SEEM2440
Engineering Economic
Lecture 03 – The Time Value of Money
Present Value, Future Value and Annuity
  Suppose you lend $100,000 to your friend, and he promises to
repay the loan and interest after 9 years at 10% interest rate,
how much you can receive after 9 years?
$F
What is this amount?
Time 0 is the base year
…
0
1
$100,000
You lend $100,000 to your friend.
Copyright (c) 2015. Gabriel Fung. All rights reserved.
2
9
This is a cash flow diagram
Present Value, Future Value and Annuity (cont’d)
  Suppose you lend $100,000 to your friend, and he promises to
repay the loan and interest to you at 10% interest rate in 9
years so that each year he will pay you $A dollars. How much
you would expect to receive in every year?
$A
$A $A
What is this amount?
…
0
1
2
9
$100,000
You lend $100,000 to your friend.
Copyright (c) 2015. Gabriel Fung. All rights reserved.
Present Value, Future Value and Annuity (cont’d)
Future Value
Annuity
$F
$A
$A $A
…
0
1
2
$P
Present Value
Copyright (c) 2015. Gabriel Fung. All rights reserved.
…
0
9
1
$P
2
9
Interest Rate
  Solving the previous problems require an understanding and
precious definition of “interest rate”
  Simple interest rate
  Compound interest rate
  Nominal interest rate
  Effective interest rate
Copyright (c) 2015. Gabriel Fung. All rights reserved.
Simple Interest
  The total interest earned is linearly proportional to the
principal (i.e., loan).
  The interest do not accumulate
I = (P)(N)(i)
P = Principle
N = number of periods (e.g., year)
i = interest rate per interest period
  Question:
  How much interest we need to pay if we borrow $1000 for 3
years with an annual simple interest rate of 10%?
Copyright (c) 2015. Gabriel Fung. All rights reserved.
Compound Interest
  The interest earned for any period is based on the remaining
principal amount plus any accumulated interest charges up to
the beginning of that period.
N
I = (Pi)∑ (1+ i )
n−1
=P[(1+ i)N − 1]
n=1
  Question:
  What is the interest we need to pay if we have borrowed $1000
for 3 years with an annual interest rate of 10%?
Year 1
Year 2
Year 3
Borrowed
$1,000
$1,100
$1,210
Interest
$100
$110
$121
Copyright (c) 2015. Gabriel Fung. All rights reserved.
Compound Interest (cont’d)
  Question:
  Suppose interest is accumulated every six months (twice a year),
what is the interest we need to pay if we have borrowed $1000
for 3 years with a 5% interest rate for every six months?
Year 0.5
Year 1
Year 1.5
Year 2
Year 2.5
Year 3
Borrowed
$1,000
$1,050
$1152.5
$1207.6
…
…
Interest
$50
$52.5
$55.1
$60.4
…
…
  Note that the interest we have to pay is more than 10% a year!
Copyright (c) 2015. Gabriel Fung. All rights reserved.
Nominal and Effective Interest Rate
  Nominal Interest Rate
  For the shake of convenient, even if the interest period is not
equal to 1 year, we usually still quote the interest rates on an
annual basis but followed by the compounding period.
  Example:
  If the interest period is 6 months and the interest rate during this
period is 6%, then we will say: “The interest rate is 12%
compounded semiannually.”
  12% is called “Nominal Interest Rate” and is usually denoted by r.
Copyright (c) 2015. Gabriel Fung. All rights reserved.
Nominal and Effective Interest Rate (cont’d)
  Effective Interest Rate
  Effective interest rate is the “truth” interest rate for 1 year. It
MUST BE expressed in a yearly basic.
  Example:
  If the interest rate is 12% compounded semiannually, then the
“real interest” is 12.36%.
  The 12.36% is known as “Effective Interest Rate”, and is usually
denoted by i.
  The relationship between i and r is (assume M compounding
period in a year):
M
⎛
r⎞
i = ⎜ 1+ ⎟ − 1
⎝ M⎠
Copyright (c) 2015. Gabriel Fung. All rights reserved.
