Tuck Bridge Finance Module 2

8/4/2019 Tuck Bridge Finance Module 2

1/21

CLASS 2

NPV ANALYSIS

Bridge Program 2005

Finance module

Finance, Bridge Program 2005 1


2/21

Contents

1 The PV formula 4

2 Assignment 2: bond pricing 11

3 Assignment 2: mortgage application 15

4 Loose ends 19

4.1 Taxes . . . . . . . . . . . . . . . . . . . . . . . . . . 194.2 Inflation . . . . . . . . . . . . . . . . . . . . . . . . 20



3/21

Recap

Main concepts from last class:

1. Effect of compounding.

2. Calculating the term structure of interest rates.

Next two classes will be applications of the idea ofdiscounting to different business situations.



4/21

1 The PV formula

The value today of a cash flow Ct occuring at date t is

PV(Ct) =Ct

(1 + rt)t;

The value of a stream of cash flows C0, C1, . . . is

PV(C0, C1, . . . , C T) = C0 +C1

1 + r1+

C2

(1 + r2)2+ . . .

=Tt=0

Ct

(1 + rt)t.



5/21

DiscountingFirst, the PV rule is obvious: we cant just add up cash flows

(money) of a project, since a dollar today may have a different

value than a dollar tomorrow. So we find the value of future

cash flows by discounting them at some rates.

The discount rates will be taken as given in the next two classes.

Easiest way to think about the projects/problems we will

look into: risk-free cash flows, so the rts are just the spot

rates from bond prices (term structure).

Loosely (will make this precise in class 6) interpretation:

the cash flows have some risk and they are therefore

discounted at a higher rate to adjust for this risk.



6/21

Perpetuities and annuities (shortcuts)

Assume flat term structure (all spot rates equal to r).

A perpetuity is a stream of cash flows that are the same each

period forever.

PVt =CFt+1

r.

Example 1. Security that pays $100 annually in perpetuitystarting next year. Discount rate 10% (annual terms). What is

its value?

V =100

0.10= 100(10) = $1000.

Example 2. Security that pays $100 annually in perpetuitystarting 5 years from today. Discount rate 10% (annual terms).

What is its value?

V =1

1.14

100

0.10

=

1000

1.14= 683.01



7/21

An annuity is a stream of (equal) cash flows for a given number

of periods. Unlike a perpetuity, payments stop after T periods.

PVt =CFt+1

r

1

1

(1 + r)T

Remembering the annuity formula:

CFt+1

r Perpetuity value

1 1

(1 + r)T

Annuity adjustment

Note: the annuity and perpetuity formulas are simple shortcuts

to summations, i.e.

PV(C1, . . . , C T) =Tt=1

C

(1 + r)t= C

Tt=1

1

(1 + r)t.



8/21

Example 3. Security that pays $100 annually for the next 4

years. Discount rate 10% (annual terms). What is its value?

V =100

0.1

1

1

1.14

= 316.98

Note: value in example 3 is equal to value in example 1 minus

value in example 2 (not by chance).Example 4. Security that pays $100 each month in perpetuity.

First compute monthly discount rate:

(1 + rm)12 = 1.10; rm 0.7974%

Therefore perpetuity value is

V =100

0.007974= 100(125.40) = $12540.54.



9/21

Example 5. Now suppose you plan to work for thenext three years. How much of your after-tax income

would you need to save to go to Business School?

Assume you need to pay $50,000 three years from now,

and $55,000 four years from now.

For simplicity assume you get paid at the end of years

t = 1, . . . , 3. Let r = 10% (annual rate).

Good practice problem: find the amount you would have

to save on a monthly basis. Answer (following similar

logic as in next slide): $2409 a month.



10/21

First, lay out the cash flows.

Time 0 1 2 3 4

Cash flow MBA - - - 50 55

Cash flow from income - x x x -

From which we see that x must satisfy:

x

0.10

1

1

(1.1)3

Value of savings

= 75.13 Value of MBA

so that x $30, 211.

Note that

75.13 =50

1.13+

55

1.14.



11/21

2 Assignment 2: bond pricing

Find the value of a 6%-coupon bond with a face value of

$100,000, which matures in May 2048. Assume coupons

are paid annually.

This bond pays $6K in May 45, May 46, and May 47,

and $106K (principal plus coupon) on May 48.

To find the value we discount these cash flows:

V = 61.0299

+ 61.0681

+ 61.1248

+ 1061.1883

= $105.983



12/21

Assignment 2: bond pricing

Find the value of a 8%-coupon bond with a face value of$500,000, which matures in February 2049. Assume the

coupons are paid semi-annually.

