Valuation of Cash Flows Chapter 3.1-3.3 Chapter 4

outline Market prices and valuation of financial assets The Time Value of Money Present value and future value Valuation of streams of cash flows Valuation of Annuities and Perpetuities Practice questions

Market prices and valuation of financial assets

Measuring value of financial assets The value of a financial asset (can be a traded asset like a stock, a project or a decision) depends on the costs and benefits involved. Advertising campaign Lowering production cost Management training program What information is required in order to measure or value the associated costs and benefits?

The Important Role of Market Prices Suppose you could trade 10 ounces of platinum and receive 20 ounces of gold. Would you trade? The price of one ounce of gold is $250 The price of one ounce of platinum is $550

Market Prices as Cash Value Suppose the jeweler can produce $10,000 worth of jewelry from 20 ounces of gold but only $6,000 worth of jewelry from 10 ounces of platinum. The Jeweler should still not trade 10 ounces of gold for 20 ounces of platinum – why? When a good is traded in a “competitive market” (which means that you are free to buy and sell the good at the same price), price determines the cash value of the good

Non-Competitive Markets A market for a particular good is said to be non-competitive if you buy and sell the good at different prices. In a non-competitive market, the value of the good is determined also by your own personal preferences (or benefit) from obtaining the good.

Non-Competitive Markets Example Suppose that just before the holidays your boss surprises you by offering all employees the choice between a brand new gardening tool-kit and $375. A brand-new tool-kit costs $450 in stores and can be sold for $400 through one of the internet websites. What would you choose?

Non-Competitive Markets Example continued If you have a garden at home and you were planning on buying the kit anyway then you would value this gift at $450. Alternatively, if you were not planning on buying the kit in the first place then you would value the gift at $400. When the market for the good is not competitive, the cash value of the economic opportunity depends on your preferences (and market prices)

Time Value of Money

How does timing affect our valuation? Timing of Payments For most decisions in life (and also financial decisions) the costs and benefits occur at different points in time. Mortgage Savings for retirement MBA How does timing affect our valuation? Lets start with this simple investment opportunity Investment of $100,000 today Payoff of $105,000 in one year Comparing the two directly is like “comparing apples to oranges”.

To compare payments in different times we need to use the interest rate Suppose that the current annual interest rate is 7%. This means that on a current deposit of $100 we will earn a total of $7 of interest over the course of the year leading to a total of $107 in our bank account. We refer to this as the risk-free interest rate and we denote this rate by rf = 7% or 1+rf=1.07 This means that we can exchange (without any risk) $1 today (or at time “0”) for $1.07 received in one year (or at time “1”)

Comparing Payments Across Time Back to our investment opportunity. We know that the benefit is $105,000 in one year. Can we calculate the cost of this investment in dollars paid in one year instead of dollars paid today in order to compare the two? By using the risk-free interest rate of 7%: The net value of the investment in terms of “dollars in one year” $105,000-107,000= - $2,000

Comparing Payments Across Time Back to our investment opportunity (again…). We will lose $2000 “dollars in one year” if we choose this investment opportunity, but how much will we loose in terms of “dollars today”? We know that the cost is $100,000 today. We can calculate the benefit of this investment in dollars paid today instead of dollars in one year: By using the risk-free interest rate of 7%: The net value of the investment in terms of “dollars today” is $98,130.84-100,000= - $1,869.16

Comparing Payments Across Time We valued the investment opportunity in terms of “dollars today” and “dollars in one year” and both led us to the conclusion not to invest in the project since it leads to a net loss. Is it always true that both approaches lead to the same conclusion? As long as we convert costs and benefits to the same point in time, we can compare them to make a decision Not surprisingly The net value of the investment in terms of “dollars today” and its value in terms of “dollars in one year” are consistent: - $1,869.16 (1.07) = - $2,000

Present Value & Future Value of Cash Flows

The Time Line The first (and very important) tool to develop is the ability to represent the stream of cash flows of an economic opportunity on a time line.

The Time Line Problem: Suppose you must pay tuition of $10,000 per year for the next two years. Your tuition payments must be made in equal installments at the start of each semester. What is the timeline of your tuition payments?

