Mod 4 - TVM - Taxes - Slides

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Mod 4 - TVM - Taxes - Slides

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  • 7/7/15

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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

    Copyright Michael R. Roberts

    Last TimeTime Value of Money Useful shortcuts

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    Copyright Michael R. Roberts

    This TimeTime Value of Money Taxes

    Taxes

    Copyright Michael R. Roberts

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    Tax Rates

    Copyright Michael R. Roberts

    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

    Copyright Michael R. Roberts

    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)

    Copyright Michael R. Roberts

    0 1 2 3 4

    354.60 100 100 100 100

    90.703

    95.238

    86.384

    82.270

    +

    +

    +

    +

    =

    RecallExample Savings (Discounting)

    Copyright Michael R. Roberts

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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)

    Copyright Michael R. Roberts

    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

    Copyright Michael R. Roberts

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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

    Copyright Michael R. Roberts

    Copyright Michael R. Roberts

    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

    Copyright Michael R. Roberts

    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

    Copyright Michael R. Roberts

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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

    +

    +

    +

    +

    =

    Copyright Michael R. Roberts

    Savings with Taxes

    0 1 2 3 4

    369.50 > 354.60 100 100 100 100

    93.804

    96.852

    90.851

    87.991

    +

    +

    +

    +

    =

    Copyright Michael R. Roberts

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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

    Copyright Michael R. Roberts

    Savings with Taxes

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

    Copyright Michael R. Roberts

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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!)

    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

    Copyright Michael R. Roberts

    Summary

    Copyright Michael R. Roberts

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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( )

    Copyright Michael R. Roberts

    Coming up next Time Value of Money

    How does inflation affect our returns and value of money?

    Copyright Michael R. Roberts

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    Problems

    Copyright Michael R. Roberts

    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

    Copyright Michael R. Roberts

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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

    Copyright Michael R. Roberts

    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

    Copyright Michael R. Roberts

    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)

    Copyright Michael R. Roberts

    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

    Copyright Michael R. Roberts

    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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    Problem Retirement 2

    Copyright Michael R. Roberts

    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

    Problem Retirement 3

    Copyright Michael R. Roberts

    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

    Retirement Years Working Years

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    Problem Retirement 4

    Copyright Michael R. Roberts

    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

    Retirement Years Working Years

    Problem Retirement 5

    Copyright Michael R. Roberts

    Recognizing the