# Exercises Time Value of Money

BusOrg (Fall 2014)

Exercises– Time Value of Money

[RA - Michael Klotz <>]

FV = (1 + i)n
future value of \$1 compounded atipercent fornperiods / PV = 1/(1 + i)n
present value of \$1 discounted atipercent fornperiods
FV = SUM [i=1 to n] (1 + i)n
future value of \$1 depositedat the endof each ofnperiods
compounded atipercent / PV = SUM [i=1 to n] 1/(1 + i)n
present value of \$1 depositedat the endofnperiods
discounted atipercent

1 - Time value of money (delayed judgment)

Your client ran over Missy's poodle. You believe that your client will be held liable in the amount of \$120,000, but you guess that it will take three years before your client will have to pay. How much should your client set aside right now and invest at 6.2% annual interest to cover this likely judgment?

1. Invest about \$80,000, and in three years you’ll have enough to cover the judgment
2. Invest about \$90,000, and in three years you’ll have enough to cover the judgment
3. Invest about \$100,000, and in three years you’ll have enough to cover the judgment
4. Put aside \$120,000 under a pillow and wait – in other words, not sure

2 - Time to double/tenfold

Your client says that he’s been offered an investment opportunity that will “certainly” double every 5 years and increase tenfold in 15 years. Is this believable? Is there anything fishy?

1. This is believable, given that the “certain” investment assumes an annual return of about 8% (what banks are now paying for long-term CDs)
2. This is hard to believe, given that the assumed “certain” investment return is about 14.5% annually
3. This is fishy – doubling your money in five years means a 14.8% return and increasing tenfold in 15 years means 16.6% -- two different returns
4. Investments are inherently fishy - not sure

3 - Differencesin post-judgment interest

Ten years ago your client was injured in a car accident caused by a Home Depot delivery van. Seven years ago your client won a \$600,000 judgment that calls for post-judgment "interest as provided by statute." The statute allows successful plaintiffs to collect interest on any judgment from the time of judgment at a "rate of 8% per annum or as determined to be reasonable by the court." Today, after all appeals have been exhausted, how much interest is your client entitled to?

1. About \$336,000 – assuming that statutory interest is compounded
2. About \$463,000 – assuming that statutory interest is not compounded
3. About \$723,000 – assuming that the trial judge finds that Home Depot’s cost of capital (to deter its dalliance) has been12.5%
4. No way is post-judgment interest more than judgment - not sure

4 - Power of compounding

You have just graduated from law school and are 25 years old. You start investing \$5,000 every year (on January 1) in a tax-deferred IRA. The IRA invests in an index stock fund that (based on the past 100 years) will have annual returns of 10%. After ten years you stop investing and let the IRA continue to grow at 10%. Meanwhile, your twin sister waits for ten years and then starts investing \$5,000 every year in the same IRA fund with the same 10% annual returns. She continues in the fund until the end of her (your) 65th year. At this point, whose IRA is larger?

1. Your sister’s IRA will be better (\$150,000 for 30 yearsis \$4.5 million), compared to yours (\$50,000 compounded for 40 years is \$2.0 million)
2. Your IRA, which will have close to \$1.7 million, while your sister’s will have just \$1 million
3. Your sister’s IRA because she invests a total of \$150,000, while you invest only \$50,000
4. Do plan to invest in low-cost, index mutual funds – but not sure

5–Time value of future payments – lottery

Your brother-in-law (now your client) asks for advice. He has won the \$20,000,000 West Virginia lottery. He will receive \$1,000,000 at the end of each year for 20 years. He has gotten offers to sell his winning ticket and has been surprised that all the offers are all for less than \$10 million. He’s been offered \$9,000,000, and asks you (disregarding taxes) whether he should sellthe ticket. (You can assume that West Virginia long-term bonds currently carry a 7.6% interest rate.)

