How to calculate present values

Money, it's a crime.Share it fairly but don't take a slice of my pie.Money, so they sayIs the root of all evil today.But if you ask for a raise it's no surprise that they're giving none away. (Waters)

Perpetuity: Constant Payment Forever

PV = PMT/i

This is the present value of receiving a constant payment forever.

C r

EXAMPLE:

• Suppose you wish to endow a chair at your old university. The aim is to provide $100,000 forever and the interest rate is 10%.

$100,000 PV = = $1,000,000 .10

A donation of $1,000,000 will provide an annual income of .10 x $1,000,000 = $100,000 forever.

PV =

Valuing perpetuities

Future Value of a Lump Sum

FV = PV * (1+i)n

Why is this formula correct?

This is the amount that will be accumulated by investing a given amount today for n periods at a given interest rate.

Simple & compound interest Simple interest rates) are calculated by multiplying the rate per

period by the number of periods. Compound interest rates recognize the opportunity to earn interest on interest.

i ii iii iv vPeriods Interest Value Annuallyper per APR after compoundedyear period (i x ii) one year interest rate

1 6% 6% 1.06 6.000%

2 3 6 1.032 = 1.0609 6.090

4 1.5 6 1.0154 = 1.06136 6.136

12 .5 6 1.00512 = 1.06168 6.168

52 .1154 6 1.00115452 = 1.06180 6.180

365 .0164 6 1.000164365 = 1.06183 6.183

Effective Annual Rate (EAR) or Yield (EAY)

EAR or EAY = (1+inom/m)m-1

This is used to calculate the compounded yearly rate. It considers interest being earned on interest.

FV With Compounding Intervals

FV of lump sum for various compounding intervals:

FV = PV * (1+i/m)n*m

where m=number of compounding periods per year

At an extreme there could be continuous compounding, then FV can be calculated as follows: FV = PV (ein) where e=2.7183...

PV of a Lump Sum

PV=FV/(1+i)n

This is the value today of a future lump sum to be received in the future after n periods of time at a given discount rate.

Present valuesexample: saving for a new computer

Suppose: - you need $3000 next year to buy a computer - the interest rate = 8% per yearHow much do you need to set aside now?

3000PV of $3000 = = 3000 x .926 = $2777.77 1.08 1- year discount factor

By end of 1 year $2777.77 grows to $2777.77 x 1.08 = $3000

Suppose you can postpone purchase until Year 2.

3000PV = = 3000 x .857 = $2572.02 1.082

2-year discount factor

PV With Compounding Intervals

PV of a lump sum for various compounding intervals is calculated as:

PV=FV/(1+i/m)n*m

where m=number of compounding periods per year

At an extreme there could be continuous discounting, then PV=FV/(ein) where e=2.7183...

PV of an Annuity

PV=A/(1+i)n = A*{(1/i) - (1/i) [1/(1+i)n]}

This is the value today of a series of equal payments to be received at the end of each period for n periods at a given interest rate.

Asset Year of payment PV

1 2 . . t t+1 . .

Perpetuity (first payment year 1)

Perpetuity (first payment year t + 1)

Annuity from year 1 to year t (1+r)

1t)

Cr(-

Cr

(1+r))rC 1

( t

Cr

An annuity is equal to the difference between two perpetuities

Using the annuity formulaExample: valuing an 'easy payment' schemeSuppose:

a car purchase involves 3 annual payments of $4000 the interest rate is 10% a year

1 1PV = $4000 x - .10 .10(1.10)3

= $4000 x 2.487 = $9947.41

ANNUITY TABLE

Number Interest Rate of years 5% 8% 10%

1 .952 .926 .909 2 1.859 1.783 1.736 3 2.723 2.577 2.487 5 4.329 3.993 3.791 10 7.722 6.710 6.145

FV of an Annuity

FV=A* (1+i)n = A*{[(1+i)n -1]/i}

This is the accumulated value of equal payments for n years at a given interest rate.

Annuity Due

Annuity due: Payments received at the beginning of each period.

