Economics 173A

The Time Value of MoneyPart 3

#1 - Is this a Good Investment?

If I invest $100 today and expect $ 150 in

10 years, is this better than the 5 percent

per year that my bank is offering me?

#1 Comparing investments.

#2 - When Do I Take My Pension?

A pension is a guaranteed, fixed, annual “receipt” of money for life. So it is a life annuity. The younger you are when you take the pension, the longer the it runs, so the smaller will be the annual payment. Conversely, the older you are when you take the pension, the shorter it runs and the larger the annual payment to you.

Do you want less for longer or more for shorter?

Quiz - When Do I Take My Social Security?

This is a decision facing several million Americans each year. It is an important question and deserves thoughtful analysis. The numbers themselves provide an objective starting-point.

Do I want less for longer – from age 62 to the end-of-my-life - or more for shorter – from age 66 to the end-of-my-life?

Starting at age 62 until

Death

Start at Age 66 until

Death

$ 1,837 / month or

$ 22,044 per year

$ 2,500 / monthor

$ 30,000 per year

My Social Security as of 2/10/14

This is the decision that I will face in 10 months.

We need an Annuity, a rate,

and a time ?

Start at age 62

Start at Age 66

Here is the Annuity $ 22,044 per year $ 30,000 per year

Rate - assume this … 4% 4%

Years - to life expectancy, say to 85 years old.

23 19

These decisions must be made now, so that means looking at this as a Present Value

decision.

Which is Greater?Do I want $ 22,044 x PVFA ( 4% , 23)starting 10 months from now

or

Do I want $ 30,000 x PVFA ( 4% , 19) starting 4 years & 10 months from now

For analysis, we can ignore the 10 months and just deal with the 4 year separation.

Remember this (from the TVM II Slides?)

Summary of the Factor Tables and their Functions

A Present Value Annuity Factor “PVFA” = (1 - PVF) / r turns an Annuity into a PV

Which is what we need …

Annuity PVFA PVOf the Annuity

$ 22,044At age 62

X 14.85 [1 point]

= $ 327,504[1 point]

$ 30,000At age 66 X 13.13

[1 point]= $ 394,000[1 point]

The Annuities

These results are not comparable because one is the PV of an annuity at age 62 and the other is the PV of an annuity at age 66. These values are 4 years apart

AnnuityPresent Value

PVF to bring age 66 to

age 62

The PV in 10 monthsOf each Annuity

$ 327,504At age 62 X 1.00 = $ 327,504

$ 394,000At age 66 X (1.04)^-4

=0.855[2 points]

= $ 336,885A PV at age 62

[1 point]

The Final Step

Both results are now comparable because each is a PV of the respective annuity at the same age.

#3 What will my Payments be on a Loan?

If you borrow money you taking a present value chunk of money and returning an annuity over a period “t” paying a rate “r”.

The present value of your annuity payments must be equal to the amount of the loan.

#3 If you borrow $30,000 for 5 years at 8 percent, what will your monthly

payments be, approximately?

We want to know what five-year, annual annuity will have a present value of $30,000 at 8 percent? We know 3 of the 4 pieces of the puzzle: PV = $ 30,000t = 5 yearsr = 8 percent.

What is a Loan?

It is an exchange of a big package of

money today in exchange for many

small packages periodically into the

future.

The big package is sold by a lender to a

borrower. The borrower pays the

lender back through loan payments.

You borrow $30,000

You Know1.$30,000 is the PV2.For 5 years = t3.At 8 percent per year interest = r

And you want to find the monthly Payments, i.e. the amount of each “Future” payment.

Monthly payments on a $30,000 loan – approximated by doing annual discounting & dividing the result by

12Solve this

Value exchanged must be the same, so:$30,000 = PV sold = PV paid

$ 30,000 = Payment x PVFA (8% , 5 years)$ 30,000 = $A x 3.99 from the Table

Solving for $A we get the annual payment = $ 7,500

Now divide by 12 to get an approximation for the monthly payment= $ 625 per month

#4 - How much do I need to Save for my Retirement?

1. The amount you will have in retirement

depends on:

2. When you start saving “t”.

3. How much you save “$A”.

4. How much you earn on your savings “r”.

#5 Should I Lease or Buy the equipment?

If you “buy” you pay the full purchase

price and you own the equipment!

If you “lease” you make a modest

down-payment followed by regular

lease payments for a few years, then

you return the equipment (because

you don’t own it).