Review
Real Estate Finance, Spring, 2017
1
Overview




Interest rates
Future value
Present value
Financial value and investment
• Net Present Value
• Internal Rate of Return
 Bond pricing
2
Interest rates
An investment in real property, or any other financial asset for
that matter, involves an exchange of current funds for claims on
future funds – in order to determine whether or not the property is
priced fairly we need to be able to compare dollars today to
dollars in the future.
 If we decide to use current funds to purchase an incomeproducing property, what are we giving up?
• Current consumption
• Other investment opportunities
3
Interest rates
The price of consumption now versus consumption later is the
interest rate
 By depositing $100 in a savings account that offers a 5% rate of interest,
you receive $105 in one year’s time – the $5 represents the price received
for giving up $100 of current consumption.
The cost of investing $100 in project A rather than project B is the
expected return on project B
 The expected return on project B represents the opportunity cost of
investment in A
4
Future value
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
FV  C0 1  r 
n
•
•
•
•
C0 = initial value
FV = future value
n = number of periods
r = periodic interest rate
5
Future value
What is the future value of $100 if invested today at a 7%
annual interest rate if the investment matures in one year?
FV  1001.07   107.00
What is the future value of $100 if invested today at a 7%
annual interest rate if the investment matures in three years?
FV  1001.07 1.07 1.07 
 1001.07 
3
 122.50
6
Future value
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 semiannually?
2
 .07 
FV  1001 
  107.12
2 

 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?
12
 .07 
FV  1001 
  107.23
 12 
7
Future value
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
 1  r n  1
FV  C 

r


• C = the periodic annuity payment
8
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?
 1.0810  1
FV  100 
  1,448.66
 .08

 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
9
Future value
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?
 1.04 20  1
FV  50 
  1,488.90
.04


 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
10
Present 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
Cn
PV 
1  r n
• 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 Cn
PV 1  r   Cn
n
• The interest rate in present value relations is often referred to as a discount rate
11
Present value
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?
12
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:
1
Cash flow
Present value (10% discount rate)
2
3
25
40
1015
22.73
33.06
762.58
Total
818.37
More formally,
T
PV  
t 1
Ct
1  r t
13
Present value
What is the relationship between present value and interest rates?
 PV is a decreasing and convex function of interest rates
PV
r
14
Present value
The present value of an annuity received for n periods is given by
C
1 
PV  1 
n
r  1  r  
What if the payments continue without end?
C
PV 
r
 An infinite sequence of constant payments is referred to as a perpetuity
15
Present value
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
n
C  1  g  
PV 
1 
n 
r  g   1  r  
What if the payments continue to grow without end?
C
PV 
rg
16
Financial value and investment
The value of any financial asset equals the present value of its
cash flows
T
Ct
V 
t
t 1 1  r 
 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
17
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
T
Ct
NPV  C0  
t
t 1 1  r 
18
Financial value and investment
Any investment offering a nonnegative NPV is an acceptable
investment
T
Ct
NPV  C0  
0
t
t 1 1  r 
 A zero net present value investment is still an acceptable investment
 Applies to now-or-never investment opportunities, ignoring future flexibility
in decision making
19
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
T
Ct
 C0  
0
t
t 1 1  IRR 
 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
20
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.
21
Financial value and investment
NPV
0
IRR
r
22
Bond pricing
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.
23
Bond pricing
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
n
Ct
M
P

t
n




1

r
1

r
t 1
 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?
24
Bond pricing
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%?
20
3
100
P

t
20




1
.
04
1
.
04
t 1
 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
25
Bond pricing
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
P
r
26
Bond pricing
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
• Yield to call
27
Bond pricing
What is the yield on a ten-year bond with $100 par value, semiannual coupon payments, 6% coupon rate and current price of $105?
20
3
100
105  

t
20
1  y 
t 1 1  y 
y  2.67, BEY  22.67   5.35
 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
28
Bond pricing
What is the yield on a ten-year bond with $100 par value, semiannual coupon payments, 6% coupon rate and current price of $95?
20
3
100
95  

t
20
1  y 
t 1 1  y 
y  3.35, BEY  23.35  6.69
 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?
29
Bond pricing
What is the yield on a ten-year zero coupon bond with $100 par
value and current price $45.00?
1 / 20
100
 100 
45 
,y

20
1  y 
 45 
BEY  2.0407   .0815
 1  .0407
 The BEY accurately reflects the yield for zero-coupon bonds as there are
no periodic cash flows and, therefore, no reinvestment of cash flows
30
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
31
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
in the near term
32
Bond pricing
33
Bond pricing
34
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.
35
Computing a Return
 You purchase a stock on Jan 1, 2017 for $100. The stock pays a
dividend of $10 on Dec 31, 2017. You sell the stock after the
dividend is paid for $105 on Dec 31, 2017. What is the one-year
return on the stock?
d t  Pt 1  Pt 
rt 
Pt
10  105  100

100
 15%
36
Computing a Return
 Notice we can split the return into income yield and growth yield
d t Pt 1  Pt
rt  
Pt
Pt
 Income yield = d/P, commonly called “cap rate” in real estate
 Growth yield = change in P/P
37
Computing a Return with a multiple-period hold
• You purchase a stock on Jan 1, 2017 for $100.
• The stock pays a dividend of $10 on 12/31/17. The stock price is $105 after
the dividend is paid.
• The stock pays a dividend of $8 on 12/31/18. The stock price is $106 after
the dividend is paid.
• What is the cumulative return to you for holding the stock?
• What is the sequence of annual returns for the stock?
38
Computing a Return with a multiple-period hold
• Cumulative return during the holding period
d t  d t 1  Pt  2  Pt  10  8  106  100
r

 24%
Pt
100
• Sequence of annual returns
d t  Pt 1  Pt  10  105  100
rt 

 15%
Pt
100
d t 1  Pt  2  Pt 1  8  106  105
rt 1 

 8.57%
Pt 1
105
• Notice that (1.15)*(1.0857) -1 = 24.86% > 24%. Why???
39