A) Read the news article

https://www.marketwatch.com/story/the-federal-reserve-is-stuck-in-quantitative-easing-hell-2020-01-16

B) Do some research

C) IN YOUR OWN WORDS, answer the questions about the Fed policy.

1. What is quantitative easing (QE)?

2. What is repurchase agreement?

3. List the monetary policy tools that the Fed can use to stimulate economy. Explain the details.

NO PLAGIARISM!!!!!!!!!!!!!!!!!!!!!!!

follow directions asap

Ch. 5 – Time Value of Money

1

Understand the time value of money

Value series of cash flows

Understand compounding

Distinguish between nominal and effective interest rates

2

Objectives

2

Suppose one year CD pay 3% interest. Then, how much would you receive one year from now if you invest $1000 in this type of CD today?
The time value of money
3

3

$1 today is more than $1 one year from now!
Money that you have today can be invested and will start earning interest immediately
The time value of money
4

Assuming that the rate of return (interest) on government bonds is 6%, how much would you pay today for a government bond that pays $1000 one year from now?
The time value of money
5

The present value of cash flow received some time in the future is equal to that cash flow multiplied by the discount factor, also called the present value factor
PV(C1)=C1×Discount factor
The discount factor is usually expressed as a rate of return
Discount factor =
Present Value

6

What is the present value of the bond we discussed before?

What is the one-year discount factor for this bond?
Present Value

7

7

If $1 today is worth more than $1 one year from today, then it makes sense that $1 one year from today is worth more than $1 two years from today.
How much would you pay for a bond that pays $1,000 two years from now, assuming that the rate of return each year is 5%?
Valuing long-lived assets

8

The present value of a cash flow 1 year from today is

The present value of a cash flow 2 years from today is
Valuing long-lived assets

9

And, in general, the present value of a cash flow t years from today is

And the discount factor is
Valuing long-lived assets

10

Assuming that the rate of return is 7%, how would you value the following stream of cash flows:

Valuing long-lived assets

$100 one year from now
$200 two years from now
$300 three years from now
11

A stream of cash flows over many periods can be valued as

Valuing cash flow over many periods

12

This formula is called the discounted cash flow (DCF) formula
A shorthand way to write this formula is

The discounted cash flow formula

13

Suppose you put $1,000 into a savings account today that will pay 11% interest for 5 years. How much will you have at the end of five years?
Future Value

14

The future value of a cash flow received some time in the future is equal to that cash flow divided by the discount factor:
Where
C0 is cash flow at date 0,
r is the appropriate interest rate, and
T is the number of periods over which the cash is invested.
Future Value

15

Future Value
Suppose that you invest $500 in a savings account that earns annual interest rate of 3%. What will your account be worth in five years?

16

16
FV = 500*(1.03)^5 = 579.64
Emphasis – 4 variables in equation…if we know 3, can always solve for the 4th

Compounding
Compounding an investment m times a year for T years provides for future value of wealth:

If you invest $50 for 3 years at 12% compounded semi-annually, what will your investment grow to?

17

17

Continuous Compounding
General formula:
Where
C0 is cash flow at date 0
r is the stated annual interest rate
T is the number of periods over which the cash is invested
e is a transcendental number approximately equal to 2.718.
{ex is a key on your calculator}
Example: You invest $1,000 at a continuously compounded rate of 10% for 2 years. How much will your investment be worth?
18

1000*e^(.1*2) = 1221.40

18
e is a transcendental number because it transcends the real numbers.
– not a solution to a polynomial
– in the limit… we go to e
1000*e^(.1*2) = 1221.40

Assuming that the discount rate is 10% per year, what is the present value of $10 paid once a year forever, starting one year from now?
Perpetuity

19

A constant stream of cash flows that lasts forever.

The formula for the present value of a perpetuity is:

Perpetuity

20

You just won a lottery that will pay you (and your estate) $100,000 per year forever, starting one year from now. The lottery organizers offered to pay you $1 million in cash today instead of the perpetuity. The discount rate is 8%. Should you accept?

Perpetuity: Example

21

21

What if the payments started today instead of one year from now?

Perpetuity

22

If the cash flows from a perpetuity begin today (not in the future), its present value is

Perpetuity

23

A constant stream of cash flows with a fixed maturity.

An ordinary annuity has the following characteristics:
The payments are always made at the end of each interval;
The interest rate compounds at the same interval as the payment interval.

