Present Value

CHAPTER 2

PRESENT VALUE – THE UNDERLYING ASSUMPTION

·        If the interest rate is 10%, we know that $100 today is worth the same as $110 next year (100 is the PV today of that future cashflow)

·        Where does this really come from?

·        Not just “you can invest $100 today and you’ll earn 10% so it will become $110 tomorrow”; there’s a little more than that

·        Value of money comes from its consumption possibilities

·        Two cashflows have the same value if and only if they have the same consumption possibilities

·        What are consumption possibilities of $100 today?

·        What are consumption possibilities of $110 tomorrow?

·        What do we assume in the second case?

=> the concept of PV assumes there is a capital market in which you can borrow at the same rate at which you can lend

 

Q: If there was no capital market (no borrowing or lending), would $100 today and $100 tomorrow have the same value?  In other words, if we have $100 tomorrow, can we say its PV today is $100 (since the interest rate is 0)?

 

REVIEW OF BASIC CONCEPTS

·        Return

·        Expected return versus required return

·        Discount rate = opportunity cost of capital

 

FOUNDATION OF NPV RULE

Consider a firm which starts with $2m in cash, and has 100,000 shares

·        Initially, stock price = $20; wealth of s/h = $2m + whatever other assets they own outside this firm

Now the firm takes a project where investment is $1m and the value of the cashflows from the project is $1.4m

·        NPV of project = 0.4m

·        Value of the firm = 1m (cash) + 1.4m (value of the assets acquired by investing 1m) = 2.4m

·        Stock price = $24

·        Wealth of s/h =  $2.4m + whatever other assets they own outside this firm

 

=> wealth of s/h increases by NPV of project

Every positive NPV project increases s/h wealth (as well as stock price)

You maximize s/h wealth by accepting all positive NPV projects (or if you have mutually exclusive projects, by accepting the one with the highest NPV)

 

NPV Rule:

1)     If projects are independent, accept all projects with positive NPV

2)     If projects are mutually exclusive, accept the one with the highest NPV

 

CHAPTER 3

 

NOTATION, TERMINOLOGY, FORMULAS

·        PV = value today of future cashflows (time 1 onwards)

·        NPV = value today of current and future cashflows (time 0 onwards)

·        Ordinary perpetuity: PV0 = C/r  

o      Payments start tomorrow

o      C/r gives you PV today, one year before the first payment

·        Ordinary Growing Perpetuity: PV0 = C1/(r-g) (once again, PV today, one year before the first payment)

·        Ordinary Annuity (t payments, first payment tomorrow): PV0 = (C/r) * [ 1 –  {1/(1+r)t}]

·        Annuity Due (t payments, first payment today): NPV0 = (PV of Ordinary Annuity)*(1+r)

 

Ordinary growing annuity example: difference of two growing perpetuities

You have the following savings plan for the next 15 years.  Your salary this year is $50,000 (will be paid in one lump sum at the end of the year), and it will increase by 5% each year.  You will save 20% of your salary each year, for the next 15 years.  If you earn 10% on your savings, what is the present value of the savings?

 

The savings stream is:

C1 = savings at time 1 = 50,000 * .2 = 10,000

C2 = 10,000 * 1.05

C3 = 10,000 * 1.052

C4 = 10,000 * 1.053

.

.

.

C15 = 10,000 * 1.0514

 

PV would be easy to compute if this was a growing perpetuity [would just go = C1/(r-g)]

Let A = a growing perpetuity which starts with 10,000 at time 1, and grows at 5%.

Then A = the 15 year saving stream + some extra CF

Call the extra cashflows B.  B is then a growing perpetuity which starts with a time 16 cashflow of 10,000*1.1515, and grows at 5%.

And the savings stream is just A - B

So, PV0 of the savings = PV0(A) - PV0(B)

PV0(A) = 10,000/(.10 - .05) = 200,000

The first cashflow in B is C16 = 10,000*1.1515 = 20,789.28

PV15(B) = C16/(r-g) = 20,789.28/(.10 - .05) = 415,785.64

PV0(B) = 415,785.64/(1.10)15 = 99,535.78

=> PV0 of the savings = 200,000 – 99,535.78 = 100,464.22

 

The general formula for a growing annuity would be:

PV0 = C1/(r-g) * [ 1  - (1+g)t/(1+r)t] = 200,000 *( 1 – 0.49768) = 100,464.22

 

NON-ANNUAL COMPOUNDING

You invest $500 today at 9% per year compounded monthly.  What is the FV at the end of the year?

o       Stated rate (APR) = 9%

o       Actual Rate = 9/12 = 0.75% each month

o       Effective annual rate = (1 + .0075)12 – 1 = 9.3807%

o       Can compute FV1 two different ways:

o      Money grows at 0.75% per month for 12 months: 500*(1.0075)12

o      Money grows at 9.3807% per year for 1 year: 500*1.093807

o      Either way, answer is $546.90

o       General formula for effective annual rate: EAR = (1 + [Stated rate]/m)m – 1, where m = # of times you compound per year

 

An annuity pays $500 at the end of each quarter for 5 years. What is the present value of the annuity, if the interest rate is 9% per year compounded monthly?

o       Since payments are made quarterly, we need to compute present value using a 3-month rate as the discount rate.

o       Actual rate = 0.75% each month    

o       Convert one-month actual rate to 3-month effective rate:  3-month effective rate = (1.0075)3 - 1 = 2.2669%

o       PV0 = 500/(*1.022669) + 500/(*1.022669)2 + . . . + 500/(*1.022669)20  = 7,968.98

 

NOMINAL VS. REAL

Nominal cashflow = actual number of $ you receive in a future year

Real cashflow = the purchasing power of those nominal cashflows

Real cashflow is just the nominal cashflow adjusted for inflation

It measures your future purchasing power in today’s dollars  

Real CFt = (Nominal CFt)/(1+i)t

Nominal return = percentage increase in actual $

Real return = percentage increase in purchasing power

Real return is the nominal return adjusted for inflation

If the nominal return is 10%, you will have 10% more $ next year; real return tells you how much more consumption these extra $ will buy you

RREAL = (1+RNOMINAL)/(1+i) - 1

 

In computing PV (e.g. in capital budgeting) should you use real cashflows or nominal cashflows? 

Only rule: be consistent

You can discount nominal cf at the nominal discount rate, or real cashflows at the real discount rate.  Will get the same answer both ways

In practice:

you always have nominal cf and nominal discount rates to begin with (the required return you get from the CAPM is a nominal rate)

converting to real cashflows and real discount rate is an extra step

do it only if it buys you some simplification

e.g. the growing annuity example was complicated because nominal cashflows were growing over time.  Suppose they were growing at the inflation rate.  In other words, the inflation rate is the same as the growth rate of 5%.

Then nominal cashflows grow at 5%, but real cashflows stay constant.

=> real cashflows are an ordinary annuity

=> computing PV is much simpler if you discount real cashflows at the real discount rate.

The nominal discount rate was 10%. The inflation rate is 5%.

Real rate would be (1.10/1.05) – 1 = 4.7619%

If you compute PV applying this discount rate to a 15-year annuity, you get 100,464.22

(compared to what we did before, this takes just two simple steps: compute the real discount rate and the constant real cashflow)  

Q: what is the real cashflow?

 

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