Lecture 2

Present Value Mechanics

Rules of Time Travel

 Financial decisions often require combining cash flows or comparing

values.

• Three rules govern these processes.

Rule 1: It is only possible to compare or combine values at the

same point in time.

Rule 2: To move cash flows forward, you must compound it.

Rule 3: To move cash flow backwards, you must discount it.

Example:

Which would you prefer?

• $1000 today

• $2000 today

• $2010 tomorrow

1. Which one can be ruled out (loại bỏ) instantly? $1000

2. Which one needs to be moved forward (compounded) (lãi kép) ? $2000

3. Which one needs to be moved backwards (discounted) (chiết khấu)?2010

4. What’s an appropriate discount rate?

Compounding

 Practically everything in finance revolves around the understanding and

manipulation of this equation.

The Compounding Formula: FV = PV × (1 + r)^n

1. This equation dictates (states) the relationship between the current value
or present

value (PV) with the future value (FV).

2. The future value (FV) of an investment is dependent on invested rate (r )

and the number of years invested (n).

 Compounding Example: Supposing we have $50,000 which will be

invested at a rate of 5% per annum for 2 years. What is its future value in
2 years' time?

PV = 50000, i = 0.05, n = 2

FV = PV × (1 + r )

n = 50000 × (1.05)

2 = 55125

Discounting

 A simple manipulation of the compounding formula gives you the

discounting

formula. This formula brings back future promises of cashflows into today’s

terms.

PV = FV × (1 + r )^-n

 Discounting Example: The future value of a 3-year investment with a yield

of 2% pa is $1,000. What is the current value of this investment?

FV = 1000, i = 0.02, n = 3

PV = FV × (1 + r )

−n = 1000 × (1.02)

−3 = 942.32

So now we know how to bring future investments (FV) back into the present

(PV) ...

When to use FV and PV?

 FV and PV are relative to where the cash flow is on the timeline.

 A Textbook Example of PV and FV

- Assume that an investment will pay you $5,000 now and $10,000 in five
years.

- The timeline would look like this:

 You can calculate the present value of the combined cash flows by
adding

their values today.

- The present value of both cash flows is $11,209.

 You can calculate the future value of the combined cash flows by
adding

their values in Year 5.

- The future value of both cash flows is $18,053.

=>

Valuing a Set of Cashflows

• A set of cashflows can yield payments at different points in time.

• This makes life difficult because we can’t compare or combine them

because

they are in different points in time (breaching Rule 1).

• In order to value cashflows, we need to either discount cashflows to today

(PV)

or compound cashflows to the future (FV).

PV of Cash Flow Sream

We can bring all the future cashflows back to today.

To summarise:

PV = C1 × (1+r)−1 +C2 × (1+r)−2 +···+Cn × (1+r)^-t

n
PV= ∑ Ct x ( 1+ r )−t
t =1

FV of Cash Flow Sream

Similarly, we can figure out the future value of our cashflows.

FV = C1 × (1+r)^(n−1) +C2 × (1+r)^(n−2)+··· +Cn

n
FV= ∑ Ct x (1+r )(n−t )
t =1

 FV example: What is the future value in three years of the following

cash flows if the
 compounding rate is 5%?

FV= 2000(1+5%)+2000(1+5%)^2+2000(1+5%)^3= 6,620

PV of an Annuity

 Assume we have a ten-year annuity(trợ cấp) as shown below with $10

payment in arrears (trả sau),

 Supposing the discount rate is r . Let us find the present value of each
cash flow payment.

• Cash flow at time 1 is 10 × (1 + r )−1 at time 0

• Cash flow at time 2 is 10 × (1 + r )−2 at time 0

• Cash flow at time 3 is 10 × (1 + r )−3 at time 0

• and so on ...

• Cash flow at time 10 is 10 × (1 + r )−10 at time 0

 We simply sum these present value cash flows to obtain the present
value of
our annuity.

PV = 10 × (1 + r )^−1 + 10 × (1 + r )^−2 + · · · + 10 × (1 + r )^−10 (1)

We can take out the 10, but this is still tedious to solve.

PV = 10 × [(1 + r )−1 + (1 + r )^−2 + · · · + (1 + r )^−10] (2)

Simplifying the tedious part:

Let us denote S to be the tedious component. Then we can write the

following.

S = (1 + r )^−1 + (1 + r )^−2 + · · · + (1 + r )^−10 (3)

It is also true that,

(1 + r )S = 1 + (1 + r )^−1 + · · · + (1 + r )^−9 (4)

because we’re simply multiplying (1 + r ) to eqn(3) on both the LHS and RHS.

Now, let us perform eqn(4) - eqn(3). You will notice that most of the terms
cancel out.

rS = 1 − (1 + r )^−10 (5)

Thus, the term S simplifies to,

S= (1-(1+r)^-10)/ r (6)

The present value of our annuity becomes,

PV= (10 x (1-(1+r)^-10)/ r) (7)

To generalise from this special case,

PV = C x (1-(1+r)^-N)/r

where C is the coupon amount, and N is the length of the annuity.

PV of an Annuity Example

You win $2 million, and can choose from the following pay-out schedules:

• $2 million now

• $172,000 per month for 12 months starting 1 month from now

Which is the best alternative? Assume your monthly interest rate is 0.5%.
PV = 172,000 x (1-(1+0.005)^-12)/0.005= 1,998,456

How would your answer change if the $172,000 payments were at the

beginning of the month instead of the end of the month?