Nominal and Effective Interest Rate (cont’d)
  The differences of some interest rates:
  7% nominal rate = 7.23% effective rate
  18% nominal rate = 19.25 effective rate
Copyright (c) 2015. Gabriel Fung. All rights reserved.
Nominal and Effective Interest Rate (cont’d)
  A credit card company claim that they charge their customers
only 1.375%/month for any unpaid balance. What is the
actual rate (effective rate) they charge?
  Answer:
  17.81% per year!
Copyright (c) 2015. Gabriel Fung. All rights reserved.
Nominal and Effective Interest Rate (cont’d)
  In the rest of the slides, unless otherwise specified, all interest
rates are “Effective Interest Rates”.
Copyright (c) 2015. Gabriel Fung. All rights reserved.
The Concept of Equivalence
  A bank wants to design the following three payment options
for their customers if they borrow $1M from her at 15%
interest rate.
  Plan 1: Pay principal and interest in one payment at Year 10.
  Plan 2: Pay the debt in equal payments for 15 years.
  Plan 3: Pay interest at the end of each year for 4 years and
principal at Year 4.
  What should be the payment for each plan?
Copyright (c) 2015. Gabriel Fung. All rights reserved.
Formulas
  Consider Plan 1 in the previous slide:
$P
1
2
…
10
0
$F
  We need to do identify $F given $P ($1M), N (10 years) and i
(15%). Hence:
  $F = $P × f (F | P, i%, N) where f (F | P, i%, N) is a function
that can help us to identify the necessary component for
computing $F when $P, i% and N are given.
  $F = $P × f (F | P, i%, N) = $P × (1 + i%)N
Copyright (c) 2015. Gabriel Fung. All rights reserved.
Formulas (cont’d)
  Consider Plan 3 in the previous slide:
$P
End of each year
1
2
…
15
0
$A
$A
$A
  We need to do identify $A given $P ($1M), N (15 years) and i
(15%). Hence:
  $A = $P × f (A | P, i%, N) where f (A | P, i%, N) is a function
that can help us to identify the necessary component for
computing $A when $P, i% and N are given.
Copyright (c) 2015. Gabriel Fung. All rights reserved.
Formulas (cont’d)
  Up to now, we have Present Value (P), Future Value (F) and
Annuity (A), hence we will have six equation:
  1. Identify the component for computing P, given F, i%, and N
  2. Identify the component for computing F, given P, i%, and N
  3. Identify the component for computing A, given P, i%, and N
  4. Identify the component for computing P, given A, i%, and N
  …
Copyright (c) 2015. Gabriel Fung. All rights reserved.
Formulas (cont’d)
(F | P,i%,N) = (1+ i)N
(P | F,i%,N) =
1
(1+ i)N
(1+ i)N − 1
i
i
(A | F,i%,N) =
(1+ i)N − 1
(F | A,i%,N) =
Single payment compound amount
Single payment present worth
Unifrom series compound amount
Uniform series present worth
(1+ i)N − 1
(P | A,i%,N) =
i(1+ i)N
Sinking fund
i(1+ i)N
(A | P,i%,N) =
(1+ i)N − 1
Capital recovery
Copyright (c) 2015. Gabriel Fung. All rights reserved.
Formulas (cont’d)
  The previous six equations is enough for us to solve most of
the problem:
1.  Draw a cash flow diagram
2.  Decompose the cash flow diagram such that it can be solved by
some of the previous equations
  Note:
  The interest rates for all equations are “effective interest rates”
Copyright (c) 2015. Gabriel Fung. All rights reserved.
A Loan Plan
  Suppose you lend $8,000 to your friend, and he promises to
repay the loan and interest after 9 years at 10% interest rate,
how much you can receive after 9 years?
  Cash flow diagram:
What is this amount?
0
1
2
$F
We need to find:
The future worth = F
We are given:
P = $8,000, i = 10%, N = 9 years
So we use:
F = $8,000 × (F / P,i%,N)
…
9
= $8,000 × (F / P,10%,9)
= $8,000(1+ i)N
$P = $8,000
= $8,000(1+ 10%)9
= $18863.58
Copyright (c) 2015. Gabriel Fung. All rights reserved.
An Investment Decision
  How much money you need to put into the bank now, with
annual interest rate 7%, in order to get $1,000,000 in 45 years?
