Corporate finance: lecture 3

calculation of present value

By Muhammad Shafiq.forshaf@gmail com

:// . . /http www slideshare net forshaf

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

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

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

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.”

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= 107If you leave your money it will grow to (107* 1.07= 114.49)

Or=(1072)

Income is growing with compound interest rateWe can put it like100 for t year i-e (1.07t) and if interest is r then = (1+rt)Higher the interest rate, the faster it grows

Future value of Rs. 100 * (1+r)t

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= 114.79/1072 = Rs100

Hence; PV= Ct/(1+r)t

Present value is the discounted value of cfDiscount factor; 1/(1+r)t

Calculating Present value: Discount factorDiscount factor; 1/(1+r)t

Discount factor is measure o PV of (1) one $ in t yearExample:

Df2= 1/(1.07)2 = 0.8734Present value = DF2* C2

PV = .8734*114.49 = 100

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

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 = 800,000/1.12and NPV = PV – Investment i-e =714286-700,000

Return = profit / investment or Returns- cost/investment= 800-700/700 =

Rule of acceptance for NPV

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 formulaCan also be write sum of Ct/(1+r)t

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 = 720,344NPV = 720,000 + 700,000 = Rs 20,000

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 =

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]

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= ?

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

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

Calculating the annual payment: examplePaying 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

786.53 215.47 315.47 100.00 1,000 1547.51 237.02 315.47 78.45 784.53 2286.79 260.72 315.47 54.75 547.51 3

0 286.79 315.47 28.68 286.79 4

Calculating the annual payment: examplePaying 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] = 3.17Now annual laon payment: 1000/3.17

= 315.47

Calculating mortgage paymentsExample:• 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

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

More short cuts- Growing perpetuities and annuitiesGrowing 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)

More short cuts- Growing perpetuities and annuitiesGrowing annuitiesExample: 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 ]

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 highExample: 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

Useful spreadsheet functionsDiscounting 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 annuity FV

Present value of single future cash flow or annuity PV

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 flow NPV