2. Learning outcome
Understanding the time value of money.
Explain the methods of calculating present and
future values.
Future value and compounding
Computation of interest rate and effective rate of
interest
3. Why Time?
“A dollar in hand today is worth more than a dollar
promised at some time in the future.”
4. Time Value of money
Time preference for money is an individual’s preference
for possession of a given amount of money now, rather
than the same amount at some future time.
Three reasons may be attributed to the individual’s time
preference for money:
risk
preference for consumption
investment opportunities
Inflation / decrease in purchasing power
5. We want to invest Nu.100. What will you have after
two years, assuming the interest rate doesn’t
change? If you leave the entire Nu.100 in the bank,
you will earn Nu.110*.10=Nu.11 in interest during
the second year. So you will have a total of Nu.110
+11=Nu121. This Nu.121 is the future value of
Nu.100 in two years at 10 percent.
6. Type of Interest
Simple Interest
Interest paid (earned) on only at the original amount, or principal,
borrowed (lent).
Compound Interest
Interest paid (earned) on any previous interest earned, as well as on
the principal borrowed (lent).
7. Simple Interest
Formula SI = PRT/100
A=I+P
SI: Simple Interest
P: Deposit today
r: Interest Rate per Period
n: Number of Time Periods
8. Question one
Calculate the interest earned at the 6% for 5 year on the Principal
balance of Nu 1000. Using Simple interest and Compound Interest.
Ans
SI=Nu. 300
CI=Nu.1338.23
9. Some Terminology
PV = Present Value (Nu.)
FV = Future Value (Nu.)
t = of Periods
r = interest rate
N=number of years
10. Time value adjustment
Two most common methods of adjusting cash
flows for time value of money:
Compounding—the process of calculating future
values of cash flows and
Discounting—the process of calculating present
values of cash flows
11. Future value
Compounding is the process of finding the
future values of cash flows by applying the
concept of compound interest.
Compound interest is the interest that is
received on the original amount (principal) as
well as on any interest earned but not
withdrawn during earlier periods.
12. Timelines show timing of cash
flows.
Time 0 is today; Time 1 is the end of Period 1; or the beginning of
Period 2.
14. Timeline for uneven CFs: -Rs.50 at
t = 0 and Rs.100, Rs.75, and Rs.50
at the end of Years 1 through 3.
15. Future value
Future value of a cash flow today is the value of the funds invested at
your opportunity cost r
Say r=5%
If you invest 100 at an interest rate of 5%. How much will you have
at the end of the year
If you invest 100 at an interest rate of 5%. How much will you have
at the end of three years
15
16. Future Value (Example)
Suppose a bank pays a 10% interest rate per year and you are given a
choice between two plans:
A: Receive Nu. 1000 today
B: Receive Nu. 1000 one year from now
Which would you prefer, why?
17. The general form of equation for calculating the future value of a
lump sum after n periods may, therefore, be written as follows:
The term (1 + i)n is the compound value factor (CVF) of a lump
sum of Re 1, and it always has a value greater than 1 for positive
i, indicating that CVF increases as i and n increase.
n
n i
P
F )
1
(
= CVF
n n,i
F P
18. FV=PV(1+r)^t
Where:
FV= How much will I have in future?
PV= How much do I need to invest now?
r = What rate of the return do I need to
earn?
t = How long will it take me to reach my
goals?
20. Question 2
Suppose you invest Nu 100 in the bank at an interest rate of 7%? How
much will you have at the end of 10 years?
Ans: Nu. 196.72
Calculate the future value, Nu. 50,000 at the end of 3 years at 12% per
annum rate of interest
=70,250
21. Question 3
What is the future value of Nu.100 if interest is compound annually at
the rate of 6% for five years.
Answer=Nu.133.82
Suppose you invest Nu. 500 in the bank at an interest rate of 7%? How
much will you have at the end of 10 years?
22. Homework
Calculate the value for 5 years hence of a deposit of Nu.1000 made
today if interest rate is:
i. 8%
ii. 10%
iii. 12%
iv. 15%
23. If you deposited Rs.100 in a bank which
was paying 10% interest rate. What will
have after:
1) One year
2) Two year
3) Three year
Future value formula;
24. What is the value of a deposit to BOB of Nu.5000
at rate of 10% at 5 year?
