Chapter 3 - Financial Decision and Law of One Price
Present Value: To move a Cashflow back in time.
𝐶
𝑃𝑟𝑒𝑠𝑒𝑛𝑡 𝑉𝑎𝑙𝑢𝑒 (𝑃𝑉) =
(1 + 𝑟𝑓 )𝑛
Future Value: To move a Cashflow forward in time.
𝐹𝑢𝑡𝑢𝑟𝑒 𝑉𝑎𝑙𝑢𝑒 (𝐹𝑉) = 𝐶0 ∗ (1 + 𝑟𝑓 )𝑛
NPV: When making an investment decision, take the alternative with the highest NPV. Choosing this
alternative is equivalent to receiving its NPV in cash today.
𝑁𝑒𝑡 𝑃𝑟𝑒𝑠𝑒𝑛𝑡 𝑉𝑎𝑙𝑢𝑒 = 𝑃𝑉 (𝐵𝑒𝑛𝑒𝑓𝑖𝑡𝑠) − 𝑃𝑉(𝐶𝑜𝑠𝑡𝑠)
Arbitrage: An opportunity to make a profit without taking any risk or making any investment.
Law of One Price: The price of an identical asset will have the same price globally, regardless of
location, when certain factors are considered.
The law of one price takes into account a frictionless market, where there are no:
- Transaction costs.
- Transportation costs.
- Legal restrictions.
- Differences in currency exchange rates are.
- Price manipulation by buyers or sellers.
Chapter 4 - The Time Value of Money
Present Value of a Cashflow stream:
𝐶1 𝐶2 𝐶𝑇
𝑃𝑉 = 1 + 2 +. . . . +
(1 + 𝑟𝑓 ) (1 + 𝑟𝑓 ) (1 + 𝑟𝑓 )𝑇
Perpetuity: A stream of equal Cashflows (C) that occurs at regular intervals and last forever.
𝐶
𝑃𝑉 (𝐶 𝑖𝑛 𝑝𝑒𝑟𝑝𝑒𝑡𝑢𝑖𝑡𝑦) =
𝑟
Note that: The perpetuity formula assumes that the first payment occurs at the end of the first
period (at date 1).
,Example:
You want to endow an annual party at work. You budget $30.000 per year (forever) for the party. If
your work earns 8% per year on the investment of the first payment, and if the first party is in one
year’s time, how much will you need to donate to endow the party?
Solution:
$30.000
𝑃𝑉 = = $375.000 𝑡𝑜𝑑𝑎𝑦
8%
This means that if you donate $375.000 today, and if your work invests it at 8% per year forever,
then they will have $30.000 every year for the party.
Annuity: Is a stream of N equal cash flows paid at regular intervals. An annuity ends after some
fixed number of payments. (e.g., car loans, mortgages, bonds)
𝐶 1
𝑃𝑉 (𝐶 𝑖𝑛 𝑎𝑛𝑛𝑢𝑖𝑡𝑦 𝑤𝑖𝑡ℎ 𝑁 𝑝𝑒𝑟𝑖𝑜𝑑𝑠) = ∗ (1 − )
𝑟 (1 + 𝑟)𝑁
Growing perpetuity: A stream of cash flows that occur at regular intervals and grow at a constant
rate forever.
𝐶
𝑃𝑉 (𝑔𝑟𝑜𝑤𝑖𝑛𝑔 𝑝𝑒𝑟𝑝𝑒𝑡𝑢𝑖𝑡𝑦) =
𝑟−𝑔
Growing annuity: A stream of N growing cash flows, paid at regular intervals.
𝐶 1+𝑔 𝑁
𝑃𝑉 (𝑔𝑟𝑜𝑤𝑖𝑛𝑔 𝑎𝑛𝑛𝑢𝑖𝑡𝑦) = ∗ (1 − ( ) )
𝑟−𝑔 1+𝑟
Future value for an annuity/perpetuity, growing annuity/perpetuity is PV * (1 + 𝑟)𝑁
Example: Mortgage
Suppose you are paying a small apartment with a 30-year mortgage loan of $100.000. Payments are
due monthly at a monthly rate of 0,5%
1. What is the value of the monthly mortgage payment?
2. What is the loan balance at the end of the first year?
- what amount would you need to pay back the bank in full at the end of the first year to pay off
the mortgage?