24
Annuity

24

The formula for the present value of an ordinary annuity is:

is called annuity factor.

25
Annuity

25

Assume that the lottery payments would start a year from now at $100,000. They would remain constant each year, but they would stop after the 5th payment. What is the PV of the lottery payments? The discount rate is 8%.
How would we deal with this problem using our knowledge of how to price perpetuities?
Annuity: Example
26

An annuity is valued as the difference between two perpetuities:
one perpetuity that starts at time 1
less a perpetuity that starts at time T + 1
27
Annuity Intuition

27

What would be the PV of our lottery payments if they started today and ended after the 5th payment?

Annuity Due

28

Annuity Due: the annuity payments are made at the beginning rather than the end of the period…
Annuity Due
AnnuityDue = AnnuityOrdinary x (1+r)

Annuity Due
Ordinary Annuity
29

So far, we assumed that interest is compounded annually. This is not what happens in reality
If we invest $1,000 today at an interest rate of 10% compounded annually, how much will we receive 2 years from today?

What if the interest were compounded semi-annually?

Nominal and effective interest rates

30

Effective interest rate is an annually compounded rate equivalent to nominal annual interest rate (also called annual percentage rate , APR) compounded more than once a year

Where m is the number of compound intervals per year
If m becomes very large (infinite), then

Nominal and effective interest rates

31

Which investment would you prefer?
An investment paying interest of 12% compounded annually?
An investment paying interest of 11.7% compounded semiannually?
An investment paying interest of 11.5% compounded continuously?

Nominal and effective interest rates

32

Time value of money
Present value and future value
Discount factor (present value factor)
Perpetuity
Ordinary annuity and annuity due
Nominal and effective interest rates

Glossary

33

r
+
1
1
9434
.
0
06
.
1
1
1
1
=
=
+
r
03
.
907
05
.
1
1000
)
1
(
2
2
2
=
=
+
=
r
C
PV
r
C
C
PV
+
=
1
)
(
1
1
2
2
2
)
1
(
)
(
r
C
C
PV
+
=
t
t
t
r
C
C
PV
)
1
(
)
(
+
=
t
r
factor
Discount
)
1
(
1
+
=
T
T
r
C
r
C
r
C
PV
)
1
(
…
)
1
(
)
1
(
2
2
1
1
+
+
+
+
+
+
=
T
T
T
t
t
t
r
C
r
C
r
C
r
C
PV
)
1
(
…
)
1
(
)
1
(
)
1
(
2
2
1
1
1
+
+
+
+
+
+
=
+
=
å
=
T
r
C
factor
Discount
C
C
FV
)
1
(
*
1
*
)
(
0
0
0
+
=
=
mT
m
r
CFV






 1
0
rT
e
C
FV
´
=
0
r
C
PV
=
L
+
+
+
+
+
+
=
3
2
)
1
(
)
1
(
)
1
(
r
C
r
C
r
C
PV
)
1
(
r
r
C
C
r
C
PV
+
=
+
=
T
r
C
r
C
r
C
r
C
PV
)
1
(
)
1
(
)
1
(
)
1
(
3
2
+
+
+
+
+
+
+
=
L
÷
÷
÷
÷
ø
ö
ç
ç
ç
ç
è
æ
+
–
=
÷
÷
ø
ö
ç
ç
è
æ
+
–
=
r
r
C
r
r
r
C
PV
T
T
)
1
(
1
1
)
1
(
1
1
÷
÷
ø
ö
ç
ç
è
æ
+
–
T
r
r
r
)
1
(
1
1
T
r
r
C
r
C
PV
)1(







)
1
(
)
1
(
1
1
)
1
(
)
1
(
1
1
r
r
r
r
C
r
C
C
r
r
r
C
PV
T
T
T
+
÷
÷
ø
ö
ç
ç
è
æ
+
–
=
+
–
+
÷
÷
ø
ö
ç
ç
è
æ
+
–
=
1
r
1
r
nominal
effective
–
÷
ø
ö
ç
è
æ
+
=
m
m
1
r
nominal
r
effective
–
=
e

Ch. 6 – Interest Rates

1

The determinants of market interest rate

Nominal and real interest rates

Term structure of interest rates

Yield curve

Pure expectations theory

Macroeconomic factors that influence interest rate

Topics

2

What four factors affect the level of interest rates?
Production opportunities
Time preferences for consumption
Risk
Expected inflation

“Nominal” vs. “Real” Rates
Nominal interest rate: r
the quoted or stated interest rate
the interest rate before taking inflation into account
Real interest rate: r*
It is the rate that would exist on a riskless security in a world where no inflation was expected.