Roll our existing 1,998,456 answer forward by 1 month.

PV due= PV (1+0.05)^1

1,998,456 × 1.005 = 2,008,449

FV of an Annuity

FV = C x ((1+r)^N-1)/r

 Example: Every year, you deposit $1000 of your savings into Savings
Bank. The

interest rate is 2%. You do this for 10 years. How much do you accumulate at

the end of the 10 years?

FV = PV × (1 + r )N = 8982.585 × 1.0210 = 10949.72

You have accumulated $10,949.72 by the end of year 10.

 An annuity that goes on forever is known as a perpetuity.

PV = C × (1 + r )−1 + C × (1 + r )^−2 + C × (1 + r )^−3 + . . .

How do we calculate something that goes on forever?

PV of a Perpetuity

PV = C/r

1. Consols: government debt issues (perpetual bonds)(trái phiếu vĩnh viễn)

2. Stocks (if you assume they pay constant dividends and the company has
no

default risk)

example: Assuming Robust Company raises capital by issuing preference

shares with a

fixed annual dividend of $10. How would you value each share if you are

using a discount rate of 7%?

PV= C/r = 5000/0.02= 250,000

-> You would need $250,000 in funding

PV of a Growing Perpetuity

PV = C/ (r-g)

Example1: You graduate from college and at the end of this year, you will be
earning

$100,000 paid up front at a sovereign wealth fund. Every year your salary

is expected to grow by 5%. Your girlfriend (boyfriend) discounts you at 10%

(this is high because she’s (he’s) pretty fed up with you).

How much are you worth to her (him)?

By the way, assume you live forever... (because I want you to use the
growing

perpetuity formula!)

PV = 100,000/( 10-5)%= 2,000,000

Example2: You are asked to value Growth Company. You note that the
company

historically increases its dividend payments by 5% per year. Their next year

dividend will be $5.

What would be ‘decent’ approximation of the stock price if you decide to

discount future cashflows of Growth Company by 8%?

PV= 166.67

Growing Perpetuities are also useful for Equity Valuation

Let’s take our growing perpetuity formula,

PV= C/(r-g)

Firstly, C is the next year dividend payments, so we can write:

PV = D1/ (r-g)

Which is the Gordon Growth Model. Rearranging this,

R = D1/P +g

Required Rate of Return = Divided Yield + Earnings Growth

=> the expected returns of stocks is positively related to:

1. Dividend Yield ( Value Investing )

2. Earnings Growth ( Growth Investing)

-> to be covered in more detail when we study equities.

Present Value of Growing Annuity

PV = C x 1/(r-g)(1-(1+g)^n/(1+r)^n)

Example: Tommy Wiseau makes $100,000 pa at the end of his first year
selling

merchandise on his online store. Since he does not need this money, he

simply deposits it in the bank, earning 5% pa interest. For the next 10 years,

his profits from his store increase by 10% pa year on year.

How much will he have at the end of the 10 years?

PV = 1,184,666

Loans and Mortgages

C = (r x PV)/ (1-(1+r)^-N)

Example1: Your friend has asked you for a $10,000 loan to start his Bitcoin
mining

business. He (she) promises to pay back in 10 equal annual instalments.

You decide to charge him (her) 5% interest on the loan. What is the annual

instalment you should demand?

->C= 1295.05

Example2: You plan to donate money to set up a fund that spends $10,000
pa on

awarding scholarships. The spendings are inflation linked, and we assume

inflation to be 2% pa. At the end of the first year, the fund expects to pay out

$10,200 and so forth. Given interest rate is 5%, what is the required donation

value?

-> Reality spending is $10,200 ( 10,000 is plan)

So C= $10,200

PV= 10,200/ 3%= 340,000

Tutorial 2 Questions

Question 1

You have the opportunity to earn $ 50,000 per annum in arrears for the next
3 years. Assuming you can earn a rate of 4% on comparable investments.
What is the present value of this investment?

PV = PV = C x (1-(1+r)^-N)/r = 138,754,55

Question 2 : Jonathan expects to save $10,000 p.a. for the next 30 years. He
will keep his savings in a bank account which earns 3% p.a. interest. How
much will Jonathan have accumulated at the end of his 30 years? (Hint: Draw
a timeline)

FV= 475,754,16

Question 3

You plan to donate money to set up a fund that spends $10,000 p.a. on
awarding scholarships. The spendings are inflation linked, and we assume
inflation to be 2% p.a. At the end of the first year, the fund expects to pay
out $10,200 and so forth. Given interest rate is 5%, what is the required
donation value?

PV = 340,000

Question 4

What is the present value of $10,000 received ...( FV= 10,000)

a) 5 years from today with interest rate of 4% p.a.

PV= 8,219

b) 10 years from today with interest rate of 3% p.a.

PV= 7.440

c) 20 years from today with interest rate of 2% p.a.

PV= 6.729

Question 5

You lend your friend $100,000 for a period of 10 years with 7% interest p.a..
If your friend agrees to pay equal annual instalments in arrears ( nợ khất lại),
how much is each instalment?

C = (r x PV)/ (1-(1+r)^-N)= 14,237,75

Question 6

Following from Question 5, you deposit each instalment in a lousy bank that
gives 1% interest. How much do you have in the bank at the end of 10
years?

FV= 148958.37

Question 7 (Not examinable, only for students interested in maths)

Derive the logic behind the growing perpetuity formula.