Download pdf - Mod 4 - TVM - Taxes - Slides

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Michael R. Roberts William H. Lawrence Professor of Finance The Wharton School, University of Pennsylvania

Time Value of Money: Taxes

Copyright © Michael R. Roberts


Last TimeTime Value of Money•  Useful shortcuts

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This TimeTime Value of Money•  Taxes

Taxes


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Tax Rates


Source: Graham, John R., Mark T. Leary, and Michael R. Roberts, 2014, “A Century of Corporate Capital Structure: The Leverage of Corporate America,” forthcoming Journal of Financial Economics

Tax Rates


How do taxes impact our returns?

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How much do you have to save today to withdraw $100 at the end of each of the next four years if you can earn 5% per annum?

Example – Savings (Discounting)


0 1 2 3 4

354.60 100 100 100 100

90.703

95.238

86.384

82.270

+

+

+

+

=

Recall…

Example – Savings (Discounting)


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Year InterestPre-Withdrawal

Balance WithdrawalPost-Withdrawal

Balance0 $354.601 $17.73 $372.32 $100.00 $272.322 $13.62 $285.94 $100.00 $185.943 $9.30 $195.24 $100.00 $95.244 $4.76 $100.00 $100.00 $0.00

Example – Savings (Account)


Savings with Taxes (Account)

Year InterestTaxes(35%)

Pre-Withdrawal Balance Withdrawal

Post-Withdrawal Balance

0 $354.601 $17.73 -$6.21 $366.12 $100.00 $266.122 $13.31 -$4.66 $274.77 $100.00 $174.773 $8.74 -$3.06 $180.45 $100.00 $80.454 $4.02 -$1.41 $83.06 $83.06 $0.00


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Savings with Taxes (Account)

We are $100 – $83.06 = $16.94 short. Taxes reduce funds available for withdrawal. We run out of money early

Year InterestTaxes(35%)

Pre-Withdrawal Balance Withdrawal

Post-Withdrawal Balance

0 $354.601 $17.73 -$6.21 $366.12 $100.00 $266.122 $13.31 -$4.66 $274.77 $100.00 $174.773 $8.74 -$3.06 $180.45 $100.00 $80.454 $4.02 -$1.41 $83.06 $83.06 $0.00



Lesson: Taxes reduce the return on our investment, R

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Rt = R x (1 – t)

•  For our example:

5% x (1 – 35%) = 3.25%

After-tax Discount Rate


Savings with Taxes

0 1 2 3 4

? 100 100 100 100

1001+ 0.0325( )2

1001+ 0.0325( )

1001+ 0.0325( )3

1001+ 0.0325( )4


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Savings with Taxes

0 1 2 3 4

369.50 100 100 100 100

93.804

96.852

90.851

87.991

+

+

+

+

=


Savings with Taxes

0 1 2 3 4

369.50 > 354.60 100 100 100 100

93.804

96.852

90.851

87.991

+

+

+

+

=


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Savings with Taxes

Year Interest TaxesPre-Withdrawal

Balance Withdrawal

Post-Withdrawal

Balance0 $369.501 $18.47 -$6.47 $381.51 $100.00 $281.512 $14.08 -$4.93 $290.66 $100.00 $190.663 $9.53 -$3.34 $196.85 $100.00 $96.854 $4.84 -$1.69 $100.00 $100.00 $0.00


Savings with Taxes

Implication: We need to save more to ($369.50 > $354.60) to withdraw $100 each year after taxes


Balance Withdrawal

Post-Withdrawal

Balance0 $369.501 $18.47 -$6.47 $381.51 $100.00 $281.512 $14.08 -$4.93 $290.66 $100.00 $190.663 $9.53 -$3.34 $196.85 $100.00 $96.854 $4.84 -$1.69 $100.00 $100.00 $0.00


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Savings with Taxes

Note: $369.50 – $354.60 = $14.90 which also equals the present value of the taxes at 5%. (Check this!)


Balance Withdrawal

Post-Withdrawal

Balance0 $369.501 $18.47 -$6.47 $381.51 $100.00 $281.512 $14.08 -$4.93 $290.66 $100.00 $190.663 $9.53 -$3.34 $196.85 $100.00 $96.854 $4.84 -$1.69 $100.00 $100.00 $0.00


Summary


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Lessons

•  Taxes reduce our dollar return

•  The after-tax return, Rt, on an investment is:

where R is the nominal return and t is the tax rate

Rt =R × 1− t( )


Coming up next

•  Time Value of Money– How does inflation affect our returns and

value of money?


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Problems


These problems are designed to test your understanding of the material and ability to apply what you have learned to situations that arise in practice – both personal and professional. I have tried to retain the spirit of what you will encounter in practice while recognizing that your knowledge to this point may be limited. As such, you may see similar problems in future modules that expand on these or incorporate important institutional features.

Know that all of the problems can be solved with what you have learned in the current and preceding modules. Good luck!

Problem Instructions


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You are preparing to buy a car that costs $36,000. You can pay for the car using an auto loan from the car manufacturer or using money from your home equity line of credit (HELOC). The auto loan charges 2.75% interest per annum. The HELOC charges 3.85% interest per annum but the interest is tax deductible. If your current tax rate is 32%, which source of funds should you use?

