23-06-2024

Aman Singhania, CFA, FRM

Aman Singhania, CFA, FRM

Quantitative methods Time Value of Money
- CFA level 1

Presented by: Aman Singhania, CFA, FRM Presented by: Aman Singhania, CFA, FRM

Aman Singhania 1 Aman Singhania 2

Aman Singhania, CFA, FRM

Aman Singhania, CFA, FRM

Time Value of Money Time Value of Money (TVM)

What is Time value of Money

Nominal risk-free rate = real risk-free rate + expected inflation rate TVM

Securities may have one or more types of risk, and each added risk increases
Future Value Present Value
the required rate of return on the security.

These types of risk are: FV of FV of a series of CFs PV of PV of a series of CFs

single CF (a.k.a. Annuity) single CF (a.k.a. Annuity)
•Default risk.
•Liquidity risk
•Maturity risk Ordinary Annuity Ordinary Annuity

Required interest rate on a security = nominal risk-free rate + default risk Annuity Due Annuity Due
premium + liquidity premium + maturity risk premium

Aman Singhania 3 Aman Singhania 4

1
 23-06-2024

Aman Singhania, CFA, FRM

Aman Singhania, CFA, FRM

FV of single CF Future Value of a single cash flow – example 1

Calculate the future value of $100 invested today for the next 3 years in a bank
account that pays 10% per annum
TVM FV = $133.10
$100 FV = ?

Future Value Present Value

10% 10% 10%
0 1 2 3
FV of FV of a series of CFs PV of PV of a series of CFs N=3
single CF (a.k.a. Annuity) single CF (a.k.a. Annuity)
I/Y = 10
PV = -100
Ordinary Annuity Ordinary Annuity PMT = 0
FV = ?

Annuity Due Annuity Due

CPT → FV = 133.10

Aman Singhania 5 Aman Singhania 6

Aman Singhania, CFA, FRM

Aman Singhania, CFA, FRM

Future Value of a single cash flow – example 2 Future Value of a single cash flow – example 3

Calculate the future value of $1M invested today for the next 6 years in a fixed Calculate the future value of $5000 invested today for the next 2 years in an
deposit that pays 6.5% per annum investment that pays 2% per quarter
FV = $1.45914M FV = $5858.29
$1M FV = ? $5000 FV = ?

6.5% 6.5% 6.5% 6.5% 6.5% 6.5% 2% 2% 2% 2% 2% 2% 2% 2%

0 1 2 3 4 5 6 0 1 2 3 4 5 6 7 8

N=6 N = 2 years = 8 quarters = 8

I/Y = 6.5 I/Y = 2
PV = -1M PV = -5000
PMT = 0 PMT = 0
FV = ? FV = ?

CPT → FV = 1.45914 CPT → FV = 5858.29

Aman Singhania 7 Aman Singhania 8

2
 23-06-2024

Aman Singhania, CFA, FRM

Aman Singhania, CFA, FRM

Future Value of a single cash flow – example 4 PV of single CF

Calculate the future value of $7500 invested today for the next 4 years in an
investment that pays 10% per annum compounded semiannually
FV = $11080.9158 TVM
$7500 FV = ?

5% 5% 5% 5% 5% 5% 5% 5%
Future Value Present Value
0 1 2 3 4 5 6 7 8

N = 4 years = 8 periods FV of FV of a series of CFs PV of PV of a series of CFs

single CF (a.k.a. Annuity) single CF (a.k.a. Annuity)
I/Y = 5
PV = -7500
PMT = 0 Ordinary Annuity Ordinary Annuity

FV = ?

Annuity Due Annuity Due

CPT → FV = 11080.9158

Aman Singhania 9 Aman Singhania 10

Aman Singhania, CFA, FRM

Aman Singhania, CFA, FRM

Present Value of a single cash flow – example 1 Present Value of a single cash flow – example 2

Calculate the present value of $100 expected to be received after 3 years. Calculate the present value of $1M expected to be received after 6 years.
Interest return = 8% per annum Interest return = 6.5% per annum
PV = $79.3832 PV = $ 0.68533 M
PV = ? $100 PV = ? $1M

6.5% 6.5% 6.5% 6.5% 6.5% 6.5%

8% 8% 8%
0 1 2 3 0 1 2 3 4 5 6
N=3 N=6
I/Y = 8 I/Y = 6.5
PV = ? PV = ?
PMT = 0 PMT = 0
FV = 100 FV = $1M

CPT → PV = -79.3832 CPT → PV = -0.68533

Aman Singhania 11 Aman Singhania 12

3
 23-06-2024

Aman Singhania, CFA, FRM

Aman Singhania, CFA, FRM

Present Value of a single cash flow – example 3 Present Value of a single cash flow – example 4

Calculate the present value of $5000 expected to be received after 2 years Calculate the present value of $9000 expected to be received after 4 years.
Interest return = 3% per quarter Interest rate = 12% per annum compounded semiannually
PV = $3947.047 PV = $5646.7113
PV = ? $5000 PV = ? $9000

