Valuation Under Certainty Investors must be concerned with: - Time - Uncertainty First, examine the effects of time for one-period assets. Money has time value. $100 in one year is not as attractive as $100 today. Rule 1: A dollar today is worth more than a dollar tomorrow, because it can be reinvested to earn more by tomorrow.

Session 1 Topics to be covered: –Time value of money –Present value, Future value –Interest rates compounding intervals –Bonds –Arbitrage

Present Value The value today of money received in the future is called the Present Value The present value represents the amount of money we would be prepared to pay today for something in the future. The interest rate, i is the price of credit in financial markets. –Interest rates are also known as discount rates.

Present Value The Present Value Factor or Discount Factor is the number we multiply by a future cash flow to calculate its present value. Present Value (PV)=Discount Factor*Future Value(FV) - Discount factor = 1/(1+i) Example (i=10%) - Discount factor = 1/(1+.10) - The present value of $200 received in 1 year is

Future Value Alternatively, we may use the interest rate, i, to convert dollars today to their value in the future. Suppose we borrow $50 today, and must repay this plus 5% interest in one year. Future Value (FV) = Present Value (PV)(1+i) FV = PV(1+i) = 50(1+.05) = $52.50

Bonds A bond is a promise from the issuer to pay the holder - the principal, or face value, at maturity. - Interest, or coupon payments, at intervals up to maturity. A $100 face value bond with a coupon rate of 7% pays $7.00 each year in interest, and $100 after a pre-specified length of time, called maturity.

Zero Coupon Bonds A zero coupon bond has no coupon payments. The holder only receives the face value of the bond at maturity. Suppose the interest rate is 10%. A zero coupon bond promises to pay the holder $1 in one year. Its price today is therefore The discount factor is just the price of a zero coupon bond with a face value of $1.

Net Present Value The Net Present Value is the present value of the payoffs minus the present value of the costs. Suppose Treasury Bills yield 10%. The present value of $110 in one year is Suppose we could guarantee this payoff by investing in a project that only costs $98 today. The NPV of this project is

NPV The formula for calculating the NPV (one-period case) is Note that C 0 is usually negative, a cost or cash outflow. In the above example, C 0 = -98 and C 1 = 110.

Rate of Return The rate of return is the interest rate expected to be earned by an investment. The rate of return for this project is We only want to invest in projects that return more than the opportunity cost of capital. The cost of capital in this case is 10%.

Decision Rules We know: 1.) This project only costs $98 to guarantee $110 in one year. In “the market”, it costs $100 to buy $110 in one year. 2.) This project returns 12.2%. In the market, our return is only 10%. This project looks good.

Decision Criteria We have equivalent decision rules for capital investment (with a ONE-PERIOD investment horizon): - Net Present Value Rule: accept investments that have a positive NPV. - Rate of Return Rule: accept investments that offer a return in excess of their opportunity cost of capital. These rules are equivalent for one period investments. These rules are NOT equivalent in more complicated settings.

Example: Market Value Continue to suppose you can borrow or lend money at 10%. Assume the price of a one-year zero-coupon bond with a FV of $110 is $98. The price of this bond is less than its present value. We may use this example to illustrate the concept of “arbitrage.”

No Arbitrage Arbitrage is a “free lunch,” a way to make money for sure, with no risk and no net cost. For example: - Buy something now for a low price and immediately sell it for a higher price. - Buy something now and sell something else such that you have no net cash flows today, but will earn positive net cash flows in the future. Assets must be priced in financial markets to rule out arbitrage.

Example (cont.) To arbitrage this opportunity, we –1.) buy the bond –2.) borrow $100 for one year. The cash flows from this strategy today and at the end of one year are: TodayOne Year Buy the bond Borrow $100 (1 yr) Net cash flow +2 0

Short Selling Suppose the price of the zero-coupon bond were $102. Our arbitrage strategy would be reversed. - Lend $100 for one year. - Short Sell the zero-coupon bond. The cash flows from this strategy would be TodayOne Year Sell the bond Lend $100 (1 yr) Net cash flow +2 0

Market Value As the above example illustrates, the only price for a bond which rules out arbitrage is $100. $100 is also the present value of the payoff of the bond. RULE 2: Assets must be priced in the market to rule out arbitrage (i.e., “no arbitrage”) –Therefore, the present value of an asset is its market price.

Compound Interest Vs. Simple Interest Next we consider assets that last more than one period. How is multi-period interest paid? Invest $100 in bonds earning 9% per year for two years: - After one year: $100(1.09) = $109 - Reinvest $109 for the second year: $109(1.09) = We do NOT earn just 9% * 2 = 18%. We earn “interest on our interest”, or COMPOUND SIMPLE INTEREST: interest paid only on the initial investment COMPOUND INTEREST: interest paid on the initial investment and on prior interest.

Example: Simple Interest $100 invested at 10% with no compounding becomes:

Example: Compound Interest $100 invested at 10% compounded annually becomes:

Compound Interest A present value $PV invested for n years at an interest rate of i per year grows to a future value (1 + i n ) n is the Compound Amount Factor. Above, the FV of $100 compounded annually at 10% for 3 years is In principle, the interest rate i n may vary with the length of the investment horizon, n. More later...

Present Value We may use the above relation to calculate the present value of an n-period investment, with compound interest: where is the discount factor, or present value factor. For example, the present value of $100 in 6 years at 10% per year with annual compounding:

Semi-Annual Compounding So far, we assume cash flows occur at annual intervals. - In Europe, most bonds pay interest annually. - In the U.S., most bonds pay interest semiannually. A $100 bond pays interest of 10% per year, but payments are semi-annual. - Half of the interest (5% or $50) is paid after 6 months. - Reinvest this $50 for the second 6 months. By the end of the year, we would have

Example This return is as if we earned if we had only received our payment at the end of the year. 10% compounded semiannually is equal to 10.25% compounded annually. - 10% is called the nominal interest rate % is called the effective interest rate.

Example Suppose you buy $100 of a 7-year Treasury note that pays interest at a nominal rate of 10% per year, compounded semiannually. Define one period as 6 months: –The interest rate per period is 5%. –There are 14 (6-month) periods until the 7-year maturity. So, we can use our general formula for future values to compute the value at maturity:

Extending the PV Formula RECALL: For a project with one cash flow, C 1, in one year, If a project produces one cash flow, C 2 after TWO years, then the present value is If a project produces one cash flow, C 1 after one year, and a second cash flow C 2 after two years, then

General Present Value By extension, the present value of an extended stream of cash flows is This is called the Discounted Cash Flow or Present Value formula: Similarly, the Net Present Value is given by

Example Suppose a project will produce $50,000 after 1 year, $10,000 after 2 years, and $210,000 after 4 years. It costs $200,000 to invest. We may earn 9% per year (compounded annually) on 1, 2, or 4 year zero-coupon bonds. The present value of this project is The NPV of this project is

Net Present Value Rule In the last example, the PV of payoffs exceeded the PV of the costs, so the project is a good one. Investment Criterion (The NPV Rule): Accept a project if the NPV is greater than 0. –This criterion is a good general rule for all types of projects. –The NPV Rule can also be used to rank projects; a project with a larger (positive) NPV is better than one with a smaller NPV.