3. Split the total payment in the first year into interest payments and amortization.
, 1. As mortgages are an annuity, we use the formula to find C.
𝐶 1
$100.000 = ∗ (1 − ) → 𝐶 = $599,6
0.005 (1 + 0.005)360
2. Loan balance at the end of the first year is the PV of all future
payments.
599,6 1
𝐿𝑜𝑎𝑛 𝐵𝑎𝑙𝑎𝑛𝑐𝑒 = ∗ (1 − ) = $98.772
0.005 (1 + 0.005)348
3.
Total payment in the first year = $599,6 * 12 = $7.194,6
Amortization = $100.000 − $98.772 = $1.228
Interest payments = $7.194,6 −$1.228 = $5.966,6
Chapter 5 - Interest Rates
Effective annual rate (EAR): is the actual amount of interest that will be earned at the end of one
year.
Annual Percentage Rate (APR): is the amount of simple interest earned in one year, without the
effect of compounding.
Example:
Suppose that JP Morgan advertises savings accounts with an interest rate of “6% APR with monthly
compounding.” In this case, you will earn 6%/12 = 0,5% every month. So, an APR with monthly
compounding is actually a monthly interest rate, rather than an annual interest rate.
Because this interest compounds every month, you will earn:
$1 ∗ (1 + 0,005)12 = $1,061678
Thus, the effective annual rate (EAR) is 6,1678%.
Converting APR to an EAR:
With “k” compounding
𝐴𝑃𝑅 𝑁 periods per year
1 + 𝐸𝐴𝑅 = (1 + )
𝑘
Annual = 1
Semiannual = 2
Monthly = 12
Daily = 365
Present Value: To move a Cashflow back in time.
𝐶
𝑃𝑟𝑒𝑠𝑒𝑛𝑡 𝑉𝑎𝑙𝑢𝑒 (𝑃𝑉) =
(1 + 𝑟𝑓 )𝑛
Future Value: To move a Cashflow forward in time.
𝐹𝑢𝑡𝑢𝑟𝑒 𝑉𝑎𝑙𝑢𝑒 (𝐹𝑉) = 𝐶0 ∗ (1 + 𝑟𝑓 )𝑛
NPV: When making an investment decision, take the alternative with the highest NPV. Choosing this
alternative is equivalent to receiving its NPV in cash today.
𝑁𝑒𝑡 𝑃𝑟𝑒𝑠𝑒𝑛𝑡 𝑉𝑎𝑙𝑢𝑒 = 𝑃𝑉 (𝐵𝑒𝑛𝑒𝑓𝑖𝑡𝑠) − 𝑃𝑉(𝐶𝑜𝑠𝑡𝑠)
Arbitrage: An opportunity to make a profit without taking any risk or making any investment.
Law of One Price: The price of an identical asset will have the same price globally, regardless of
location, when certain factors are considered.
The law of one price takes into account a frictionless market, where there are no:
- Transaction costs.
- Transportation costs.
- Legal restrictions.
- Differences in currency exchange rates are.
- Price manipulation by buyers or sellers.
Chapter 4 - The Time Value of Money
Present Value of a Cashflow stream:
𝐶1 𝐶2 𝐶𝑇
𝑃𝑉 = 1 + 2 +. . . . +
(1 + 𝑟𝑓 ) (1 + 𝑟𝑓 ) (1 + 𝑟𝑓 )𝑇
Perpetuity: A stream of equal Cashflows (C) that occurs at regular intervals and last forever.