Will be worth more (higher PV) since it gets payments sooner.

Will have higher FV since it has one extra period to earn interest.

Calculations are the same as before except now we multiply by (1+i).

Solving for Annuity Payments (Present Value)

Recall that

PV=A*{(1/i) - (1/i) [1/(1+i)n]}, then

A=PV/{(1/i) - (1/i) [1/(1+i)n]}

• A is the payment necessary for n years at given interest rate to amortize a present (loan) amount.

Solving for Annuity Payments (Future Value)

Recall that

FV=A*{[(1+i)n-1]/i}, then

A=FV/{[(1+i)n-1]/i} •A is the amount needed to be invested each period at a given interest rate to accumulate a desired future amount at the end of n years.

Solving for Rate of Return (i)

For Lump Sum Case:

Since PV=FV/(1+i)n, then

(1+i)n=FV/PV, and it follows that

(1+i) = (FV/PV)1/n, and therefore

i= (FV/PV)1/n-1

Solving for Rate of Return (i)

Annuities:

• In the annuity case, you could also solve for i using annuity relationship once you know the annuity.

• You do not need a cash flow register.

Solving for Rate of Return (r) with uneven cash flows

0 = C0 + + + . . .

•Spreadsheets (use financial function =IRR)

•Financial calculators (IRR using cash flow register)

•Manual (Trial and error until PV of all cash flows equal zero)

C1 C2

(1 + r)1 (1 + r)2

Solving for Number of Periods (n)

Since PV=FV/(1+i)n, then

(1+i)n=FV/PV, and it follows that

nLN(1+i)=LN(FV/PV), and therefore

n=LN(FV/PV)/LN(1+i)

Net Present Value in the General discounted cash flow

formula

NPV = C0 + + + . . .

Note: It is today’s cost of capital that matters

C1 C2

(1 + r)1 (1 + r)2

ExampleIf C0 = -500, C1 = +400, C2 = +400

r1 = r2 = .12

NPV = -500 + +

= -500 + 400 (.893) + 400 (.794)

= -500 + 357.20 + 318.80 = +176

400 400

1.12 (1.12)2

Growing perpetuitiesPV =

EXAMPLE:

Next year’s cash flow = $100

Constant expected growth rate = 10%

Cost of capital = 15%

PV =

= = 2000

C

(r - g)

next year’s cash flow

cost of capital - growth rate

100

.15 - .10

Valuing bondsSome definitions: BOND Long-term debt issued by government or firms COUPON Regular interest payment on bond FACE VALUE Amount repaid at maturity (usually $1000)

Example: A Treasury 9% coupon bond matures in 1998. In 1993 each bond offered these cash flows:

1994 1995 1996 1997 1998 $90 $90 $90 $90 $1090

The interest rate on similar bonds was 5.3%.Therefore the value of 9% Treasuries was

$90 $90 $90 $90 $1090PV = + + + + (1 + r) (1 + r)2 (1 + r)3 (1 + r)4 (1 + r)5

$90 $90 $90 $90 $1090 = + + + + 1.053 (1.053)2 (1.053)3 (1.053)4 (1.053)5

= $1158.87

Calculating bond yields

If the price of the 9% Treasury bond is $1158.87,what return do investors expect? Return is usuallymeasured by the yield to maturity. This is the discountrate, r, that makes a bond's present value equal to price.

$90 $90 $90 $90 $1090PV = + + + + (1 + r) (1 + r)2 (1 + r)3 (1 + r)4 (1 + r)5

= $1158.87

Yield to maturity (r) = .053 or 5.3%.

How bond prices vary with interestrate (yield to maturity)

0

200

400

600

800

1000

1200

1400

1600

0 2 4 6 8 10 12 14

5 Year 9% Bond 1 Year 9% Bond

Yield

Pri

ce

How bond prices vary with interestrate (yield to maturity)

• Inverse relationship.

As yields go up (down) prices on existing bonds go down (up).

• The longer the maturity, the more sensitive the bond prices are to interest rate changes.