Real vs. Nominal & Net Present Value (Topic 4) Objectives: Real vs. Nominal Interest Rates Define Net Present Value (NPV) Text: Section 5.5

Objectives:

Real vs. Nominal Interest Rates

Define Net Present Value (NPV)

Text:

Section 5.5

REAL VS. NOMINAL

Real vs. Nominal Cash Flows Because of inflation, a dollar tomorrow typically does not buy as much as a dollar today If the inflation rate is 5%, a bundle of goods and services that costs $100 this year will cost $105 next year The Consumer Price Index (CPI) tracks the cost of goods and services through time Year 1947 1970 1980 1990 1995 2000 2001 CPI $100 $170 $370 $572 $656 $744 $756 Therefore, a bundle of goods and services that cost $1.00 in 1970 cost approximately $4.45 in 2001; conversely, $1 in 2001 was worth approximately $0.22 in 1970

Because of inflation, a dollar tomorrow typically does not buy as much as a dollar today

If the inflation rate is 5%, a bundle of goods and services that costs $100 this year will cost $105 next year

The Consumer Price Index (CPI) tracks the cost of goods and services through time

Example: Cost of Correspondence In 1970, airmail cost $0.50. In 2001, the price of airmail had risen to $1. In nominal terms, the price rose by 100%. But a dollar in 1970 is worth more than a dollar in 2001. What was the change in the real cost of communicating via airmail?

In 1970, airmail cost $0.50. In 2001, the price of airmail had risen to $1. In nominal terms, the price rose by 100%. But a dollar in 1970 is worth more than a dollar in 2001. What was the change in the real cost of communicating via airmail?

Measuring Inflation What was the annual inflation rate between 1970 and 2001? What was the annual inflation rate between 2000 and 2001?

Real vs. Nominal Interest Rates Definitions: Nominal Interest Rate (r): borrowing cost in nominal $ due Inflation Rate (i): rate at which prices increase in economy Real Interest Rate (r * ): growth in purchasing power of your cash where

Definitions:

Nominal Interest Rate (r): borrowing cost in nominal $ due

Inflation Rate (i): rate at which prices increase in economy

Real Interest Rate (r * ): growth in purchasing power of your cash

where

Determining r, r * , and i You currently have $100, enough to purchase 100 oranges for $1 each. You are willing to lend the $100 to Joe provided he repays you enough to purchase 105 oranges next year. The price of oranges is expected to increase to $1.10 by the end of the year. How much must Joe repay you at the end of the year? Number of Oranges: 100 105 Price of Oranges $1.00 $1.10 Loan Amount $100 Repayment Amount $115.50 Therefore:

You currently have $100, enough to purchase 100 oranges for $1 each. You are willing to lend the $100 to Joe provided he repays you enough to purchase 105 oranges next year. The price of oranges is expected to increase to $1.10 by the end of the year. How much must Joe repay you at the end of the year?

Number of Oranges: 100 105

Price of Oranges $1.00 $1.10

Loan Amount $100

Repayment Amount $115.50

Therefore:

Present Value Summary Let’s calculate the present value of FV n using different interest rates and discuss the meaning of each present value; assume r, i, and r* measured as effective annual rates (so that we can assume annual compounding) number of dollars you need today to buy the same bundle of stuff as FV in year n; when inflation rate is zero, the present value simply equals the future value number of dollars you need to invest today at r annual to have a nominal balance of FV in year n; if I ask you to calculate a PV and do not talk about inflation or specify which rate to use, this is the default number of dollars you need to invest today to be able to buy the same bundle of stuff in year n that you can buy today with FV; if cash flows measured in real dollars, you should use r* when calculating the PV

Let’s calculate the present value of FV n using different interest rates and discuss the meaning of each present value; assume r, i, and r* measured as effective annual rates (so that we can assume annual compounding)

Bolivia Revisited Recall that you moved to Bolivia with $1 million, determined to spend the next 40 years living large off your reward from Dr. Evil. Further recall that r = 6% in Bolivia and that you want to have $0 left in 40 years. Last lecture, we ignored inflation and calculated that you could live off of $66,461.52 each year.