The bond has 9 coupon payments of $20K each, plus a

payment of principal and the last coupon, amounting to

$520K at maturity.

V =

20

1.0090.25 +

20

1.020.75 +

20

1.0251.25 + +

520

1.044.75

= $583.838



13/21

An aside on discounting

Note in previous two slides I discounted (1) using total holding

returns, (2) using annualized returns (and accounting for when

cash flows occured). They are equivalent.

Yet one more way: use discount factors, the price of $1 at time t

pt =1

(1 + rt)t

PV formula can be written as

PV =T

t=1ptCtFor example, for 6% coupon bond

V = 6(0.9709) + 6(0.9362) + 6(0.8891) + 106(0.8416) = 105.983



14/21

Assignment 2: bond pricingImagine you want to issue a coupon bond with face value of

$300,000, which matures in May 2047, and which pays an

annual coupon of x%. For what coupon x would the bond sell at

par (i.e. its price would equal its face value)?

This bond has cash flows of 300x in May 45 and May 46, and of

300(1 + x) in May 47.

V =300x

1.0299

+300x

1.0681

+300(1 + x)

1.1248

.

The above is the value of the bond, given x. Setting V = 300

(its face value), we can solve for x = 3.967%.



15/21

3 Assignment 2: mortgage application

$300000 mortgage loan, with 6% annual interest rate. Payable

in equal monthly installments.

What is the monthly payment the bank will ask for?

First we convert the annual rate into a monthly rate:

rm = 0.06/12 = 0.5%.

Using the annuity formula we can find the payment as before bysolving

300000

House cost (financed)=

C

0.005

1

1

(1 + 0.005)240

Value of cash flows to the bankso that

300000 = 139.58C; C = 2149.29



16/21

Payments of interest and principal

Each year calculate what portion of the $2149 payment is

interest, and what portion is principal repayment.

Month Balance Interest Principal Balance

(before payment) (after payment)

1 300,000.00 1,500.00 649.29 299,350.71

2 299,350.71 1,496.75 652.54 298,698.17

3 298,698.17 1,493.49 655.80 298,042.36

4 298,042.36 1,490.21 659.08 297,383.28

Interest is simply (0.5%)(Outstanding balance).

The outstanding balance at the end of the month equals thebeginning balance minus the principal payment.

You pay more interest than principal early on, and pay more

principal than interest when the loan approaches its maturity

date.



17/21

Amortization schedule

Months

Intere

standprincipalpayments

0 50 100 150 200

0

500

1000

1500

2000

2500



18/21

Tax effects

Mortgate payments are deductible from federal taxes (at

least). Therefore, for an investor who buys a house, theactual yearly cash flow due to the loan is actually lower

than $2149.

The actual after-tax amount will depend on their marginal

tax rate, and can be easily obtained from the calculations

outlined above once this tax rate is known

A simple approximation subtracts

(tax rate) (Interest expense) to the actual mortgage

payment.

In month 1 for example, the after-tax payment on themortgage with a marginal tax rate of 30% would be

2149 (0.3)1500 = 2149 450 = 1699.

In month 20 (see spreadsheet) we have an after-tax

payment of 2149 (0.3)1435 = 1718.



19/21

4 Loose ends

4.1 Taxes

We care about what we consume: after-tax dollars.

Important concepts:

Marginal tax rates. Most decisions are incremental: will gettaxed at marginal tax rate.

Capital gains taxes versus ordinary income taxes. Capital

gains taxes are only paid when an asset is sold (at 15%). In

contrast, interest income is taxed on the year you receive it

(at ordinary income tax rate, say 34%).

Taxes fill up hundreds of volumes of books.

You should know the basics discussed in chapter 6.



20/21

4.2 Inflation

Nominal interest rate (rn): the return you receive in dollars.

Real interest rate (rr): the return you receive in consumption

units.

Inflation rate (): the increase in the dollar price of a particular

basket of goods.

Their relationship

(1 + rn) = (1 + rr)(1 + ).

Think about the above formula as compounding effect ofinflation.

Rule of thumb: discount real CFs at real rate, nominal CFs at

nominal rate.



21/21

Main topics for class 2

The NPV rule.

Annuities and perpetuities.

Mortgage calculations and after-tax effects.

Taxes and inflation.

Main topics for class 3

More applications of the NPV rule.

Assignment 3: evaluating projects (accounting plusdiscounting).


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Tuck Bridge Finance Module 2