3% 3% 3% 3% 3% 3% 3% 3% 6% 6% 6% 6% 6% 6% 6% 6%
0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8

N = 2 years = 8 quarters = 8 N = 4 years = 8 periods

I/Y = 3 I/Y = 6
PV = ? PV = ?
PMT = 0 PMT = 0
FV = 5000 FV = 9000

CPT → PV = 3947.047 CPT → PV = 5646.7113

Aman Singhania 13 Aman Singhania 14

Aman Singhania, CFA, FRM

Aman Singhania, CFA, FRM

Annuities Annuities

A series of equal cash-flows at regular intervals

TVM Types of Annuities

1. Ordinary annuity – End of the period
2. Annuity Due – Beginning of the period
Future Value Present Value

FV of FV of a series of CFs PV of PV of a series of CFs

single CF (a.k.a. Annuity) single CF (a.k.a. Annuity)

Ordinary Annuity Ordinary Annuity

Annuity Due Annuity Due

Aman Singhania 15 Aman Singhania 16

4
 23-06-2024

Aman Singhania, CFA, FRM

Aman Singhania, CFA, FRM

Annuities Ordinary Annuity – example 1 - Future Value

Calculate the future value of ordinary annuity that deposits $100 per year at the end of
each year for the next 3 years in a bank account that pays 10% per annum
FV = $331
TVM FV = ?
-$100 -$100 -$100

Future Value Present Value

10% 10% 10%
0 1 2 3
FV of FV of a series of CFs PV of PV of a series of CFs N=3
single CF (a.k.a. Annuity) single CF (a.k.a. Annuity)
I/Y = 10
PV = 0
Ordinary Annuity Ordinary Annuity PMT = -100
FV = ?

Annuity Due Annuity Due

CPT → FV = 331

Aman Singhania 17 Aman Singhania 18

Aman Singhania, CFA, FRM

Aman Singhania, CFA, FRM

Ordinary Annuity - example 2 – Present Value Ordinary Annuity – example 3 – future & present value

Calculate the present value of ordinary annuity that pays $100 per year at the end of Calculate the future & present value of an ordinary annuity of $5000 deposited
each year for the next 3 years. interest rate = 10% per annum at the end of each quarter for the next 2 years. Interest return = 3% per quarter
FV = $331
PV = $248.6852
PV = $ 35098.46 FV = $44461.68
PV = ? FV = ?
PV = ?
$100 $100 $100 $5000 $5000 $5000 $5000 $5000 $5000 $5000 $5000

10% 10% 10% 3% 3% 3% 3% 3% 3% 3% 3%

0 1 2 3 0 1 2 3 4 5 6 7 8

N=3
N = 2 years = 8 quarters = 8 N = 2 years = 8 quarters = 8
I/Y = 10
I/Y = 3 I/Y = 3
PV = ?
PV = ? PV = 0
PMT = 100
PMT = -5000 PMT = -5000
FV = 0
FV = 0 FV = ?

CPT → PV = -248.6852
CPT → PV = 35098.46 CPT → FV = 44461.68

Aman Singhania 19 Aman Singhania 20

5
 23-06-2024

Aman Singhania, CFA, FRM

Aman Singhania, CFA, FRM

Annuities Annuity Due – example 1 - Future Value

Calculate the future value of annuity due that deposits $100 per year at the beginning of
each year for the next 3 years in a bank account that pays 10% per annum
FV = $364.10
TVM FV = ?
-$100 -$100 -$100

Future Value Present Value

10% 10% 10%
0 1 2 3
FV of FV of a series of CFs PV of PV of a series of CFs N=3
single CF (a.k.a. Annuity) single CF (a.k.a. Annuity)
I/Y = 10
PV = 0
Ordinary Annuity Ordinary Annuity PMT = -100
FV = ?

Annuity Due Annuity Due

CPT → FV = 364.10 OR FVB = FV E x (1 + r %) = 331 x ( 1 + 0.10) = 364.10

Aman Singhania 21 Aman Singhania 22

Aman Singhania, CFA, FRM

Aman Singhania, CFA, FRM

Annuity Due – example 2 – present value Annuity – example

Calculate the present value of annuity due that pays $100 per year at the beginning of What is the present value of four $100 end-of-year payments if the first payment is to be
each year for the next 3 years. Interest rate = 10% per annum received three years from today and the appropriate rate of return is 9% per annum?
PV = $ 273.55
PV = ? PV = $ 272.68
PV = ? PV = $ 323.97
$100 $100 $100 PV = ?
$100 $100 $100 $100

9% 9% 9% 9% 9% 9%
10% 10% 10%
0 1 2 3 0 1 2 3 4 5 6
N=3 N=2 N=4
I/Y = 10 I/Y = 9 I/Y = 9
PV = ? PV = ? PV = ?
PMT = 100 PMT = 0 PMT = -100
FV = 0 FV = 323.97 FV = 0

