Solutions to A2StudyPack2012

F370 Solutions to A2 Study Pack -- Page 1© 2010 Dan Greiner

Solutions to the A2 Numerical Problems

I. Basic Problems

Lump Sums

1. If you save a present value of $340 and plan to earn a 6.5% annual return, then what future value do you expect to have after 9 years?

2. If you invest $420 and expect an annual return of 11% and you want to grow this amount into $2,230.58, then how long should you expect to have to wait until you hit your target future value of $2,230.58?

3. If you invest $210 and want it to grow into $264.21 over the next five years, then what annual return must you earn?


4. If you want to have $50,000 in your investment account exactly 12 years from today, then how much present value should you deposit right now if you expect to earn 13% return per year?

5. ** If you save a present value of $3,600 in an account that pays a 7% return for 3 years and then a 10% return thereafter, then what will be your future value after 8 years?


6. ** How long will it take you to triple an initial deposit if you are using an account that pays a 12% return for the first 4 years and then an 8.5% return thereafter?

7. ** Suppose that you invest $1,000 and it earns 9% a year for 5 years. You then add $X to this future amount and proceed to grow that sum for 10 more years at a 6% rate. If your final account level is $4,044.85 after the full 15 years, then what must $X (your time-5 deposit) have been?


Uneven Streams

8. What is the present value of a stream of cash flows that pays $500 at timepoint one, $750 at timepoint two, $170 at timepoint three and $920 at timepoint four? The stream has a level of risk that leads you to believe that its discount rate should be 8.15%.

9. What is the present value of a stream of cash flows that pays $6,000 at timepoint two, $9,500 at timepoint four and $5,400 at timepoint six? The stream has a level of risk that leads you to believe that its discount rate should be 7.3%.


10.The second flow is smudged in the following uneven stream whose discount rate is 9%. If the full stream has a present value of $1,725.16, then what is the number that was smudged out?


Annuities

11. What is the present value of a 3 year annuity with payments (flows) of $650 and a discount rate of 11.2%? Find the answer first the long way by using the lump sum approach as in the above problem…then find it the short way using the annuity keystrokes.

12. What would be the present value of the annuity in the previous question if (all else equal) it were actually an annuity due instead of an ordinary annuity?


13. If a 7-year annuity with an annual payment level of $222 has a present value of $1,221.86, then what must be its discount rate?

14. If an annuity has a present value of $6,700, a discount rate of 9.3% and annual payments of $875.08, then how many flows are in this stream?

15. If a 48-month annuity has a present value of $6,700 and an annual discount rate of 5.8%, then how large are its monthly payments?


16. If a 38-month annuity due has a present value of $12,120 and an annual discount rate of 10.6%, then how large are its monthly payments?

17. Suppose that you save $500 a year for the next 6 years in an account that has an average rate of return of 7.2%. If your first payment is at the end of year one (i.e., at timepoint one), then what future value do you expect to have in the account after six years (i.e., at timepoint six)?

Perpetuities

18. What is the size of a perpetuity’s payment given that is has a discount rate of 7% and a present value of $43.40?


19. What is the present value of a perpetuity with annual flows of $1200 and an annual discount rate of 10%?

20. What is the present value of a growing perpetuity with a time 1 cash flow of $1200, annual flows growing at a rate of 4% and a discount rate of 14%?

Formulas

21. Consider the following five formulas:a) PV = PMT / r

b) PV = FV / (1+r)n

c) PV = ∑ CFi / (1+r)i

d) PV = (PMT / r) (1 - 1/(1+r)n)

e) PV = PMT1 / (r-g)

Which formula gives you the present value of an uneven stream?

Which formula gives you the present value of a perpetuity?

Which formula gives you the present value of a lump sum?

Which formula gives you the present value of an annuity due?

Which formula gives you the present value of an annuity?

Which formula gives you the present value of a growing perpetuity?


II. Implied Return Problems for Real-World Contracts

Please find the implied annual return for the following real-world contracts or situations.