  Cash flow diagram:
$F = $1,000,000
0
1
2
…
30
$P
Answer: P = $47,613
Copyright (c) 2015. Gabriel Fung. All rights reserved.
Purchasing a Land
  An investor has an option to purchase a land that will be
worth $10,000 in six years. If the value of the land increases at
8% each year, how much should the investor be willing to pay
now?
  Cash flow diagram:
Answer: P = $6,302
Copyright (c) 2015. Gabriel Fung. All rights reserved.
An Investment Plan
  If you deposit $1,000 in a bank every year starting from the
next year. How much would this amount be after 15 years if
the interest rate is 5% p.a.?
  Cash flow diagram:
$F
1
0
2
14
15
…
Note
When using (F / A, i%, N), the last
deposit and F are coincident at the
same time. Also, A must begin from
the first year (not now).
A = $10,000
Answer: F = $21,578.60
Copyright (c) 2015. Gabriel Fung. All rights reserved.
Become A Millionaire in 30 Years
  How much do you need to invest every year if you want to
yield $1,000,000 in 45 years? Assume the interest rate is 7%.
  Cash flow diagram:
Answer: A = $3,500
Copyright (c) 2015. Gabriel Fung. All rights reserved.
Become a Millionaire By Saving $1 a Day
  If you invest $1 each day, how many years would the money
become $1,000,000? Assume that the interest rate is 10%.
  Cash flow diagram:
Answer: N = 58.93
Copyright (c) 2015. Gabriel Fung. All rights reserved.
Become a Millionaire By Saving $1 a Day (cont’d)
  The Excel function used to solve for N is
  NPER(rate, pmt, pv), which will compute the number of
payments of magnitude pmt required to pay off a present
amount (pv) at a fixed interest rate (rate).
Copyright (c) 2015. Gabriel Fung. All rights reserved.
Borrowing Money
  Your friend borrow money from you and agrees to pay you
$20,000 each year with annual interest rate of 15% for 5 years.
How much money should you lend to him?
  Cash flow diagram:
Answer: P = $67,044
Copyright (c) 2015. Gabriel Fung. All rights reserved.
Mortgage Plan
  You want to buy an apartment at the price of $4,000,000. You
will do this with a mortgage from a Bank at (P – 2%) for 30
years. What is your monthly payment?
  Cash flow diagram:
Answer: A = $X (Monthly payment = $X/12)
Copyright (c) 2015. Gabriel Fung. All rights reserved.
When N is large…
  For (A | P, i%, N):
i(1+ i)N
(A | P,i%,N) =
(1+ i)N − 1
  When N is large, the annual payment will be less and less, and
eventually:
A ≈ Pi
Copyright (c) 2015. Gabriel Fung. All rights reserved.
More About the Formulas
  For most of the problems, they are very complex, and we have
to “combine the formulas” and “reformulate the cash flow
diagram” to solve these problems.
  That’s why cash flow diagram is very useful.
Copyright (c) 2015. Gabriel Fung. All rights reserved.
Investment Plan
  A father, on the day his son is born, wishes to determine what
amount would have to be put into an account with 12%
interest rate, so that he can withdraw $2,000 during his son’s
18th, 19th, 20th and 21st birthdays.
  Cash flow diagram:
$F’
$A = $2,000
$F’
$A = $2,000
Step 1
0
1
2
… 17
Step 2
0
18
19
20
21
17
18
19
1
2
…
17
21
20
$P
$P
Answer: P = $884.46
Copyright (c) 2015. Gabriel Fung. All rights reserved.
A Loan Question
  Smith borrowed $4,000 four years ago when the interest rate
was 4.06%/year. $5,000 was borrowed three years ago at
3.42%/year. Two years ago, she borrowed $6,000 at 5.23%/
year, and last year $7,000 was borrowed at 6.03%/year. Now
she want to consolidate her debt into a single 20-year loan
within 5% fixed annual interest rate. If Smith makes annual
payment (starting in one year later) to repay her total debt,
what is the amount of each payment?
Copyright (c) 2015. Gabriel Fung. All rights reserved.
