Present Perfect
This financial concept is the reverse of compounding; it involves taking a future amount and shrinking it down to what it is worth today.
Discounting (or Present Value)
Explanation: Discounting removes the interest that would have been earned over time, bringing a future sum back to its value in current dollars.
Future Value relies on this powerful concept, often described as "earning interest on your interest".
Compounding.
Compounding occurs when interest generated adds to the principal, forming a larger base for the next interest calculation.
This type of annuity pays cash flows at the end of each period, which is the standard default for most financial calculations.
Ordinary Annuity
An Ordinary Annuity is defined by cash flows that occur at the end of each period, distinguishing it from an Annuity Due where payments are made at the beginning.
This rate, known as the APR, is the simple nominal rate that ignores compounding within the year.
Annual Percentage Rate.
The Annual Percentage Rate (APR) is the nominal interest rate stated by lenders, which does not account for the effects of compounding that occurs more frequently than once a year.
You want to buy a vintage Mustang 5 years from now. You estimate it will cost $80,000 then. If you can earn 8% on your money, this is how much cash you need to set aside today to afford it.
PV of a Lump Sum (Scenario)
$54,447
To determine the amount needed today, discount the future cost of $80,000 by dividing it by (1.08) ^5 (1.4693), which yields a present requirement of $54,447.
You have a future payment of $1,210 due in exactly 2 years. If the relevant discount rate is 10%, this is the Present Value of that liability today.
Present Value of a Single Lump Sum
$1,000
Formula: PV = FV/(1+r) ^n
1210 / 1.21 = 1,000.
You invest $2,500 today in a bond fund returning 4% annually. In 5 years, your account balance will grow to this amount.
Future Value of a Lump Sum
$3,041.63
You pay rent of $1,000 at the beginning of every year. If you sign a 3-year lease and the discount rate is 10%, this is the Present Value of your total commitment today.
Present Value of Annuity Due
$2,735.54
First calculate the ordinary annuity PV of $1,000 for 3 years ($2,486.85), then multiply by (1+r) or 1.10 to adjust for payments made at the beginning of the year, resulting in $2,735.54
An investment offers 12% APR, but it compounds quarterly. This is the Effective Annual Rate (EAR) you actually earn.
- Effective Annual Rate (EAR)
12.55%
Divide the 12% annual rate by 4 to get a 3% quarterly rate, then raise 1.03 to the 4th power and subtract 1 to find the Effective Annual Rate of 12.55%.
A philanthropist wants to create a prize that awards $20,000 every year forever. If the trust fund earns 4% interest, this is the check they must write today to fund it.
Perpetuity (Scenario)
$500,000
Since the scholarship pays forever, use the perpetuity formula by dividing the annual payment of $20,000 by the interest rate of 0.04 to find the required endowment of $500,000.
You are promised a bonus of $5,000 exactly 4 years from today. If the interest rate is 6%, this is what that bonus is worth right now.
Present Value of a Single Lump Sum
$3,960.47
Formula: PV = FV/(1+r) ^n
FV = 5000, r = 0.06, n = 4.
5000 / 1.2625 = 3,960.47.
You deposit $100 at the end of Year 1 and $200 at the end of Year 2. If the interest rate is 10%, this is the total value of your account at the end of Year 3.
Future Value of Unequal Cash Flows
$341
To find the total future value, compound each deposit forward to the end of Year 3: the $100 from Year 1 grows for 2 years to become $121 ($100 X 1.21), and the $200 from Year 2 grows for 1 year to become $220 ($200 X 1.10), resulting in a combined total of $341.
You want to save $20,000 for a car in 5 years. If you can earn 6% annually, this is the amount you must save at the end of each year to reach your goal.
Solving for Payment (Annuity)
$3,547.93
Determine the future value annuity factor for 5 years at 6% (5.6371) and divide the $20,000 goal by this factor to determine that an annual savings of $3,547.93 is required.
In an amortized loan, like a mortgage, the early payments are primarily composed of this, while later payments are primarily principal.
Interest
Formula: Interest = Balance X Rate
In an amortized loan, interest is calculated on the remaining balance; since the balance is highest at the start of the loan, the early payments consist primarily of interest rather than principal.
A project costs $20,000 today. It returns $10,000 (Yr 1), $8,000 (Yr 2), and $5,000 (Yr 3). If your required rate is 10%, you should make this investment decision (Invest or Do Not Invest).
Net Present Value (NPV)
Do Not Invest
Calculate the total PV of inflows ($9,091 + $6,611 + $3,756 = $19,458) and compare it to the $20,000 cost; since the cost exceeds the value by over $500, you should not invest.
