This document provides an overview of key concepts for calculating present and future values, including: 1) How to calculate present and future values of single cash flows using discount factors and compound interest formulas. 2) How to calculate present value for a stream of multiple cash flows by summing the discounted cash flows. 3) Examples are provided to illustrate calculating present value for investments, loans, perpetuities, annuities, and growing cash flows. 4) Shortcuts for calculating perpetuities and annuities are explained. 5) The differences between nominal interest rates, effective interest rates, and how interest is quoted are defined. 6) Useful spreadsheet functions for present value calculations are listed.

1. Corporate finance: lecture 3
calculation of present value
By Muhammad Shafiq
forshaf@gmail.com
http://www.slideshare.net/forshaf

2. Sequence
• How to calculate present value
• Calculating future and present values
• Looking for shortcuts- perpetuities and annuities
• More shortcuts-growing perpetuities and annuities
• How interest is paid and quoted
• Use of spreadsheet functions

3. How to calculate present value:
Introduction
• Company invests in many things e-g tangible and intangible assets
• You are investing in your education. What you hope for?
• In this regard, borrowing with the promise to repay (plus interest)
• So money you have today and in the future, makes point to think
•Lets start

4. Basic Concepts
• Future Value: compounding or growth over time
• Present Value: discounting to today’s value
• Single cash flows & series of cash flows can be considered
• Time lines are used to illustrate these relationships

5. Future value and present values:
calculating future values
• Money can be invested to earn interest;
• So Rs 100 today or 100 next year is common example every finance teacher
asks from students
• Financial managers say money has time value as it is said:
“a rupee today is more worth-while than a rupee tomorrow”.

6. Calculating future values: example
• Suppose you invested Rs. 100 in a bank account that pays interst of
7% (r= 0.07) a year
first year you will earn interest of 100*.07= 107
If you leave your money it will grow to (107* 1.07= 114.49)
Or
=(107
2
)
Income is growing with compound interest rate
We can put it like
100 for t year i-e (1.07
t
) and if interest is r then = (1+r
t
)
Higher the interest rate, the faster it grows
Future value of Rs. 100 * (1+r)
t

7. Calculating Present value
• Lets think as opposite of this, “how much we need to invest today to
get Rs. 114.79 at end of second year?
• In other words, what is the present value (PV) of the 114.79?
• Just turn it and find out: PV= 114.79/107
2
= Rs100
Hence; PV= Ct/(1+r)
t
Present value is the discounted value of cf
Discount factor; 1/(1+r)
t

8. Calculating Present value:
Discount factor
Discount factor; 1/(1+r)
t
Discount factor is measure o PV of (1) one $ in t year
Example:
Df2= 1/(1.07)2 = 0.8734
Present value = DF2* C2
PV = .8734*114.49 = 100

9. Net Present Value
• Example: building worth 700,000 it can be sold for Rs. 800,000 so its
present value is Rs.747,664
• But actual or net present value is different
• NPV = PV minus Investment i-e 747664-700,000= 47664
• NPV= C0 + C1/ 1+ r
• That is: NPV = 700,000 + 747664= 47664
• 800,000/1.07

10. Risk and present value: example
• Office building is more risky than govt securities but less riskier than
investing in the stock market it provides 12% return. So the
opportunities cost of capital for project. If we decides to invest in
stock at 12% with 800,000 the present value will be:
PV = 800,000/1.12
and NPV = PV – Investment i-e =714286-700,000
Return = profit / investment or Returns- cost/investment
= 800-700/700 =
Rule of acceptance for NPV

11. Calculating CF when multiple CF
• It is stream of CF and the formula is:
PV= C1/1+r + C2/(1+r)2 . . . + Cn/(1+r)n
This is discounted CF formula
Can also be write sum of Ct/(1+r)t

12. Calculating CF when multiple CF:
Example
• A property dealer suggests that if an office building is rented,it will
give Rs 30,000 and next if sold Rs 840,000. If opportunity cost 12%
then the calucalation will be as follow:
PV= C1/1+r + C2/(1+r)2 . . . + Cn/(1+r)n
PV= 30,000/1.12 + 840,000/ (1.12)2 = 720,344
NPV = 720,000 + 700,000 = Rs 20,000

13. Opportunity cost of Capital
• It is giving up opportunity
Suppose:
A bank find your business safe and provide you loan of
Rs700,000 at 8%. If you earn Rs 30,000 first year and
870,000 second year then Calculate NPV:
PV= 30,000/1.08 + 840,000/ (1.08)2 =

14. Looking for shortcuts:
perpetuities and annuities
• How to value perpetuities
• Example:
• British Government issued perpertuity bond during war still it exist
• Formula to calculate it:
Returns = CF (promise annual Amount)
PV
or
= C/r
• More clear view:
PV = 1 Billion * 1/r * 1/ (1 + r)3
• How to value annuities: PV of t year annuity = C [ 1/r – 1/ r (1 + r)t]

15. Costing an installment plan:
An Example
• Loan obtained for car purchase on yearly installment plan paid at
end of each next five years of Rs. 360,000. what is the car really
costing to you, if interest rate is 10%?
• PV of t year annuity = C [ 1/r – 1/ r (1 + r)t] hence:
PV = Cash flow [1/0.1 – 1/ 0.1 (1.1)5]
PV= 360,000* 3.79
Answer= ?