1. “Well, Ethan, if you sell you’d have to hope to invest the money at more than 12% to get annual income of \$1,000,000 -- that’s going to be tough.”
2. “Well, Ethan, that’s a mighty fine deal because those payments in the future have a total value today of about \$7,000,000.”
3. “Well, Ethan, it might be best to just take the payments, they’re worth a bit more than \$10 million.”
4. Your hourly rate just went through the roof – but not sure

6–Pre-nupvaluation

Your 50-year-old client is considering a pre-nuptial agreement. She wants to be sure that if the marriage fails she will have enough to live comfortably. She has been offered two options by her (very wealthy) husband-to-be: Option 1 - Payment to her of \$80,000 per year for 50 years, should they divorce. And if he dies, his estate would assume this payment obligation; and if your client dies before the term ends, her estate would receive the annuity's value at death. Option 2 - Payment of a lump sum of \$1,500,000, should they divorce. Ignoring taxes, which is the better deal – assuming that the husband-to-be has a payment risk comparable to that of a safe corporate bond, now paying6.5%?

1. Take the lump-sum option, it’s worth about \$300,000 more than the payout plan
2. Take the payout plan, 50 times \$80,000 is \$4 million –much more than the \$1.5 million lump sum
3. Take the payout plan, the \$1.5 million could only be invested safely at 4% return, generating only \$60,000 per year
4. Sorry, family law is not my area - not sure

7 - Loan amortization

You graduate from law school, find work, and buy a condo. Congrats! You are now thinking of refinancing your condo. You will take out a loan for \$120,000. What would be your monthly payments with a 30-year loan that carries an annual interest rate of 4.5%? How much interest would you pay, and thus be able to deduct, during the first year of the loan? Assuming you are in a 28% tax bracket, what is the after-tax rate you are paying in the first year for this 4.5% loan?

1. This is good to know: \$845 monthly payment / \$7,360 in interest / \$593 monthly payment after taxes
2. This is good to know: \$806 monthly payment / \$6,360 in interest / \$547 monthly payment after taxes
3. This is good to know: \$608 monthly payment / \$5,360 in interest / \$483 monthly payment after taxes
4. Oh, boy, renting sounds much easier - not sure

Your client is interested in buying a business. You help structure the transaction and draw up the appropriate documents, including an escrow account should business earnings and other assumptions be different from what’s been promised. According to the seller, the business in the first five years is expected to have cash flows of \$1,200,000, \$1,400,000, \$1,600,000, \$1,750,000 and \$1,900,00 and then thereafter cash flows of \$2,000,000 annually. The seller says that the business cash flows have a discount rate of 20%. What is the value of the business under these assumptions? How would things change if the discount rate were a more realistic 30%?

1. Seller’s assumptions lead to a business valuation of \$8.5 million, while with a 30% discount rate the business is worth \$5.4 million
2. Seller’s assumptions lead to a business valuation of \$14.5 million, while with a 30% discount rate the business is worth \$10.3 million
3. Seller’s assumptions are unrealistic, because a discount rate of 20% assumes a very high return on investment
4. This is a matter for business moguls, like Warren Buffett – not sure

Bonus – Mutual fund investing

You start working at a law firm and are offered to participate in the law firm’s 401(k) plan – terrific! The firm will match your contributions that will then be invested in mutual funds of your choice. You recently heard of a study that stock mutual funds (which invest in a portfolio of US equity securities) generally do not outperform the market. That is, there’s no point in choosing a managed fund that has had above-average past returns, because those returns statistically will not continue.

So you come down to two choices for your 401(k): Fund #1 is an index fund that just tracks the broad US equity markets and has 0.20% annual management fees, but no other fees. Fund #2 is a managed fund that says it tries to beat the US equity markets and has management fees of 1.2% (the industry average for managed funds) and a low marketing 12b-1 fee of 0.3%. Assuming you invest \$2,500 (and the firm kicks in another \$2,500) for the next 40 years and that the stock market returns its historical average of 9.7% per year over this period, how much will be in Fund #1 compared to Fund #2?

1. Fund #1 will have about \$275,000 – about 1.3% more than Fund #2
2. Fund #1 will have about \$550,000 – about 25% more than Fund #2
3. Fund #1 will have about \$2.1 million – about 50% more than Fund #2
4. Albert Einstein is wrong when he said: “The most powerful force in the universe is compound interest.”