# PV & FV of a lump sum, PV & FV of an annuity. Differences?

There are four basic calculations: PV of a lump sum, FV of a lump sum, PV of an annuity, and FV of an annuity. Discuss the differences between each. Use examples to help explain.

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#### Solution Preview

First, bear in mind the concept of the time value of money: that an investor (or most any one) would rather have the money today as opposed to some future date. If a person concludes they want the money today rather than a later date, the money will be discounted from the future back to today.

Why? Because the one who will pay the money can invest the funds at interest in the interim period between today and the future date. The payer does not have to put up as much money at the front because part of the future payment will come ...

#### Solution Summary

The solution presents a very practical response in explanation of the terms together with common real life examples.

Finance: PV of cash, annuities, FV of uneven cash flow, expected returns, CAPM, SML

Problem: PV of cash Flow stream.

A rookie quarterback is negotiating his first NFL contract. His opportunity cost is 10%. He has been offered three possible 4-year contracts. Payments are guaranteed and they would be made at the end of each year. Terms of each contract are as follows:

Year 1 2 3 4

Contract 1 $3,000,000 $3,000,000 $3,000,000 $3,000,000

Contract 2 $2,000,000 $3,000,000 $4,000,000 $5,000,000

Contract 3 $7,000,000 $1,000,000 $1,000,000 $1,000,000

As his advisor, which contract would you recommend that he should accept?

Problem: FV of uneven cash flow

You want to buy a house within 3 years, and you are certainly saving for the down payment. You plan to save $5000 at the end of first year, and you anticipate that your annual savings will increase by 10% annually thereafter. Your expected annual return is 7%. How much will you have for a down payment at the end of year 3?

Problem: Evaluating lump sums and annuities

Crissie just won the lottery, and she must choose between three award options. She can elect to receive a lump sum today $61 million, to receive 10 end-of-year payments of $9.5 million, or to receive 30 end-of-year payments of $5.5 million.

a.If she thinks she can earn 7% annually, which should she choose?

b.If she thinks she can earn 8% annually, which should she choose?

c.If she thinks she can earn 9% annually, which should she choose?

d.Explain how interest rates influence the optimal choice.

Problem: PV and Loan eligibility

You have saved $4000 for a down payment on a new car. The largest monthly payment you can afford is $350. The loan will have a 12% APR based on end-of-months payments. What is the most expensive car you can afford if you finance it for 48 months? For 60 months?

Problem: Expected Returns

Stock X and Y have the following probability distributions of expected future returns:

Probability X Y

0.1 (10%) (35%)

0.2 2 0

0.4 12 20

0.2 20 25

0.1 38 45

a. Calculate the expected rate of return, ry, for stock Y (rx=12%)

b. Calculate the standard deviation of expected returns for stock X (S.D for stock Y=20.35%). Now calculate the coefficient of variation for stock Y. Is it possible that most will regard stock Y as being less risky than stock X? Explain.

Problem: CAPM and required return

Bradford manufacturing company has a beta of 1.45, while Farley Industries has a beta of 0.85. The required return on an index fund that holds the entire stock market is 12%. The risk free rate of interest is 5%. By how much does Bradford's required return exceed Farley's required return?

Problem: Required Rate of return

Suppose rRF=9%, rM=14% and bi=1.3

a.What is ri, the required rate of return on Stock i?

b.Now suppose that rRF(1) increases to 10% or (2) decreases to 8%. The slope of SML remains constant. How much this affect rM and ri?

c.Now assume that rRF remains at 9% but rM (1) increases to 16% or (2) falls to 13%. The slope of SML does not remain constant. How would these changes affect ri?

Problem: Security Market Line

You plan to invest in Kish Hedge fund, which has total capital of $500 million invested in 5 stocks:

Sock Investment Stock's Beta coefficient

A $160 million 0.5

B $120 million 1.2

C $80 million 1.8

D $80 million 1.0

E $60 million 1.6

Kish's beta coefficient can be found as a weighted average of its stocks' betas. The risk free rate is 6%, and you believe the following probability distribution for future market returns is realistic:

Probability Market Return

0.1 -28%

0.2 0

0.4 12%

0.2 30%

0.1 50%

a.What is the equation for Security Market Line (SML)? (Hint : First determine the expected market return)

b.Calculate Kish's required rate of return.

c.Suppose Rick Fish, the president, receives a proposal from a company seeking new capital. The amount needed to take a position in the stock is $50 million, it has an expected return of 15%, and its estimated beta is 1.5. should Kish invest in the new company? At what expected rate of return should Kish be indifferent to purchasing the stock?