4

Determinants of Interest Rates
r = r* + IP + DRP + LP + MRP
r = required return on a debt security, nominal rate
r* = real risk-free rate of interest
IP = inflation premium
DRP = default risk premium
LP = liquidity premium
MRP = maturity risk premium

5

Default (Credit ) risk
Definition: The risk that a security’s issuer will default on that security by being late on or missing an interest or principal payment
Investors must consider the creditworthiness of the security issuer
Can use bond ratings of rating agencies
The higher the rating, the lower the perceived credit risk
Ratings can change over time as economic conditions change

6

Default (Credit ) risk
  Ratings Assigned by:
Description of Security Moody’s Standard and Poor’s
Highest quality Aaa AAA
High quality Aa AA
High-medium quality A A
Medium quality Baa BBB
Medium-low quality Ba BB
Low quality (speculative) B B
Poor quality Caa CCC
Very poor quality Ca CC
Lowest quality (in default) C DDD, D

Speculative-grade
Investment-grade

7

Liquidity
Definition: the degree to which the securities can easily be converted to cash without a loss in value
If all other characteristics are equal, securities with less liquidity will have to offer a higher yield to be preferred.

8

Term to maturity
Maturity dates will differ between debt securities
Maturity risk premium: to compensate investors for taking on the risk of holding bonds over a lengthy period of time.

9

Premiums Added to r* for Different Types of Debt
IP MRP DRP LP
S-T Treasury 
L-T Treasury  
S-T Corporate   
L-T Corporate    

10

Yield Curve and the Term Structure of Interest Rates
Term structure: relationship between interest rates (or yields) and maturities.
The yield curve is a graph of the term structure.
The March 2010 Treasury yield curve is shown at the right.

11

1 5 10 30 1 5 10 30 1 5 10 30 0.38000000000000045 2.42 3.68 4.5999999999999996 Years to Maturity
Interest Rate

Treasury Yield Curve and Yield Curves for Corporate Issues
Corporate yield curves are higher than that of Treasury securities, though not necessarily parallel to the Treasury curve.
The spread between corporate and Treasury yield curves widens as the corporate bond rating decreases.
Since corporate yields include a default risk premium (DRP) and a liquidity premium (LP), the corporate bond yield spread can be calculated as:

12

Pure Expectations Theory
Pure expectations theory suggests that the shape of the yield curve is determined solely by expectations of future interest rates
The yield curve will become upward sloping if interest rates are expected to rise
The yield curve will become downward sloping if interest rates are expected to decline
The yield curve will become flat if interest rates are expected to remain the same

13

Macroeconomic Factors That Influence Interest Rate
Federal reserve policy
When the Fed reduces the money supply, it reduces the supply of loanable funds, putting upward pressure on interest rates.
Federal budget deficits or surpluses
A high deficit means a high demand for loanable funds by the government
Shifts the demand schedule outward (to the right)
Interest rates increase
International factors
Level of business activity

14

LP

DRP

yield

bond
Treasury

yield

bond

Corporate
spread

yield
bond

Corporate
+
=
–
=

TimeValue of Money

*Financial calculator functions:

N I/Y PV PMT FV

Enter four of the five keys, and solve the 5th one.

*Excel functions:

Future value FV

FV(rate, nper, pmt, pv)

Present value PV

PV(rate, nper, pmt, fv)

Number of periods nper

nper(rate, pmt, pv, fv)

Interest rate rate

rate(nper, pmt, pv, fv)

Payment pmt

pmt(rate, nper, pv, fv)

FV, PV, rate, nper, pmt

Enter four of the five parameters, and solve the 5th one.

Calculate Future Value

(1) Compute the future of $1000 compounded annually for 10 years at 6 percent.

1. Use the mathematical formula

To find the FV of a lump sum, we use:

FV = PV(1 + r)t

FV = $1,000(1.06)10 = $1,790.85

2. Use financial calculator

Enter 10 6 -1,000 0

N I/Y PV PMT FV

Solve for 1790.85

3. Use Excel Spreadsheet

FV(rate, nper, pmt, pv)

FV(0.06, 10, 0, -1000)= 1790.85

(2) Compute the future of $1000 compounded semiannually for 10 years at 6 percent.

1. Use the mathematical formula
To find the FV of a lump sum, we use:

?? = ??(1 +
?