𝐶
𝑃𝑉 (𝐶 𝑖𝑛 𝑝𝑒𝑟𝑝𝑒𝑡𝑢𝑖𝑡𝑦) =
𝑟
Note that: The perpetuity formula assumes that the first payment occurs at the end of the first
period (at date 1).
,Example:
You want to endow an annual party at work. You budget $30.000 per year (forever) for the party. If
your work earns 8% per year on the investment of the first payment, and if the first party is in one
year’s time, how much will you need to donate to endow the party?
Solution:
$30.000
𝑃𝑉 = = $375.000 𝑡𝑜𝑑𝑎𝑦
8%
This means that if you donate $375.000 today, and if your work invests it at 8% per year forever,
then they will have $30.000 every year for the party.
Annuity: Is a stream of N equal cash flows paid at regular intervals. An annuity ends after some
fixed number of payments. (e.g., car loans, mortgages, bonds)
𝐶 1
𝑃𝑉 (𝐶 𝑖𝑛 𝑎𝑛𝑛𝑢𝑖𝑡𝑦 𝑤𝑖𝑡ℎ 𝑁 𝑝𝑒𝑟𝑖𝑜𝑑𝑠) = ∗ (1 − )
𝑟 (1 + 𝑟)𝑁
Growing perpetuity: A stream of cash flows that occur at regular intervals and grow at a constant
rate forever.
𝐶
𝑃𝑉 (𝑔𝑟𝑜𝑤𝑖𝑛𝑔 𝑝𝑒𝑟𝑝𝑒𝑡𝑢𝑖𝑡𝑦) =
𝑟−𝑔
Growing annuity: A stream of N growing cash flows, paid at regular intervals.
𝐶 1+𝑔 𝑁
𝑃𝑉 (𝑔𝑟𝑜𝑤𝑖𝑛𝑔 𝑎𝑛𝑛𝑢𝑖𝑡𝑦) = ∗ (1 − ( ) )
𝑟−𝑔 1+𝑟
Future value for an annuity/perpetuity, growing annuity/perpetuity is PV * (1 + 𝑟)𝑁
Example: Mortgage
Suppose you are paying a small apartment with a 30-year mortgage loan of $100.000. Payments are
due monthly at a monthly rate of 0,5%
1. What is the value of the monthly mortgage payment?
2. What is the loan balance at the end of the first year?
- what amount would you need to pay back the bank in full at the end of the first year to pay off
the mortgage?
3. Split the total payment in the first year into interest payments and amortization.
, 1. As mortgages are an annuity, we use the formula to find C.
𝐶 1
$100.000 = ∗ (1 − ) → 𝐶 = $599,6
0.005 (1 + 0.005)360
2. Loan balance at the end of the first year is the PV of all future
payments.
599,6 1
𝐿𝑜𝑎𝑛 𝐵𝑎𝑙𝑎𝑛𝑐𝑒 = ∗ (1 − ) = $98.772
0.005 (1 + 0.005)348
3.
Total payment in the first year = $599,6 * 12 = $7.194,6
Amortization = $100.000 − $98.772 = $1.228
Interest payments = $7.194,6 −$1.228 = $5.966,6
Chapter 5 - Interest Rates
Effective annual rate (EAR): is the actual amount of interest that will be earned at the end of one
year.
Annual Percentage Rate (APR): is the amount of simple interest earned in one year, without the
effect of compounding.
Example:
Suppose that JP Morgan advertises savings accounts with an interest rate of “6% APR with monthly
compounding.” In this case, you will earn 6%/12 = 0,5% every month. So, an APR with monthly
compounding is actually a monthly interest rate, rather than an annual interest rate.
Because this interest compounds every month, you will earn:
$1 ∗ (1 + 0,005)12 = $1,061678
Thus, the effective annual rate (EAR) is 6,1678%.
Converting APR to an EAR:
With “k” compounding
𝐴𝑃𝑅 𝑁 periods per year
1 + 𝐸𝐴𝑅 = (1 + )
𝑘
Annual = 1
Semiannual = 2
Monthly = 12
Daily = 365