Recall that you moved to Bolivia with $1 million, determined to spend the next 40 years living large off your reward from Dr. Evil. Further recall that r = 6% in Bolivia and that you want to have $0 left in 40 years.

Last lecture, we ignored inflation and calculated that you could live off of $66,461.52 each year.

Bolivia is Bueno! Named after independence fighter Simon Bolivar, Bolivia broke away from Spanish rule in 1825 Population: 8,586,443 (July 2003 est.) Labor force: 2.5 million Languages: Spanish (official), Quechua (official), Aymara (official) Capital: La Paz (seat of government); Sucre (legal capital and seat of judiciary) Natural resources: tin, natural gas, petroleum, zinc, tungsten, antimony, silver, iron, lead, gold, timber, hydropower Inflation rate (consumer prices): 2% (2001 est.) Geography: landlocked ; shares control of Lago Titicaca, world's highest navigable lake (elevation 3,805 m), with Peru Source: CIA's The World Factbook

Named after independence fighter Simon Bolivar, Bolivia broke away from Spanish rule in 1825

Population: 8,586,443 (July 2003 est.)

Labor force: 2.5 million

Languages: Spanish (official), Quechua (official), Aymara (official)

Capital: La Paz (seat of government); Sucre (legal capital and seat of judiciary)

Natural resources: tin, natural gas, petroleum, zinc, tungsten, antimony, silver, iron, lead, gold, timber, hydropower

Inflation rate (consumer prices): 2% (2001 est.)

Geography: landlocked ; shares control of Lago Titicaca, world's highest navigable lake (elevation 3,805 m), with Peru

Source: CIA's The World Factbook

Bolivia Revisited How much can you live on if you want the real value of your consumption to be constant? First, we need to solve for the real interest rate: Now what? (Hint: The real interest rate is the rate at which your purchasing power increases over time.)

How much can you live on if you want the real value of your consumption to be constant?

Bolivia Revisited (cont.) Second, we use our present value of an annuity formula again. Since $1,000,000 is measured in 2004 dollar, it is the real value of money in 2004. And 3.92% is the rate at which our purchasing power grows. So we use 3.92% as our discount rate.

Second, we use our present value of an annuity formula again. Since $1,000,000 is measured in 2004 dollar, it is the real value of money in 2004. And 3.92% is the rate at which our purchasing power grows. So we use 3.92% as our discount rate.

Bolivia Revisited (cont.) Following table shows nominal spending in column (1) and real spending in column (2). Both streams of payments have a PV of $1,000,000.

Following table shows nominal spending in column (1) and real spending in column (2). Both streams of payments have a PV of $1,000,000.

Life in Bolivia Revisited (cont.) Finally, assume that you want to throw a big party at the end of your stay in Bolivia. How much do you need to invest today to have the equivalent of $100,000 in 2004 dollars left over in 40 years? Two approaches…

Finally, assume that you want to throw a big party at the end of your stay in Bolivia. How much do you need to invest today to have the equivalent of $100,000 in 2004 dollars left over in 40 years? Two approaches…

Cryonics Savings Accounts Wall Street Journal reported in Jan ‘06 on a group of people who want to be frozen at the time of death and unfrozen in the distant future when medical science is able to heal them. The financial angle is that these people want to leave money to themselves when they die. You’ve been asked to help one of these people set up a cyronics savings account (CSA). Your client is going to save $24,000 per year for the next 20 years and then die. He expects to be unfrozen 120 years from today. Assume the nominal interest rate is a constant 6.0% and the inflation rate is a constant 2.0% (both with annual compounding). What is the present value of his savings?

Wall Street Journal reported in Jan ‘06 on a group of people who want to be frozen at the time of death and unfrozen in the distant future when medical science is able to heal them. The financial angle is that these people want to leave money to themselves when they die.