Dechen has deposited 500 in saving account at
interest rate of 7% compounded annually. How
much will she get at the end of three years?
Sonam try to sell a land at Gedu. Yesterday he was
offered Nu.10,000. He was about to accept the
offer when another person offered him Nu.11,424.
However the second offer was to be paid a year
later. Rate is 12%
25. Class work
Dechen, a financial analyst at Patil Developers, a leading real estate
firm, is thinking about recommeding that Patil Developers invest in a
piece of land that costs Nu.85000. she is certain that next year the
land will be worth Nu.91000, a sure Nu.6000 gain. Given that the
interest rate in similar alternative investment is 10 percent, should
Patil Developers undertae the investment in land?
28. Present Value
What is the present value of receiving Nu 110
one year from now if the interest rate is 10%.?
Ans: Nu.100
Suppose you will inherit Nu.121,000 two years
from now and the interest rate is r = 10%.
What is the value today to you?
Ans: Nu.100,000
29. Assume that you need Nu.1,000 in 2 years. Let’s
examine the process to determine how much
you need to deposit today at a discount rate of
7% compounded annually?
Ans:Nu.873.44
30. You invest Nu.3000 today and get Nu.10,000 after 6 years. What is the
implicate interest rate.
Solution:
PV=Nu.3000
FV=Nu.10000
t= 6years
r=?
31.
32. Present Value and Discounting
Dema will receive Nu.10,000 three years from now. She can earn 8% on
her investment, so the appropriate discount rate is 8 percent. What is the
present value of her future cash flow.
Answer: Nu.7938
33. An investor deposited Nu. 100 in bank account
for 5 years at 8% interest rate. Find the
amount which he will have in his account, if
compounded:
i. Annually
ii. Semi -annually
iii. Quarterly
iv. Continuously
Ans 1. Nu.146.93 2. Nu.148.02 3. Nu.148.59
4. Nu.149.18
34. Continuous compounding
Formula for continuous compounding
=𝑐𝑜 × 𝑒𝑟𝑇
Where 𝑐𝑜 is the initial investment, r is the stated annual
interest rate. And T is the number of years over which the
investment runs.
The e is the constant and is approximately equal to 2.718.
it is not an unknown like 𝑐𝑜 ,r and T
35. Dechen invested Nu.1000 at continuously compounded rate of 10%
for one year. What is the value of her wealth at the end of one year?
=1105.20
36. I have a following choice of payment to be received
a)Received now Nu.1,00,000
b)Received NU. 1,50,000 after 2 year
c)Received Nu. 3,50,000 after 5 years
d)Received Nu. 30,000 after 1 year and 1,50,000
after 5 years
e)Received Nu. 10,000 now and 70,000 after 3 years
and 50,000 after 5 years
My required rate of return is 12%. Which is the best
choice?
38. Class work
I have following choice of payment to be made
1.Pay now 125000
2.Pay now 50000 and 150000 after 3 years
3.Pay 40000 at the end of year 1, year 2 and year 3
4.Pay 20000 now and 25000 in 3 year
Rate 13%
I have a choice to invest certain sum in an avenue which promises return from year 1 to 5 as
following. 40000,70,000,120000,130000 and 150000. if the rate is 12%. What is the maximum
amount you are going to pay to buy these investment
If the investment is available for the following amount
300000
344680
400000
Suggest whether we buy or not
39. Group work
1. Professional Artworks, Inc., is a firm that speculates in modern paintings. The manager is thinking of buying an original
Picasso for Nu.400,000 with the intention of selling is at the end of one year. The manager expects that the painting
will be worth Nu.480,000 in one year. Suppose the interest rate granted by banks is 10%. Should the firm purchase
the piece of art or not?
2. Carl Voigt, who recently won Nu.10,000 in the lottery, wants to buy a car in five years. Carl estimates that the car will
cost Nu.16105 at that time. What interest rate must he earn to be able to afford the car?
