Real Estate Finance, January XX, 2016

Review

The interest rate can be thought of as the price of consumption now rather than later• If you deposit $100 in a savings account offering a 5% rate of interest,

in one year’s time, you receive $105 – the $5 represents the price you received for giving up $100 of current consumption.

Why do we generally find positive rates of interest in market-based economies?• Productivity of capital. Capital goods can be used to generate

additional output - if we add to our stock of capital goods instead of consuming today, we can consume tomorrow not only those capital goods but also the additional output they produce.• As long as individuals have the option of using funds not spent on current

consumption to increase their own capital stock and, thereby, their future income, they would never loan these funds to someone else at an interest rate of zero.

The interest rate

If interest rates are positive, money invested today will result in a future value greater than its present value The future value of C0 invested today is given by

• C0 = initial value• FV = future value• n = number of periods• r = interest rate

Future value

nrCFV 10

What is the future value of $100 if invested today at a 7% annual interest rate if the investment matures in one year?

What is the future value of $100 if invested today at a 7% annual interest rate if the investment matures in three years?

Future value

00.10707.1100 FV

50.12207.1100

07.107.107.11003

FV

For a given maturity and annual interest rate, future values increase with more frequent compounding What is the future value of $100 if invested today at a 7%

annual interest rate if the investment matures in one year and is compounded semi-annually?

What is the future value of $100 if invested today at a 7% annual interest rate if the investment matures in one year and is compounded monthly?

Future value

23.1071207.1100

12

FV

12.107207.1100

2

FV

An annuity is an investment offering a series of constant periodic payments for a given amount of time• The future value of an annuity is given by

• C = the periodic annuity payment

Future value

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Future value What is the future value of an annuity offering

payments of $100 a year for 10 years assuming an 8% annual interest rate?

– This equals the total dollar amount to be paid, $1000, plus the gain to reinvesting the payments in another investment offering an 8% annual return, $448.66

66.448,108.

108.110010

FV

What is the future value of an annuity offering payments of $50 on a semi-annual basis for 10 years assuming an 8% annual interest rate?

– This equals the total dollar amount to be paid, $1000, plus the gain to reinvesting the payments in another investment offering an 4% semi-annual return, $488.90

90.488,104.

104.15020

FV

Future value

An investment can be thought of as an exchange of current income for the right to receive future income – in order to arrive at some measure of the value of an investment, we must be able to compare future income to current income. For a given interest rate r > 0, the present value of a

cash flow received n periods from now is given by

• The present value is the current income that would be have to be invested today at given interest rate r in order to receive future cash flow

• The interest rate in present value relations is often referred to as a discount rate

Present value

nn

rCPV

1

nC

nn CrPV 1

If r = 5%, what is the present value of $250 to be received in one year’s time?

If r = 5%, what is the present value of $250 to be received in two year’s time?

If r = 5%, what is the present value of $250 to be received in five year’s time?

Present value

The present value associated with an investment offering a sequence of cash flows over a given time horizon can be found by determining the present value of each individual cash flow and then summing:

More formally,

Present value

1 2 3Cash flow 25 40 1015Present value (10% discount rate)

22.73 33.06 762.58

Total 818.37

T

tt

t

rCPV

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What is the relationship between present value and interest rates?• PV is a decreasing and convex function of interest rates

Present value

PV

r

The present value of an annuity received for n periods is given by

What if the payments continue without end?

• An infinite sequence of constant payments is referred to as a perpetuity

Present value

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CPV111

rCPV

If the annuity payment is growing at constant rate g < r, the present value of a growing annuity received for n periods is given by

What if the payments continue to grow without end?

Present value

n

n

rg

grCPV

111

grCPV

The value of any financial asset equals the present value of its cash flows

The interest rate used to discount future cash flows in this context is often referred to as the required return on investment and represents the investor’s opportunity cost.– If an investor’s financial resources are limited, the

financial capital required to invest in one particular project means that some other investment opportunity is foregone – the return on the investor’s next-best investment alternative is the opportunity cost of their capital

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rCV

1 1

Financial value and investment

The net present value (NPV) of an investment opportunity is the difference between the current income needed to acquire the investment and its present value

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rCCNPV

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Financial value and investment

Any investment offering a nonnegative NPV is an acceptable investment

– A zero net present value investment is still an acceptable investment

– Applies to now-or-never investment opportunities, ignoring future flexibility in decision making

0

110

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rCCNPV

Financial value and investment

An investment’s internal rate of return (IRR) is the return at which an investor is indifferent between accepting and rejecting the investment

– If an investment offers an IRR greater than or equal to the investor’s required return, the investment offers an acceptable return.