You receive nothing in Year 1, $2,000 in Year 2, and $3,000 in Year 3. If the discount rate is 8%, this is the Present Value today (t=0).
PV of Unequal Cash Flows
$4,096.18
Topic: PV of Unequal Cash Flows
Explanation: Discount each cash flow back to today individually.
Year 1: 0
Year 2: 2000 / (1.08) ^2 = 2000 / 1.1664 = 1,714.68
Year 3: 3000 / (1.08) ^3 = 3000 / 1.2597 = 2,381.50
Total: 1,714.68 + 2,381.50 = $4,096.18.
You want to buy a boat that costs $10,000 in 5 years. You have $6,000 today. To reach your goal without adding any more money, you must earn this annual rate of return.
Solving for Rate (r)
10.76%
To find the required annual rate, divide the future goal of $10,000 by the current $6,000, raise the result (1.6667) to the power of 1/5 (representing the 5-year period), and subtract 1, which reveals a return of 10.76%.
You have a loan of $10,000 at 5% interest to be repaid over 3 years. This is the annual payment amount (PMT).
Loan Amortization (Solving for PMT)
$3,672.08
Using the amortization formula, multiply the principal ($10,000) by the rate (0.05) and divide by [1 - (1.05) ^3 (0.1362) to calculate the annual fixed payment of $3,672.08.
You want to triple your investment. If you earn 9% annually, it will take this many years.
Solving for Time (n)
12.75 Years
Ratio FV/PV = 3 (Tripling).
Numerator: ln3 = 1.0986.
Denominator: ln(1.09) = 0.0861$.
Solve: $1.0986 / 0.0861 = 12.75$ years.
You save $3,000 at the beginning of every year for a big trip 4 years from now. The account earns 5%. This is the total amount you will have for your vacation.
FV of Annuity Due (Scenario)
$13,576.50
Calculate the future value of an ordinary annuity ($12,930.30) and multiply by 1.05 to adjust for the payments being made at the start of the year, resulting in a total of $13,576.50.
An investment pays $500 at the end of Year 1 and $1,000 at the end of Year 4 (nothing in years 2 and 3). At a 10% rate, this is the total Present Value.
PV of Unequal Cash Flows (with Gaps)
$1,137.56
Formula: PV = CF /(1+r) ^1
PV of Year 1: $500 / (1.10) ^1 = 454.55
PV of Year 4: $1000 / (1.10) ^4 = 1000 / 1.4641 = 683.01
Total: 454.55 + 683.01 = 1,137.56.
You deposit $1,000 today (t=0) and another $1,000 at the end of Year 2. If the account earns 5%, this is your total balance at the end of Year 3.
FV of Unequal Cash Flows
$2,207.63
Explanation: $1,000 grows for 3 years: $1000 X (1.05) ^3 = 1,157.63. (2) $1,000 (deposited end of Year 2) grows for 1 year (Year 2 to Year 3): $1000 X (1.05) ^1 = 1,050.00.
Total: $1,157.63 + 1,050.00 = $2,207.63.
You take out a standalone business loan of $25,000 at an 8% annual interest rate. You are required to make a fixed annual payment of $3,000. This is the exact amount of Principal you pay off in the first year.
Amortization Schedule (Principal Component)
$1,000
Formula: Principal Paid = Total Payment - (Balance X Rate)
In an amortized loan, every payment is split between interest and principal. First, calculate the interest due on the full balance by multiplying $25,000 by 8%, which equals $2,000. Then, subtract this interest cost from the total payment of $3,000 to find that exactly $1,000 is applied toward reducing the principal balance.
Bank A offers 6.0% compounded monthly. Bank B offers 6.1% compounded semi-annually. This is the difference in EAR between the two banks (in percentage points).
EAR Comparison
0.03% (Bank B is higher by 0.03%)
Explanation: Bank A's 6% monthly compounds to an EAR of 6.167%, while Bank B's 6.1% semi-annual rate compounds to an EAR of 6.193%; the difference between these effective rates is approximately 0.03%.
You save $5,000 at the end of every year for 10 years at 8%. You then stop contributing but leave the money in the account to grow for another 10 years at 8%. This is your final balance at Year 20.
Multi-Stage Compounding (Annuity + Lump Sum)
$156,377
First, the annuity grows to $72,432 over the contribution period; then, this lump sum compounds for another 10 years at 8% (factor 2.1589) to reach a final balance of $156,377.