?

)?×?

m=2, r =6%, t=10, PV=1000

FV = $1,000(1.03)20 = $1,806.11

2. Use financial calculator

Enter 20 3 -1,000 0

N I/Y PV PMT FV

Solve for 1806.11

3. Use Excel Spreadsheet
FV(rate, nper, pmt, pv)

FV(0.03, 20, 0, -1000)= 1806.11

Calculate Present Value

What is the present value of $500,000 to be received ten years from today if the discount rate is

6% annually?

1. Use the mathematical formula

To find the PV of a lump sum, we use:

?? =
??

(1+?)?

PV = 500000 / (1.06)10 = $279,197

2. Use financial calculator

Enter 10 6 0 500000

N I/Y PV PMT FV

Solve for -279,197

3. Use Excel Spreadsheet
PV(rate, nper, pmt, fv)

PV(0.06, 10, 0, 500000)= -279,197

Present value is $279,197.

Solve for the number of periods

How many years will it take for your initial investment of $7,752 to grow to $20,000 if it is

invested so that it earns 9% compounded annually?

1. Use the mathematical formula
FV = PV(1 + r)t

20,000 = 7,752(1.06)t = $1,790.85

FV=PV (1+i)n

N =ln (FV/PV) /ln (1+i)

N=ln (20,000/7,752) /ln (1.09)

N=11 years

2. Use financial calculator

Enter 9 -7,752 0 20,000

N I/Y PV PMT FV

Solve for 11

3. Use Excel Spreadsheet
nper(rate, pmt, pv, fv)

nper(0.09, 0, -7752, 20000)= 11

Solve for the interest rate, i

At what rate must your initial investment of $7,752 be compounded annually for it to grow to

$20,000 in 11 years?

1. Use the mathematical formula
FV = PV(1 + r)t

20,000 = 7,752(1+r)11

(1+r)11 = 2.58

1 + ? = √2.58
11

= 1.09

? = 0.09

2. Use financial calculator

Enter 11 -7,752 0 20,000

N I/Y PV PMT FV

Solve for 9

3. Use Excel Spreadsheet
rate(nper, pmt, pv, fv)

rate(11, 0, -7752, 20000) = 9%

Answer: The interst rate is 9%.

Orninary Annuity

Future value of an ordinary annuity:

??? = ??? [
(1 + ?)? − 1

?
]

Present value of an ordinary annuity:

??? = ??? [
1 −

1
(1 + ?)?

?
]

Annuity Due

Future value of an ordinary annuity:

??? = ??? [
(1 + ?)? − 1

?
] (1 + ?)

Present value of an ordinary annuity:

??? = ??? [
1 −
1
(1 + ?)?
?
] (1 + ?)

Perpetuities

PV of level perpetuity

?? =
???

?

PV of growing perpetuity

?? =
???

? − ?

Effective Annual Rate: EAR

APR: Annual Percentage Rate, Quoated Annual Rate

m: compounding periods per year

??? = (1 +
???

?
)

?

− 1

Ordinary Annuity Examples

(1) You’ve taken your first job and you plan to same $5000 each year for the next five years for

your grad school fund. How much money will you accumulate by the end of year five? The

rate of interest is 6% annually.

1. Use the mathematical formula
??? = ??? [
(1 + ?)? − 1
?
]

??? = 5000 × [
(1 + 0.06)5 − 1

0.06
] = 5000 × 5.63709296 = 28,185.46

2. Use financial calculator

Enter 5 6 0 -5000

N I/Y PV PMT FV

Solve for 28,185.46

2. Use Excel Spreadsheet

FV(rate, nper, pmt, pv)

FV(0.06, 5, -5000, 0) = 28,185.46

Answer: The prevent value is $28,185.46.

Ordinary Annuity Examples

(2) How much of the annual payment must you deposit in a savings account earning 6% annual

interest in order to accumulate $5000 at then end of 5 years?

1. Use the mathematical formula
??? = ??? [
(1 + ?)? − 1
?
]

5000 = ??? × [
(1 + 0.06)5 − 1

0.06
]

5000 = ??? × 5.63709296

??? = 886.98
2. Use financial calculator

Enter 5 6 0 5000

N I/Y PV PMT FV

Solve for -886.98

3. Use Excel Spreadsheet
pmt(rate, nper, pv, fv)

pmt(0.06, 5, 0, 5000) = -$886.98

Answer: The annual payment is $886.98.