– Assumes that all periodic cash flows are reinvested in other assets offering returns equal to the IRR

0

110

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IRRCC

Financial value and investment

In most cases, both the NPV and IRR investment rules lead to the same decision: If NPV ≥ 0, then IRR ≥ r If IRR ≥ r, then NPV ≥ 0

• One potential problem with the IRR is that there may be multiple IRRs that are solutions to NPV = 0 and no reliable way to choose from among the set of solutions.

Financial value and investment

NPV

r

Financial value and investment

0IRR

A bond is a debt instrument requiring the issuer to repay the lender the amount borrowed plus interest on a periodic basis for a pre-specified length of time The amount borrowed is referred to as the principal value,

par value, face value, or redemption value. The periodic interest payments are referred to as coupon

payments and are determined by the coupon rate on the bond.

Bond pricing

If a ten-year bond with a par value of $100 makes semi-annual interest payments based on an annual coupon rate of 6%, what are the corresponding cash flows? Coupon payment? Repayment of principal?

For a given discount rate, r, the price of a bond is given by the present value of the annuity corresponding to its coupon payments plus the present value of the repayment of principal at maturity

Bond pricing

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111

What is the price of a ten-year bond with a par value of $100 which provides semi-annual interest payments based on an annual coupon rate of 6% if the corresponding discount rate is 8%?

What if the discount rate is 6%– Priced at par value

What if the discount rate is 4%– Priced at a premium to par value

Bond pricing

2020

1 04.1100

04.13

t

tP

What is the relationship between a bond’s price and it’s discount rate? • The bond price is given by the present value of its future cash

flows, so the relationship has the same characteristics as the PV as a function of r

Bond pricing

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r

A bond’s yield to maturity is the required return setting the current price equal to the present value of future cash flows. Effectively an IRR for bonds Applies only to non-callable bonds

Bond pricing

What is the yield on a ten-year bond with $100 par value, semi-annual coupon payments, 6% coupon rate and current price of $105?

The convention involved in calculating the annual yield by simply doubling the semi-annual yield is referred to as the bond-equivalent yield (BEY) or simple annualized rate.– The BEY allows for better comparisons between bonds with

different underlying characteristics, but does not consider the interest generated by reinvesting periodic interest payments and, therefore, understates the bond’s true yield

The bond is priced at a premium to par, so the BEY is less than the coupon rate

Bond pricing

35.567.22,67.21100

13105 20

20

1

BEYyyyt

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What is the yield on a ten-year bond with $100 par value, semi-annual coupon payments, 6% coupon rate and current price of $95?

The bond is priced at a discount to par, so the BEY is greater than the coupon rate.– What is the yield on this bond if the price equals par value?

Bond pricing

69.635.32,35.31100

1395 20

20

1

BEYyyyt

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What is the yield on a ten-year zero coupon bond with $100 par value and current price $45.00?

The BEY accurately reflects the yield for zero-coupon bonds as there are no periodic cash flows and, therefore, no reinvestment of cash flows

0815.0407.2

0407.145100,

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BEY

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Bond pricing

What are the underlying determinants of a bond’s yield? The yield on the corresponding Treasury note, bill or

bond.• Term to maturity or duration

Compensation for risk• Likelihood of full payments to principal and interest• Timing of payments to principal and interest• Interest rate risk

• Reinvestment risk• Price risk

• Liquidity

Bond pricing

The relationship between yield and time to maturity for Treasury bonds is summarized in the yield curve. The yield curve is typically upward sloping, meaning

that investors require higher returns to hold longer term securities.

The yield curve provides some information regarding investor expectations about future short-term interest rates– An inverted yield curve, where the yield on short term bonds

exceed those for longer terms bonds, suggests that investors expect short term rates to increase

Bond pricing

Bond pricing

Bond pricing

The yield curve is frequently used as a benchmark for pricing other fixed-income securities – the risk premium for a bond equal’s the bond’s yield to maturity less the yield on a Treasury bond with the same maturity Treasury securities are backed by the “full faith and

credit” of the US government and are therefore treated as being effectively free of the risk of default.– The securities underlying the yield curve are not coupon

paying securities, but “stripped” securities that offer no coupon payments.

Bond pricing