3. Yeshi deposit Nu.10,000 in a Bank of Bhutan for a period of 1 year. The bank offered two
options: --
(a) To receive interest at 12% p.a. compounded monthly or
(b) To receive interest at 12.25% p.a. compounded half-yearly.
Which option should Yeshi accept?
4. ABC Corporation offered some securities for sale to the public in March 2020 under the terms of the deal that the
company will repay the owners of the securities Nu. 100,000 only at the end of 30 years but investors would receive
nothing until 30 years. Investors paid Nu.24,099 for each of the securities.
a) Why would ABC Corporation willing to accept small amount today in exchange for a promise to repay in 30 years?
b) Would you be willing to pay Nu. 24,099 today in exchange for Nu.1,00,000 in 30 years if discount rate is 10%?
39
40. . Imprudential, Inc., has an unfunded pension liability of Nu.360 million that must be paid in 20 years. To assess
the value of the firm’s stock, financial analysts want to discount this liability back to the present. If the relevant
discount rate is 7.1 percent, what is the value of this liability?
(i) Bank of Bhutan pays 6.5 percent simple interest on its saving and Bhutan National bank pays 6.5 on
compounded annually. If you deposit Nu.6000 in each bank.
a) How much money would you earn from each bank at the end of 10 years
b) Which banks offer is better and how much more money would you earn?
c) Examine the reason for Time preference for money.
40
42. What is Perpetuity ?
Perpetuity refers to an infinite amount of time. In finance, perpetuity is a constant
stream of cash flows with no end.
example: British bond called “consols” . An investor purchasing a consol is entitled to
receive yearly interest from the British government forever.
43.
44. Question: Consider a perpetuity paying $100 a year. If
the relevant interest rate is 8 percent. What is the value
of the consol?
Answer: $1,250
Suppose interest rate 6 percent.
Answer : $1,666.67
Note: Value of perpetuity rise when rate decrease
45. Question: Company “Rich” pays Nu.2 dividends annually and
estimates they will pay it indefinitely. How much are investors
willing to pay for the dividend with a required rate of return of 5%?
PV = 2/5% = Nu.40
An investor will consider investing in the company if it’s worth
Nu.40 or less.
46. Perpetuity with growth formula
Formula: PV = C / (r – g)
Where:
PV = Present value
C = Amount of continuous cash payment
r = Interest rate or yield
g = Growth Rate
47. Taking the above example, imagine if the Nu.2 dividend is expected
to grow annually by 2%.
PV = 2 / (5% – 2%) = Nu.66.67
48.
49. What is an Annuity
An annuity is a financial product that pays out a fixed stream of
payments to an individual, primarily used as an income stream
for retirees.
Annuities are created and sold by financial institutions, which
accept and invest funds from individuals and then, upon
annuitization, issue a stream of payments at a later point in time.
The period of time when an annuity is being funded and before
payouts begin is referred to as the accumulation phase. Once
payments commence, the contract is in the annuitization phase.
50. Future Value Annuity
Future value is the value of a sum of cash to be paid on a specific date
in the future.
An ordinary annuity is a series of payments made at the end of each
period in the series.
Therefore, the formula for the future value of an ordinary annuity
refers to the value on a specific future date of a series of periodic
payments, where each payment is made at the end of a period.
52. Question1: I have a choice to invest in certain sum in
an avenue which promise return from year 1 to 5 for
Nu.50,000 per year
if the rate is 12 percent. What is the maximum
amount you are going to pay to buy these investment.
If the investment is available for:
a) Nu.1,80,550
b) Nu.1,50,000
c)Nu.2,00,000
53. Question 2: I am 60 and about to retire, my employer
gives me following choice of receipt of future
payment.
a) Nu.1,50,000 lump sum now
b) Nu.1,80,000 per year for next 15 year
c) Get Nu. 5,00,000 each in year 65,70,75 and
80
If the required rate of return or market rate is 15
percent. Which option should I choose
54. Question 3: I have a choice to invest certain sum in
an avenue which promise return from 1 to 5 as 40,000,
70,000, 1,20,000, 1,30,000 and 1,50,000.
