The time value of money
The Timeline
Timeline: a linear representation of the timing of potential cash flows
The Three Rules of Time Travel
•It is only possible to compare or combine values at the same point in time.
•To move a cash flow forward in time, you must compound it.
•To move a cash flow backward in time, you must discount it.
Valuing a Stream of Cash Flows
Interest rate: “exchange rate” between earlier money and later money = Discount rate = Cost of capital = Opportunity cost of capital = Required return
future values
simple interest: no interest on interest
compounding interest: interest is earned on interest
present values
PV = FV / (1 + r)t ( Note: FV = PV*(1 + r)^t)
Present-Value Interest Factor (PVIF):1 / (1 + r)^t, which is the present value of one dollar received t years from now at r discount rate
Calculating the Net Present Value
net present value (NPV): NPV = PV(benefits) - PV(costs) = PV(benefits-costs)
Perpetuities and Annuities
Perpetuities
When a constant cash flow will occur at regular intervals forever it is called a perpetuity
Annuities
When a constant cash flow will occur at regular intervals for a finite number of N periods, it is called an annuity
Growing Cash Flows
Growing Perpetuity: Assume you expect the amount of your perpetual payment to increase at a constant rate,g
Growing Annuity: The present value of a growing annuity with the initial cash flow c, growth rate g
Non-Annual Cash Flows
•The same time value of money concepts apply if the cash flows occur at intervals other than annually.
•The interest and number of periods must be adjusted to reflect the new time period.
The Internal Rate of Return
internal rate of return (IRR): the interest rate that sets the net present value of the cash flows equal to zero
Interest rates
Interest Rate Quotes and Adjustments
The Effective Annual Rate (EAR): Indicates the total amount of interest that will be earned at the end of one year; Considers the effect of compounding; Also referred to as the effective annual yield (EAY) or annual percentage yield (APY)
The annual percentage rate (APR): indicates the amount of simple interest earned in one year
Application: Discount Rates and Loans
Amortizing loans: Payments are made at a set interval, typically monthly; Each payment made includes the interest on the loan plus some part of the loan balance; All payments are equal and the loan is fully repaid with the final payment.
(e.g.mortgages and car loans)
The Determinants of Interest Rates
Nominal Interest Rate: The rates quoted by financial institutions and used for discounting or compounding cash flows
Real Interest Rate: The rate of growth of your purchasing power, after adjusting for inflation
Term structure of interest rate: The relationship between investment term and the interest rate; determined by future interest rate only
Yield Curve: A graph of the term structure
term structure can be used to compute the present and future values of a risk-free cash flow over different investment horizons
Risk and Taxes
Taxes reduce the amount of interest an investor can keep, and we refer to this reduced amount as the after-tax interest rate.
Interest on savings: taxable
Interest on loans: tax deductable
Valuing bonds
Bond Cash Flows, Prices, and Yields
terminology
Bond Certificate: States the terms of the bond
Maturity Date: Final repayment date
Term: The time remaining until the repayment date
Coupon: Promised interest payments
Face Value: Notional amount used to compute the interest payments
Coupon Rate: Determines the amount of each coupon payment, expressed as an APR
Yield to Maturity (YTM): The discount rate that sets the present value of the promised bond payments equal to the current market price of the bond
Zero-Coupon Bond
Does not make coupon payments
Always sells at a discount(a price lower than face value), so they are also called pure
discount bonds
Treasury Bills are U.S. government zero-coupon bonds with a maturity of up to one year
Risk-Free Interest Rates: A default-free zero-coupon bond that matures on date n provides a risk-free return over the same period. Thus, the Law of One Price guarantees that the risk-free interest rate equals the yield to maturity on such a bond.
Spot Interest Rate: Another term for a default-free, zero-coupon yield
Zero-Coupon Yield Curve: A plot of the yield of risk-free zero-coupon bonds as a function of the bond’s maturity date
Coupon Bonds
Pay face value at maturity
Pay regular coupon interest payments
Treasury Notes: U.S. Treasury coupon security with original maturities of 1–10 years
Treasury Bonds: U.S. Treasury coupon security with original maturities over 10 years
Yield to Maturity: The YTM is the single discount rate that equates the present value of the bond’s remaining cash flows to its current price
Dynamic Behavior of Bond Prices
discount: price < face value
par: price = face value
premium: price > face value
Time and Bond Prices
Holding all other things constant, the price of discount or premium bond will move
towards par value over time.
If a bond’s yield to maturity has not changed, then the IRR of an investment in the
bond equals its yield to maturity even if you sell the bond early.
Interest Rate Changes and Bond Prices
an inverse relationship: As interest rates and bond yields rise, bond prices fall; As interest rates and bond yields fall, bond prices rise.
The sensitivity of a bond’s price to changes in interest rates is measured by the
bond’s duration: All else equal, bonds with longer maturity / lower coupon rates tend to have higher duration
The Yield Curve and Bond Arbitrage
Using the Law of One Price and the yields of default-free zero-coupon bonds, one can
determine the price and yield of any other default-free bond.
The yield curve provides sufficient information to evaluate all such bonds.
Corporate Bonds
Issued by corporations
Have credit Risk (risk of default: expected return will be less than the yield to maturity & higher yield to maturity does not necessarily imply a higher expected return)
Bond Ratings
Investment Grade Bonds (AAA, AA, A, BBB): low default risk
Speculative Bonds (BB, B, CCC, CC, C, D): high default risk
default spread (credit spread): The difference between the yield on corporate bonds and Treasury yields
Sovereign Bonds
Bonds issued by national governments
Appendix
Computing Forward Rates
A forward interest rate (or forward rate): an interest rate that we can guarantee today for a loan or investment that will occur in the future
Computing Bond Yields from Forward Rates
Forward Rates and Future Interest Rates
a break-even rate.
is a good predictor only when investors do not care about risk
Expected Future Spot Interest Rate = Forward Interest Rate + Risk Premium
