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.

Future value of a cash flow:

•To move a cash flow backward in time, you must discount it.

Present value of a cash flow

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

the composition of interest over time

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

Present Value of 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

Present Value of an Annuity

Growing Cash Flows

Growing Perpetuity: Assume you expect the amount of your perpetual payment to increase at a constant rate,g

Present Value of a Growing Perpetuity

Growing Annuity: The present value of a growing annuity with the initial cash flow c, growth rate g

Present Value of a Growing Annuity

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

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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

Converting an APR to an EAR

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

real vs. nominal interest rate

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 of Risk-Free U.S. Interest Rates, November 2006, 2007, and 2008

term structure can be used to compute the present and future values of a risk-free cash flow over different investment horizons

Present Value of a Cash Flow Stream Using a Term Structure of Discount Rates

Risk and Taxes

Higher default risk → Higher return

Taxes reduce the amount of interest an investor can keep, and we refer to this reduced amount as the after-tax interest rate.

τ = tax rate

Interest on savings: taxable

Interest on loans: tax deductable

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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

Coupon Payment

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

Yield to Maturity of an n-Year Zero-Coupon Bond

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.

Risk-Free Interest Rate with Maturityn

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

Yield to Maturity of a Coupon Bond

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.

The Effect of Time on Bond Prices

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.

Valuing Coupon Bond Using Zero-Coupon Yields

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

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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

general formula for the forward interest rate

Computing Bond Yields from Forward Rates

zero-coupon yields from the forward interest 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