Moving Cash Flows Across Time When comparing cash flows obtained in different points in time we need to first convert all cash flows to one point in time Rule 1 It is only possible to compare or combine values at the same point in time Rule 2 To move a cash flow forward in time, we must compound it Rule 3 To move a cash flow back in time, we must discount it

Compounding Cash Flows Suppose we have $1,000 today and we want to determine the equivalent amount in one year’s time. If the annual risk-free interest rate is 10% then we can calculate: ($1000 today)(1.10) = $1100 in one year What if we were interested in the equivalent amount in two year’s time? Given that the annual risk-free rate for two years is also 10% then we can calculate:

Future Value The value of a cash flow that is moved forward in time is known as its future value When interest rate is 10% then, $1210 is the future value of $1000 two years from today $1100 is the future value of $1000 one year from today

Future Value If we were interested in the future value of $1000 three years from today we would simply multiply one more period Future Value of a Cash Flow “$C” “n” years from today, given interest “r” The future value is increasing in the interest rate “r” and the number of periods “n”.

Discounting Cash Flows Suppose we anticipate having $1,000 in one year and we want to compute the value today of this payment. If the annual risk-free interest rate is 10% then we can calculate: ($1000 in one year)/(1.10) = $909.09 today Suppose you anticipate receiving $1000 in two years. Given that the annual risk-free rate for two years is also 10% then we can calculate:

Present Value If we were interested in the present value of $1000 received three years from today we would simply multiply one more period Present Value of a Cash Flow “$C” Received “n” years from today, given interest “r” The present value is decreasing in the interest rate “r” and the number of periods “n”

Stream of Cash Flows Valuation

Future Value of a stream of CF’s Question: Suppose you plan to save $1000 annually in the next three years starting from today. Time line of deposits: If the interest rate on your savings account is 10%, then how much money will you have in your account three years from now? How much would you alternatively need to deposit at time 0 in order to obtain the same amount at time 3 with a single investment?

solution We are asked to find the future value of the stream of cash flows. The future value at time 3 of the payment of $1000 received at time 0: The future value at time 3 of the payment of $1000 received at time 1: The future value at time 3 of the payment of $1000 received at time 2: The future value of the stream of cash flows:

solution An alternative way to find the future value of the stream of cash flows is to follow the balance on our saving account. At time 0 we have …………. in our bank account. At time 1 we have ………… in our bank account. At time 2 we have …………. in our bank account. At time 3 we have accumulated …………….

solution Now suppose that you wish to invest all the money now instead of $1000 each year. What size investment is required in order to obtain the exact same amount of accumulated funds three years from now? You are asked to find the present value of the stream of cash flows. The present value of the stream of cash flows is …………….

Present Value of Stream of Cash Flows The present value of a stream of “n” cash flows of arbitrary amounts is the sum of the present value of all cash flows.

Practice Question Car loan: Suppose that you need money to purchase a new car. Your uncle will lend you the money so long as you agree to pay him back within four years, and you offer to pay him the rate of interest that he would otherwise get by putting his money in a savings account. Based on your yearnings and living expenses, you think you will be able to pay him $5000 in one year (time t=1), and then $8000 each year for the next three years. If your uncle would otherwise earn 6% per year on his savings, how much can you borrow from him?

solution You are asked about the present value of the stream of cash flows you plan to pay your uncle. The present value of the stream of cash flows is the sum of present values of all individual cash flows

solution So you are asking you uncle to give up $24,890.65 now and in return receive a stream of cash flows. Your calculation indicated that your uncle should be indifferent between the two alternatives. To verify lets compare the future wealth of your uncle if he had invested his money in a savings account instead of lending it to you $24,890.65. What shall we compare this to?

solution There are two ways we can go: First we can discount this future value using the relation: Second we can alternatively calculate the balance on the saving account of the uncle if he was to save the payments for the next four years by calculating the sum of future values of all individual installments:

The Net Present Value Decision Rule The term Present Value or PV refers to the fact that value is calculated in today’s dollars The Net Present Value (NPV) of a project or investment opportunity is the difference between the present value of benefits and the present value of costs Net Present Value (NPV) If we use positive cash-flows to represent benefits and negative cash flows to represent costs then We accept projects with positive NPV

Practice the NPV concept Question: Consider the following investment opportunity: If you invest $1000 today , you will receive $500 at the end of each of the next three years. If you could otherwise earn 10% per year on your money, should you undertake the investment opportunity?