16. Winning a national lottery: Example
• Suppose friends of yours win lottery of Rs. 360 Million. However, they
suspect the amount is less what actually was promised (Rs 360). The
amount would be paid in 30 installments of Rs 12.5 million each. At
that time interest rate was 6%. What would be the payment at end of
time?
• PV = c * installment so 12.5 m * 30
• PV of t year annuity = C [ 1/r – 1/ r (1 + r)t] hence:
PV = 12.5 [1/0.06 – 1/ 0.06 (1.06)30]
• PV = 12.5*13.765
• Answer: whether it is below or more

17. Valuing annuities DUE
• Remind the last example of Winning a national lottery. We assume
that payment is made at the end of the year
• In Annuity due, it is in the beginning of year
• For computation we increase the present value by multiplying (1 + r )
• In the last example we would multiply the answer as
Annuity due = Answer * 1.06

18. Calculating the annual payment: example
Paying off a bank Loan
• You borrowed a four year loan of Rs. 1000 at 10%. bank requires you to pay
it in evenly installments in four years.
** first line calculation explanation:
year end interest= 1000*.10 (interest rate) amortization of loan 315.47 – 100 (Interest amount)
End of year balance = 1000 - 315.47
End of year
Balance
Amortization
of loan
Total year-end
payment
Year-end
interest
Balance
Beginning of year
balance
year
786.53215.47315.47100.001,0001
547.51237.02315.4778.45784.532
286.79260.72315.4754.75547.513
0286.79315.4728.68286.794

19. Calculating the annual payment: example
Paying off a bank Loan…
• Four year annuity factor = [ 1/r – 1/ r (1 + r)t] hence:
[1/0.1 – 1/ 0.1 (1.1)4] = 3.17
Now annual laon payment: 1000/3.17
= 315.47

20. Calculating mortgage payments
Example:
• You got a house on mortgaged at Rs 5,00,000 at 12% interest for 30
years. Bank requires you to pay in equal installment, how come?
a Rough idea: Annual mortgage payment / 30 year annuity
500,000/ 30 = 16.66
30 year annuity factor = [ 1/r – 1/ r (1 + r)t] hence:
[1/0.12 – 1/ 0.12 (1.12)30] = 8.055
so 5,00,000/8.055 = 620, 732

21. Future value of an annuity:
Example
• You want to change your car buying Mercedes but some serious
saving. You know that once start work, you can save Rs 500,000 a year
of your income and 8% on saving. How much you may be able to
spend for car after five year?
• PV of t year annuity = C [ 1/r – 1/ r (1 + r)t] hence:
PV = 500,000 [1/0.08 – 1/ 0.08 (1.08)5]
PV= ? (calculate it please). When it calculated easy to compute future
value by multiplying (1.08)5 as:
PV answer* (1.08)5

22. More short cuts- Growing perpetuities and
annuities
Growing perpetuities
• Sometimes you want to know the streams of CFs grows at constant rate
Example: you want to donate Rs 10 Million a year regularly to cancer Hospital for salaries
and other costs but have no allowance for the growth in salaries of 4%. You can calculate as follow
as the presently the discount rate is 10%:
PV= C1/1+r + C2/(1+r)2 . . . + Cn/(1+r)n
You can change the formula little for facilitation as:
PV= C1(1 + g)/1+r + C2(1+g)/(1+r)2 . . . + Cn(1 + g)/(1+r)n
Or more specifically:
PV: C1/r-g = 10 m/.10-.04 = ? (find the answer)

23. More short cuts- Growing perpetuities and
annuities
Growing annuities
Example: you contemplating a membership in Badminton Club which
you have to pay at end of the year, fee of Rs. 5,000, however; if you pay
Rs. 12,500 you will be exempted for next three year. The discount rate
is 10% whereas the fee charges growing with 6% constantly.
Formula: PV= C* 1/r-g [1- (1+g)t / (1+ r)t ]
PV= 5000* 1/.10-.06 [1- (1+.06)3 / (1+ .10)3 ]

24. How interest is paid and quoted:
Nominal rate and effective rate
• There is difference between interest paid annually, semi-annually,
quarterly and monthly
• More the rate is paid in pieces in a year, the effective rate will be high
Example: A bank offers you car financing Loan at an annual percentage
rate (APR) of 12% with interest will be paid monthly. It means you are
paying 1% each month i-e quoted rate is 12% and effective rate is:
[1+ (r/m)]m -1
[1+ .12/12]12 -1 = .1268 or 12.68%
For weekly we
we m= 52 and daily m = 365

25. Useful spreadsheet functions
Discounting cash flow
• Find the function by pressing fx on the excel toolbar.
• List of few functions for DCF problem:
Future value of single investment or annuityFV
Present value of single future cash flow or annuityPV
Interest rate (or rat of return) needed to produce given
future value or annuity
RATE
Amount of annuity payment with a given present or future
value
PMT
Calculate the value of stream of negative, positive cash flowNPV