A Loan Question (cont’d)
  Cash flow diagram:
Copyright (c) 2015. Gabriel Fung. All rights reserved.
Mortgage Plans
  Given the following plans:
  Bank 1: (P – 2%) for the whole duration
  Bank 2: (P – 0.5%) for the whole duration + 4% cash rebate
now.
  Bank 3: (P – 2.5%) for the fist 15 years + (P – 1.5%) for the rest
of the time
  Developer: (P – 2%) for the whole duration for the first 70% of
the price + (P + 2%) for the rest of the purchasing price
  Which plan should you choose if you need to borrow
$5,000,000 mortgage for 30 years? Assume the prime rate P is
fixed at 5%.
Copyright (c) 2015. Gabriel Fung. All rights reserved.
Mortgage Plans (Cont’d)
  Cash flow diagrams:
Copyright (c) 2015. Gabriel Fung. All rights reserved.
Two Saving Plans
  Plan 1: You have to deposit $2,000 each year when you are 32
until 65. The annual interest rate is 10%.
  Plan 2: You have to deposit $1,000 each year for 15 years. The
interest rate is 10%. You can take your money out when you
are 65. You can join this plan when you are 22.
Answer: $444,500 vs. $504,010
Copyright (c) 2015. Gabriel Fung. All rights reserved.
Equivalent Cash Flow Diagram
  Determine Q in terms of H, so as to make both cash flow
diagram equivalent
Copyright (c) 2015. Gabriel Fung. All rights reserved.
Uniform Gradient Series
  Sometimes, we face the problems that involve receipts or
expenses that are projected to increase or decrease by a uniform
amount each period.
(N - 1)G
(N - 2)G
(N - 3)G
3G
..
.
..
.
..
.
N-2
N-1
N
2G
G
0
1
2
3
4
…
Both Time 0 and Time 1 DO NOT contain any cash flows
Copyright (c) 2015. Gabriel Fung. All rights reserved.
Uniform Gradient Series (Cont’d)
  The formulas for the uniform gradient series:
(F | G,i%,N) = (F | A,i%,N − 1) + (F | A,i%,N − 2) +!+ (F | A,i%,1)
(1+ i)N−1 − 1 (1+ i)N−2 − 1
(1+ i)1 − 1
+
+!+
i
i
i
N−1
1
N
= ∑ (1+ i)n −
i n=0
i
=
1
N
[(1+ i)N − 1] −
2
i
i
(P | G,i%,N) = (F | G,i%,N)(P | F,i%,N)
=
1 ⎡1 ⎛
1⎞ 1 ⎤
= ⎢ −⎜N + ⎟
⎥
i ⎣i ⎝
i ⎠ (1+ i)N ⎦
(A | G,i%,N) = (F | G,i%,N)(A | F,i%,N)
⎡1
⎤
N
=⎢ −
⎥
N
⎣ i (1+ i) − 1 ⎦
Copyright (c) 2015. Gabriel Fung. All rights reserved.
A Simple Example
  Suppose the expenses are: $1,000 for the second year, $2,000
for the third year and $3,000 for the forth year. What is the
present worth of the expenses if the interest rate is 15%?
  Cash flow diagram:
Answer: P = $3,790 / A = $1,326.3
Copyright (c) 2015. Gabriel Fung. All rights reserved.
A Complicated Example
  A project will have expenses $8,000, $7,000, $6,000 and
$5,000 in Year 1, 2, 3 and 4, respectively. The interest rate is
15%. What is the total present expense of the project?
  Cash flow:
0
1
2
3
4
$5,000
$6,000
$7,000
$8,000
$P
Answer: P = -$19,050
Copyright (c) 2015. Gabriel Fung. All rights reserved.
Geometric Gradient Series
  Similar to the idea of uniform gradient series, the geometric
gradient series is used to model the situation that a fixed
percentage is increased in every period.
  The following table presents a geometric gradient series. It
begins at the end of year 1 and has a rate of growth 20%.
End of Year
Cash Flows ($)
1
1,000
2
1,200
3
1,440
4
1,728
Copyright (c) 2015. Gabriel Fung. All rights reserved.