CPT → PV = -273.55 OR PVB = PV E x (1 + r %) = 248.6852 x ( 1 + 0.10) = 273.55 CPT → PV = -272.68 CPT → PV = 323.97

Aman Singhania 23 Aman Singhania 24

6
 23-06-2024

Aman Singhania, CFA, FRM

Aman Singhania, CFA, FRM

Present value of a perpetuity Present value of a perpetuity – example 1

• A perpetuity is a financial instrument that pays a fixed amount of money at Kodon Corporation issues preferred stock that will pay $4.50 per year in annual
set intervals over an infinite period of time. dividends beginning next year and plans to follow this dividend policy forever.
• In essence, a perpetuity is a perpetual annuity Given an 8% rate of return, what is the value of Kodon's preferred stock
today?

Answer:
Given that the value of the stock is the present value of future dividends, we
have:

= 4.50 / 0.08 = $56.25

Aman Singhania 25 Aman Singhania 26

Aman Singhania, CFA, FRM

Aman Singhania, CFA, FRM

Rate of return for a perpetuity – example 2 Loan Amortization

Using the Kodon preferred stock described in the preceding example, determine Construct an amortization schedule to show the interest and principal components of
the end-of-year payments for a 10%, 5-year, $10,000 loan.
the rate of return that an investor would realize if she paid $75 per share for
the stock?
$10000 X X X X X
Answer:
10% 10% 10% 10% 10%
0 1 2 3 4 5

N=5
I/Y = 10
PV = 10000
PMT = ?
FV = 0

CPT → PMT = 2637.97

Aman Singhania 27 Aman Singhania 28

7
 23-06-2024

Aman Singhania, CFA, FRM

Aman Singhania, CFA, FRM

Loan Amortization Schedule TVM – example 1

Construct an amortization schedule to show the interest and principal components of At an expected rate of return of 7%, how much must be deposited at the end
the end-of-year payments for a 10%, 5-year, $10,000 loan. of each year for the next 15 years to accumulate $3,000?

N = 15
I/Y = 7
PV = 0
PMT = ?
FV = 3000

CPT → PMT = -119.38

Aman Singhania 29 Aman Singhania 30

Aman Singhania, CFA, FRM

Aman Singhania, CFA, FRM

TVM – example 2 TVM – example 3

How many $100 end of year payments are required to accumulate $920 if the Suppose you have the opportunity to invest $100 at the end of each of the
discount rate is 9% per annum? next 5 years in exchange for $600 at the end of the fifth year. What is the
annual rate of return on this investment?
N=? N=5
I/Y = 9 I/Y = ?
PV = 0 PV = 0
PMT = -100 PMT = -100
FV = 920 FV = 600

CPT → N = 7 CPT → I/Y = 9.13

Aman Singhania 31 Aman Singhania 32

8
 23-06-2024

Aman Singhania, CFA, FRM

Aman Singhania, CFA, FRM

TVM – example 4 TVM – example 5

What rate of return you earn on an ordinary annuity that requires a $700 How many years will it take for an investment of $1000 to grow to $2000 at
deposit today and promises to pay $100 per year at the end of each of the an annual compound rate of 14.87%?
next ten years?
N = 10 N=?
I/Y = ? I/Y = 14.87
PV = -700 PV = -1000
PMT = 100 PMT = 0
FV = 0 FV = 2000

CPT → I/Y = 7.07% CPT → N = 4.99999

Aman Singhania 33 Aman Singhania 34

Aman Singhania, CFA, FRM

Aman Singhania, CFA, FRM

TVM – example 6 Effective Annual Rate (EAR)

Suppose you must make five annual $1000 payments, the first one starting at the EAR represents the annual rate of return actually being earned after adjustments have
beginning of year 4 (end of year 3). To accumulate the money to make these payments, been made for different compounding periods
you want to make three equal payments into an investment account, the first to be made
one year from today. Assuming a 10% rate of return, what is the amount of these three
payments? For example 12% per annum compounded quarterly

EAR = (1 + r ) f - 1
f
With BGN mode Where,
With END mode
N=5 r = stated annual return
N=3
I/Y = 10 f = frequency of compounding
I/Y = 10
PV = ?
PV = 0
PMT = -1000
PMT = ?
FV = 0
FV = 4169.87

CPT → PV = 4169.87
CPT → PMT = 1260

Aman Singhania 35 Aman Singhania 36

9
 23-06-2024

Aman Singhania, CFA, FRM

Effective Annual Rate (EAR) – Example 1

Compute EAR if the stated annual rate is 12%, compounded quarterly.

EAR = (1 + r ) f - 1
f

EAR = (1 + 12% ) 4 - 1
4
EAR = (1 + 3%) 4 – 1 = 1.034 – 1 = 1.1255 -1 = 0.1255 = 12.55%

Aman Singhania 37

10