You’ve been asked to help one of these people set up a cyronics savings account (CSA). Your client is going to save $24,000 per year for the next 20 years and then die. He expects to be unfrozen 120 years from today. Assume the nominal interest rate is a constant 6.0% and the inflation rate is a constant 2.0% (both with annual compounding).

What is the present value of his savings?

Cryonics Savings Accounts (cont.) What is the nominal value of his savings at the end of 20 years? What is the nominal value of his savings at the end of 120 years?

What is the nominal value of his savings at the end of 20 years?

What is the nominal value of his savings at the end of 120 years?

Cryonics Savings Accounts (cont.) What is the real value of his savings at the end of 120 years (measured in today’s dollars)? What is the real interest rate?

What is the real value of his savings at the end of 120 years (measured in today’s dollars)?

What is the real interest rate?

Cryonics Savings Accounts (cont.) Once unfrozen, your client plans to purchase a 40-year annuity that allows him to buy the same bundle of goods and services each year, beginning immediately. What is real value of these payments (measured in today’s dollars)?

Once unfrozen, your client plans to purchase a 40-year annuity that allows him to buy the same bundle of goods and services each year, beginning immediately. What is real value of these payments (measured in today’s dollars)?

(Brief) Introduction to NET PRESENT VALUE

Net Present Value Net present value: present value of all expected cash flows associated with a project; typically, the initial cash flow is negative (reflecting the current cost of the project) and subsequent cash flows are positive (reflecting the expected future benefits of the project) where CF n is the expected cash flow in year n and discount rate r is chosen to reflect the systematic risk of project’s cash flows NPV > 0  stream of cash flows are a “good deal” in that the PV of benefits exceeds the PV of costs; “you’re doing better than just investing at r” NPV < 0  stream of cash flows are a “bad deal” We have more to say about NPV (Topic 5) and discount rates (Topic 9)

Net present value: present value of all expected cash flows associated with a project; typically, the initial cash flow is negative (reflecting the current cost of the project) and subsequent cash flows are positive (reflecting the expected future benefits of the project)

where CF n is the expected cash flow in year n and discount rate r is chosen to reflect the systematic risk of project’s cash flows

NPV > 0  stream of cash flows are a “good deal” in that the PV of benefits exceeds the PV of costs; “you’re doing better than just investing at r”

NPV < 0  stream of cash flows are a “bad deal”

We have more to say about NPV (Topic 5) and discount rates (Topic 9)

First NPV Example Watching Brewster’s Millions gave you a brilliant idea. You want to pilot an iceberg to Africa and sell lots of ice cold, very refreshing aqua! $10 million (paid today) buys you a huge iceberg with a huge propeller and another $1 million (paid next year) covers gasoline and salaries You expect $9 million in water sales next year and $3 million in water sales the following year What is the NPV of this project if r annual = 15%?

Watching Brewster’s Millions gave you a brilliant idea. You want to pilot an iceberg to Africa and sell lots of ice cold, very refreshing aqua!

$10 million (paid today) buys you a huge iceberg with a huge propeller and another $1 million (paid next year) covers gasoline and salaries

You expect $9 million in water sales next year and $3 million in water sales the following year

What is the NPV of this project if r annual = 15%?

Second NPV Example Your firm, Mad Cheddar, plans to host a celebrity golf and cheese tasting event one year from today. The event will cost you $60,000 today and $100,000 next year. If it does not rain, your firm will earn $500,000. If it does rain, your firm will earn $100,000. Assume the probability of rain is 80 percent and the discount rate is 6%. What is the NPV?

Your firm, Mad Cheddar, plans to host a celebrity golf and cheese tasting event one year from today. The event will cost you $60,000 today and $100,000 next year. If it does not rain, your firm will earn $500,000. If it does rain, your firm will earn $100,000. Assume the probability of rain is 80 percent and the discount rate is 6%. What is the NPV?