If the market rate is 12 percent. What is the maximum
amount you are going to pay to buy these investment’
If the investment is available for ;
a) 300,000
b) 3,44,680
c) 4,00,000
Answer=3,44,680
55. Students’ task
Calculate future value of an Investment of
Nu. 400,000 at 5% p.a. compounding
monthly for:
a. 3 years
b. 5 years
c. 7 years
56. Question: Dechen won a lottery paying $50,000 a
year for 20 years. She is to receive her first payment
a year from now. This is an Million Dollar lottery
because $50,000 * 20=$1000,000. if the rate of
interest is 8 percent, what is the present value of the
lolttery?
Answer= $4,90,750/490
57. Stated Annual Interest Rate and Effective
Annual Rate
Check Into allow you to write a check for $115 in the
future, for which they give you $100 today. Find rate.
Answer=15%
Calculate the following
APA No. of compounding EAR
7% Quarterly ?
16% Monthly ?
? Semi-annually 9.8%
? Monthly 19.6%
58. Formula for EAR and APR
EAR=(𝟏 +
𝑸𝑹
𝒎
)𝒎
−𝟏
APR=m((𝟏 + 𝑬𝑨𝑹)𝟏/𝑴
−𝟏)
QR= Quoted rate
EAR=Effective rate
60. APR=m((𝟏 + 𝑬𝑨𝑹)𝟏/𝑴
−𝟏)
EAR=9.8% Compounded semi-annually
=2((1 + 9.8%)1/2−1)
=2((1 + 0.098)1/2
−1)
=9.571%
Effective rate is always higher than quoted rate
TRY TO SOLVE THE REMAINING PROBLEM IN THE TABLE?
62. Ordinary Annuity
An annuity is a series of equal payments that are
made at the end of equidistant points in time such as
monthly, quarterly, or annually over a finite period of
time.
Payment or receipt occur at the end of each period,
the annuity is referred to as ordinary annuity
Payment or receipt occur at the beginning of each
period, the annuity is referred to as annuity due
63.
64. Mr. X deposits Nu.10,000 at the end of every year for 5 yrs. In his saving
a/c paying 5% p.a interest. How much money will he get at the end of 5
yrs.
Ans: Nu.55256.31
Suppose that a firm deposits Nu. 5,000 at the end of each year for four
years at 6 percent rate of interest. How much would this annuity
accumulate at the end of fourth year?
Solution: 21,873.08
65. Annuities Due
Annuity due is an annuity in which all the cash flows occur at the beginning of the period.
66. Question
You deposit Nu.12000 at the beginning of every year for 10years. If interest being paid
at 8%. How much will you have in 10 years?
Answer:187745.85
67. PV of annuity due (Homework)
Dorji receives an annuity of Nu.5000 for four years. If the rate of interest
is 10%. What is the present value of annuity due?
Answer=17435
An Investment promises to pay Nu.4000/- at the beginning of each
year for the next 6 yrs. If your required rate is 13%. Find the future
value of annuity due
Ans: Nu. 37618.63
68. Mortgages- Amortized Loans
An amortized loan is a loan paid off in equal payments –
consequently, the loan payments are an annuity.
Examples: Home mortgage loans, Auto loans (two
wheeler/car loan)
In an amortized loan, the present value can be thought of as
the amount borrowed, n is the number of periods the loan
lasts for, i is the interest rate per period, future value takes
on zero because the loan will be paid of after n periods, and
payment is the loan payment that is made.
69. Finding PMT
Suppose you would like to have $25,000 saved 6 years from now to pay towards your
down payment on a new house. If you are going to make equal annual end-of-year
payments to an investment account that pays 7 percent, how big do these annual
payments need to be?
Answer=3495
70. Amortization Example
Suppose you plan to get a Nu90,000 loan from a furniture
dealer at 18% annual interest with annual payments that you
will pay off in over five years.
What will your annual payments be on this loan?
Using a Financial Calculator
• Enter
– N = 5
– i/y = 18.0
– PV = 90000
– FV = 0
– PMT = Nu. 28,780.00
yearly