Geometric Gradient Series (cont’d)
  Find P when you are given A:
P = A1 (P | F,i%,1) + A 2 (P | F,i%,2) +!+ A N (P | F,i%,N)
= A1 (1+ i)−1 + A 2 (1+ i)−2 +!+ A N (1+ i)− N
= A1 (1+ i)−1 + A1 (1+ f )(1+ i)−2 +!+ A N (1+ f )N−1 (1+ i)− N
N
= A1 ∑ (1+ f )n−1 (1+ i)− n
n=1
=
A1 N ⎛ 1+ f ⎞
∑
1+ i n=1 ⎜⎝ 1+ i ⎟⎠
n−1
⎧ A1[1− (1+ i)− N (1+ f )N ]
⎪
f ≠i
i− f
⎪
=⎨
A1N
⎪
f =i
⎪
(1+ i)
⎩
⎧ A1[1− (P / F,i%,N)(F / P,i%,N)]
⎪
i− f
=⎨
⎪
A1N(P / F,i%,1)
⎩
f ≠i
f =i
Copyright (c) 2015. Gabriel Fung. All rights reserved.
A Simple Example
  A company predicted that the revenues this year are $1.1
million, and will increase 15% per year for the next 5 years.
What are the present value and equivalent annual amount for
the anticipated revenues if the interest rate is 20%?
  Solution:
  Use the geometric gradient formula to find the present value,
then convert the present amount to an annual amount.
P0 =
$1, 100, 000[1
(P/F, 20%, 5)(F/P, 20%, 5)]
= $4, 216, 974
0.20 0.15
A = $4, 216, 974(A/P, 20%, 5) = $1, 410, 071
Copyright (c) 2015. Gabriel Fung. All rights reserved.
Continuous Interest Rate
  So far, we assume that the interest period is a fixed interval
  E.g. 1 year, 6 months, etc.
  How about if…
  The interest is accumulated in every minute, every second, every
mini-second,... Or “continuously” (The interest period is
toooooooooo short)?
  Question:
  If we borrowed $1,000 with a nominal interest rate of 20%
compound continuously, then what should we paid 10 years later?
Copyright (c) 2015. Gabriel Fung. All rights reserved.
Continuous Interest Rate (cont’d)
  Let the nominal interest rate be r. Assume that there are M
periods.
  For one year:
F = P(F | P,
For N years:
r
%,M)
M
⎛
r⎞
= P ⎜ 1+ ⎟
⎝ M⎠
F = Pe rN
M
M
⎛
r⎞
= lim P ⎜ 1+ ⎟ ,since M is continuous
M→∞ ⎝
M⎠
= Pe r
Copyright (c) 2015. Gabriel Fung. All rights reserved.
Formulas for Continuous Compounding
(F | P,r%,N) = e rN
1
e rN
e rN − 1
(F | A,r%,N) = r
e −1
er − 1
(A | F,r%,N) = rN
e −1
e rN − 1
(P | A,r%,N) = rN r
e (e − 1)
(P | F,r%,N) =
e rN (e r − 1)
(A | P,r%,N) = rN
e −1
Single payment compound amount
Single payment present worth
Unifrom series compound amount
Uniform series present worth
Sinking fund
Capital recovery
Copyright (c) 2015. Gabriel Fung. All rights reserved.
Compounded Quarterly
  If $100 is invested for 10 years at a nominal interest rate of 6%
compounded quarterly. How much is it worth at the end of
the 10th year?
  Solution 1:
  There are four compounding periods per year.
  Total interest periods are 40 (4 x 10)
  The interest rate per interest period is (6% / 4) = 1.5%
  So, F = P (F | P, 1.5%, 40) = …
  Solution 2:
  The effective interest rate is 6.14%.
  So, F = (F | P, 6.14%, 10) = …
Copyright (c) 2015. Gabriel Fung. All rights reserved.
Computing a Monthly Auto Payment
  There is a loan of $10,000 to be repaid in equal end-of-month
installments for five years with a nominal interest rate of 12%
compounded monthly. What is the amount of payment?
  Solution:
  Number of installment, N =
  Interest rate per month, i% =
  A = P ( A | P, i%, 60) =
Copyright (c) 2015. Gabriel Fung. All rights reserved.