Problem – HELOC


HELOC after-tax interest rate, Rt =R × 1− t( ) = 0.0385 × 1− 0.32( ) = 2.618%

Use the HELOC. The after-tax rate is less than the after-tax rate on the auto loan. (though this is a no-no in practice…)

The interest on some municipal bonds is tax free, in contrast to the interest on corporate bonds. If the current annual interest rates on otherwise similar (i.e., maturity, credit risk, liquidity) municipal bonds are 1.48% and 1.80%, what is the implied tax rate?

Problem – Municipal Bonds


Rt =R × 1− t( )⇒ t = 1− Rt /R( ) = 1− (1.48 / 1.80) = 17.78%

This tax rate reflects the tax rate of the marginal investor, not the highest statutory tax rate. It also reflects differences in liquidity and perceived credit quality.

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Problem – Retirement (Setup)


You are currently 27 years old and plan on retiring at age 67. Based on current life tables and family history, you anticipate living another 23 years after retirement (morbid but important to consider). You estimate that you will need $157,212 in after-tax income per year in retirement, during which time all withdrawals from your retirement savings will be taxed at 25%. Assume that you can earn 5% per annum on your investments.

Answer the following questions using this information.

Problem – Retirement 1


How much pre-tax income do you need in retirement each year in order to meet your retirement needs?

After-tax income = Pre-tax income × 1− tax rate( )⇒ Pre-tax income = After-tax income1− tax rate( )

= $157,2121− 0.25( ) = $209,616

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How much money do you need at the start of your retirement to meet the pre-tax income demands during retirement? Assume that you will live on your last year of income in the first year of retirement and then draw down your nest egg at the start of the second year (equivalently, the end of the first year).

PV40 =$209,6160.05

1− 1+ 0.05( )−23⎡⎣ ⎤⎦ = $2,827,420.9033

0 1 2 39 40 41

$210k $210k $210k

62 63

27 28 29 66 67 68 89 90 Age Period

Compare with $2,120,565 ignoring taxes

Retirement Years Working Years



How much money do you need today to meet your pre-tax income demands during retirement if you will live on your last year of income in the first year of retirement and then draw down your nest egg at the start of the second year (equivalently, the end of the first year)?

PV40 =$209,6160.05

1− 1+ 0.05( )−23⎡⎣ ⎤⎦ = $2,827,420.9033

0 1 2 39 40 41

$157k $157k $157k

62 63

27 28 29 66 67 68 89 90 Age Period

Compare with $301,217 ignoring taxes

PV0 =$2,827,420.901+ 0.05( )40

= $401,622.93


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Assuming you do not currently have any savings, how much do you have to save each year during your working years to ensure that you will have enough for your retirement after accounting for the taxes in retirement?

Compare with $17,554.38 ignoring taxes

0 1 2 39 40 41 62 63

27 28 29 66 67 68 89 90 Age Period

? $301,217 ? ? ?

401,622.93= C0.05

1− 1+ 0.05( )−40⎡⎣ ⎤⎦⇒C = $401,622.93× 0.051− 1+ 0.05( )−40⎡⎣ ⎤⎦

= $23,405.85




Recognizing the importance of inflation, you realize that while you will need $157,212 in your first year of retirement, this need will grow at approximately 2% per year. Recalculate your annual savings during your working years to meet this increased after-tax retirement burden assuming you do not have any savings today?

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Problem – Retirement 5 (Cont)


Compare with $21,109.43 ignoring taxes

PV40 =$209,6160.05 − 0.02

1− 1+ 0.051+ 0.02

⎛⎝⎜

⎞⎠⎟−23⎡

⎣⎢

⎤

⎦⎥ = $3,400,019.36

0 1 2 39 40 41

$210k $210k x 1.0221 $210k x 1.0222

62 63

PV0 =$3,400,019.361+ 0.05( )40

= $482,958.07

482,958.07 = C0.05

1− 1+ 0.05( )−40⎡⎣ ⎤⎦⇒C = $482,958.07 × 0.051− 1+ 0.05( )−40⎡⎣ ⎤⎦

= $28,145.91


27 28 29 66 67 68 89 90 Age Period

How would your answer to the previous problem change if your age today was 37, instead of 27, and all other information remains unchanged. That is, instead of having 40 years to work and save until retirement, you only have 30.



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Problem – Retirement 6 (Cont)


0 1 2 29 30 31

$157k $157k x 1.0221 $157k x 1.0222

52 53

$210k $210k x 1.0221 $210k x 1.0222 Before-Tax

After-Tax Before Tax = After Tax / (1 – Tax Rate) ÷ (1-0.25) ÷ (1-0.25) ÷ (1-0.25)


PV0 =$3,400,019.361+ 0.05( )30

= $786,687.80

CF = $786,687.80 × 0.05 × 1− 1+ 0.05( )−30( )−1 = $51,175.17

PV30 =$209,6160.05 − 0.02

1− 1+ 0.051+ 0.02

⎛⎝⎜

⎞⎠⎟−23⎡

⎣⎢

⎤

⎦⎥ = $3,400,019.36

Compare with $28,145.91 if you start saving at 27

37 38 39 66 67 68 89 90 Age Period