Special Stream of Cash Flows Valuation

Annuities and Perpetuities Several of real-life situations involve streams of cash flows with “standard” structure. Mortgages Car loans Retirement savings Corporate and treasury bonds stocks We will study the valuation of Annuity: periodic stream of cash flows with finite life (Growing annuity has growing cash flows) Perpetuity: annuity with infinite life (Growing perpetuity has growing cash flows)

Perpetuity The PV of a Perpetuity equals the amount of funds required today in order to generate this perpetuity. How much is required today in order to generate an annual payment of $1 starting one year from now and continuing forever if the interest rate is 8%? The present value of a perpetuity is:

Perpetuities

Perpetuities

Annuities The PV of an Annuity equals the amount of funds required today in order to generate this annuity. How much is required today in order to generate an annual payment of $1 starting one year from now and continuing for 10 years if the interest rate is 8%? The present value of an “n” year annuity is:

Annuities

Annuities

Annuities

Annuities

Growing Perpetuity A growing perpetuity with growth rate “g” : The Present Value of a Growing Perpetuity

Growing Perpetuities

Growing Perpetuities

Growing Annuities The present value of an “n” year growing annuity is:

Growing Annuities

Growing Annuities

Practice questions

Loan Payment

Loan Payment

Loan Term

Loan Term

Mortgage Example Q21(2nd edition): When you purchased your house, you took our a 30-year annual-payment mortgage with an interest rate of 6% per year. The annual payment of the mortgage is $12,000. You have just made a payment and have now decided to pay the mortgage off by repaying the outstanding balance. What is the payoff amount if: You have lived in the house for 12 years (there are 18 years left on the mortgage) You have lived in the house for 20 years (there are 10 years left on the mortgage) You have lived in the house for 12 years (there are 18 years left on the mortgage) and you decide to pay off the mortgage immediately before the twelfth payment is due?

Amortization Schedule a. The PV(18 remaining payments)= b. The PV(10 remaining payments)= c. The PV(remaining payments just before 12th payment)=

Cost of schooling Q27(2nd edition): [modified] Your oldest daughter is about to start kindergarten at a private school. Tuition is $10,000 per year, payable at the beginning of the school year. You expect to keep your daughter in private school through high school. You expect tuition to increase at a rate of 5% per year over the 13 years of her schooling. What is the present value of the tuition payments if the interest rate is 6% per year? How much would you need to have in the bank now to fund all 13 years of tuition? What would your answers to the above questions be if interest rate is 5% per year?

solution When the interest rate is 6% you would need to have in the bank: If interest rate is 5% we cannot use the standard formula

Growing annuity when growth rate is larger than interest rate solution Growing annuity when growth rate is larger than interest rate The standard formula is applicable for all cases in which the interest rate is either larger or smaller than the growth rate

Investment in Art The interest rate over a period of 2 years is: Q33(2nd edition): You are thinking about buying a piece of art that costs $50,000. The art dealer is proposing the following deal: He will lend you the money, and you will repay the loan by making the same payment every two years for the next 20 years (i.e., a total of 10 payments). If the interest rate is 4%, how much will you have to pay every two years? The interest rate over a period of 2 years is: Calculating biennial payment:

Retirement Calculating annual installment: Q35(2nd edition): You are saving for retirement. To live comfortably, you decide you will need to save $2 million by the time you are 65. Today is your 30th birthday, and you decide, starting today and continuing on every birthday up to and including your 65th birthday, that you will put the same amount into a savings account. If the interest rate is 5%, how much must you set aside each year to make sure that you will have $2 million in the account on your 65th birthday? The present value of $2 million received at age 65 Calculating annual installment: