity premium reflects the opportunity cost of lending for different time periods.

• ★Collateral makes a loan risk-free.25:47
• ★Real interest rate measures improvement in quality of life over time.25:55
• ★Nominal interest rate = real rate + inflation rate.26:15
• ★Default risk premium is based on borrower creditworthiness.26:30
• ★Liquidity premium reflects opportunity cost of lending.26:45

Example: If I charge a 2% real rate and 10% inflation, the total interest charged is 12%.26:20

When lending money, you give up access to that capital, leading to a liquidity premium. For example, you might charge 1% for a one-month loan and 4% for a one-year loan. This illustrates how interest rates can vary significantly based on the duration of the loan.

The maturity risk premium reflects the additional risk associated with longer loan durations. If you need to sell your loan in the secondary market, a one-month loan will typically sell for a higher price than a one-year loan due to this risk.

The overall interest rate can be calculated using several components: the real risk-free rate, expected inflation premium, default risk premium, liquidity premium, and maturity premium. The formula is $r=\text{Real risk free interest rate}+\text{Inflation premium}+\text{Default risk premium}+\text{Liquidity premium}+\text{Maturity premium}$.

The effective annual interest rate (EAR) accounts for different compounding frequencies. The formula is $EAR=(1+\text{Periodic rate}{)}^m-1=(1+\frac{\text{Stated annual rate}}{m}{)}^m-1$.

• ★Liquidity premium is charged for giving up access to capital.30:48
• ★Interest rates vary by loan duration, e.g., 1% for one month and 4% for one year.30:55
• ★Maturity risk premium reflects the risk of longer loan durations.31:02
• ★The overall interest rate includes multiple premiums: $r=\text{Real risk free interest rate}+\text{Inflation premium}+\text{Default risk premium}+\text{Liquidity premium}+\text{Maturity premium}$.31:10
• ★Effective annual interest rate (EAR) is calculated from stated annual rate and compounding frequency.31:25

Example: For a stated interest rate of 10% compounded quarterly, the effective annual rate is $EAR={\left(1+\frac{0.10}{4}\right)}^4-1=0.10381$ or 10.38%.30:55

The effective annual rate (EAR) increases with more frequent compounding. For example, compounding quarterly gives an EAR of 10.38%, while compounding monthly results in an EAR of 10.47%. This illustrates that the more frequent the compounding, the higher the effective annual rate.

The formula for future value is given by $F{V}_N=PV\times (1+r{)}^N$, where $r$ is the interest rate in decimal form and $N$ is the number of compounding periods. This formula shows how an initial amount grows over time due to compounding.

To find present value, rearrange the future value formula: $PV=\frac{F{V}_N}{(1+r{)}^N}$. This allows us to determine how much needs to be saved today to achieve a desired future amount.

When dealing with different compounding periods, adjustments are necessary. The formula becomes $PV=\frac{F{V}_n}{(1+\frac{r}{m}{)}^{m\cdot N}}$, where $m$ is the number of compounding periods per year. This accounts for the frequency of compounding.

• ★The effective annual rate (EAR) increases with more frequent compounding.35:45
• ★Future Value formula: $F{V}_N=PV\times (1+r{)}^N$.35:46
• ★Present Value formula: $PV=\frac{F{V}_N}{(1+r{)}^N}$.35:47
• ★Adjust for different compounding periods with $PV=\frac{F{V}_n}{(1+\frac{r}{m}{)}^{m\cdot N}}$.35:48
• ★Compounding allows initial savings to grow over time.35:49

Example: To have $10,000 at the end of year three with a 10% annual return compounded quarterly, calculate the present value using the adjusted formula.35:49

When calculating present value with quarterly compounding, adjust the annual interest rate by dividing by four. For example, an annual rate of 10\text{%} becomes 2.5\text{%} per quarter.

The number of periods for three years of quarterly compounding is $12$ quarters. Thus, set $n=12$ and i/y = 2.5\text{%} in your financial calculator to compute the present value accurately.

• ★Adjust the denominator for quarterly calculations by dividing the annual rate by $4$.40:42
• ★Set $n=12$ and i/y = 2.5\text{%} for quarterly compounding over three years.40:42
• ★Changing the payments per year (p/y) setting can lead to errors if not reverted.40:42
• ★Present value can be calculated using both the time value factor formula and a financial calculator with adjustments.40:42

Example: Using a financial calculator with $n=12$ and i/y = 2.5\text{%} yields a present value of $7,435$.40:42

The future value (FV) of a loan can be calculated using the formula $FV=PV\times (1+i{)}^n$. For a $50,000loanover12yearsat1$209,530$.

An annuity is a series of consecutive equal cash flows over a fixed period. An ordinary annuity consists of equal payments starting one period from today, while an annuity due consists of equal payments starting today.

Mortgage payments are an example of an ordinary annuity, where payments are made monthly. Insurance premiums are an example of an annuity due, where the first payment is made at the beginning of the period.

To adjust the present value and future value of an annuity due, multiply by $(1+r)$. The calculation for an annuity due is more complex than for an ordinary annuity.

• ★The future value formula is $FV=PV\times (1+i{)}^n$.43:27
• ★An ordinary annuity starts payments one period from today.43:35
• ★An annuity due starts payments today.43:42
• ★Mortgage payments are ordinary annuities; insurance premiums are annuity dues.43:50
• ★Multiply by $(1+r)$ to adjust for annuity due calculations.44:00

Example: For an annuity of $200 per year over 15 years, calculate the future value using the annuity formulas.44:40

To determine the premium for an annuity, identify if it is an ordinary annuity or an annuity due. An ordinary annuity has payments that begin one year from today, while an annuity due has payments that begin today.

Using the time value of money formula, the calculated premium for the annuity is $1430$. To calculate using a BA II Plus calculator, switch from end mode to beginning mode using the second function.

After solving an annuity due problem, remember to switch the calculator back to end mode. This prevents errors in subsequent calculations.

Constructing a timeline helps visualize cash flows. For example, cash flows of $100$, $150$, $250$, and $300$ over five periods with a 5\text{%} interest rate can be represented on a timeline.

• ★An ordinary annuity has payments starting one year from today.48:25
• ★An annuity due has payments starting today.48:25
• ★The premium for the annuity calculated is $1430$.48:25
• ★Switch calculator to beginning mode for annuity due calculations.48:25
• ★Always switch back to end mode after solving an annuity due problem.48:25
• ★Timelines help visualize cash flows, especially for unequal amounts.48:25

Example: Cash flows of $100$, $150$, $250$, and $300$ over five periods at 5\text{%} interest rate can be visualized on a timeline.48:25

Know the components of an interest rate. Key concepts include liquidity, maturity, and opportunity cost. These will be revisited throughout the CFA Level I readings.

Be familiar with the steps leading to calculator outputs. This knowledge is essential for exam success.

• ★Liquidity is a key concept in interest rates.53:25
• ★Maturity affects the interest rate.53:26
• ★Opportunity cost is a fundamental concept in finance.53:27
• ★Familiarity with calculator outputs is necessary.53:28