22. A 1-year T-Bill with a face value of $1000 and a current market price of P0 = $968.22.

…oops should be 3.28%

23. A 2-year T-Note with a face value of $1,000, no coupon payments and a current market price of P0 = $932.76.

24. A 10-year corporate bond with a coupon rate of 7%, a face value of $1,000 and a current market price of P0 = $1,032.50.


25. A mortgage loan with a loan amount of 200,000 and 360 monthly loan payments of $1,465 each

26. Next, consider the mortgage loan in the previous question, but now assume that two years have passed and only 336 payments of $1,465 each are left in the loan. Suppose that the loan sells in the secondary market for a market price P0 of $245,665.66 today. What implied return will the buyer earn if they hold the loan and collect its payments until it matures?

27. A pre-existing 36-month car lease that was just sold on the secondary markets for P0 = $10,519.19. It currently has 30 monthly payments of $389 remaining before it expires. The holder of the contracts gets the 30 remaining payments, but they do not gain possession of the car at the end of the lease….possession of the car goes to the original lease writer.


28. A share of preferred stock paying annual dividends of $1.55 and having a current market price of P0 = $19.38.

29. A share of common stock with a current market price P0 = $27.60 and next year’s annual dividend expected to be $2.76. You believe that the stock’s current dividend growth rate is 3.25%.

30. A share of common stock that you expect to hold for three years. You anticipate that the next three annual dividends will be 3.15, 3.40 and 3.65, respectively and you expect to sell this share for $72.65 right after you receive the third dividend. The share’s current market price is P0 = $61.21.


Advanced Problems

31. What is the value today (at timepoint zero) of a deferred annuity that features 8 payments of $400 and a discount rate of 9.2%? The annuity starts in 6 years …which means its first cash flow occurs at timepoint 7.

32. What is the cash flow growth rate g for a growing perpetuity (annual flows) with a time 1 cash flow of $200, a discount rate of 9% and a present value of $2,857.14?


33. If a growing annual-payment perpetuity has a discount rate of 11%, a cash flow growth rate of 3% and a present value of $6,250, the what is the nominal value of the cash flow located at timepoint 3 on its timeline?

34. What is the present value of the following stream of cash flows?


35. What is the present value of the following stream of cash flows?


36. Consider a cash flow stream that looks sort of like a perpetuity with annual $200

payments. But, it isn’t quite a perpetuity because every 4th flow (the 4th 8th 12th 16th

and so on) is nothing ($0). See the timeline given below. If the stream’s discount rate is 8%, then what is its present value?

same pattern repeats

0 1 2 3 4 5 6 7 8 9 10 11 12 13… goes forever

Answers for the A2 Conceptual Questions

1. What is the difference between the Bring-it-Back perspective and the Push-it-Forward perspective?


2. What does it mean to capitalize a cash flow and how is this related to growing a flow? How is the term discount rate related to the term rate of return?

3. Can the present value of any stream of future cash flows, be evaluated as the sum of all its capitalized parts?


4. In the notes it says that the present value of an annuity due will equal (1+r) times the present value of a similar ordinary annuity. Why does this make sense?


Some Conceptual Questions about Answers you derived for some Numerical Problems

5. Consider your answers to numerical problems #19 and #20. You should have gotten $12,000 as a present value for both of these two streams. How can this be …both had a future time one flow of $125 that you put into the PMT key and 10.0 in the [ I ] key on your calculator… but yet they are not the same stream. What is going on here?

6. Consider your answers to numerical problems #22 and #23. Let’s assume that the market prices for both of these Treasury securities reflect a trade that was made during the lat two minutes. Yet, the 1-year bill has an implied return of 3.25% and the 2-year note has an implied return of 3.54% per year. Shouldn’t these two securities have the same implied return or yield to maturity?


7. Consider your answers to numerical problems #25 and #26. When this mortgage loan was first issued for a loan amount of $200,000, the original lender (probably a bank) made an implied return of 7.98%. Now, two years later, the buyer of the contract will only make an implied return of 5.70% How could this make sense given that it is the same loan in both problems?


Study Pack for the A2 Lecture

Vocabulary for the A2 Lecture

Risk Free Rate, Risk Premium

Cash Flow Uncertainty

Time Value of Money

Present Value [PV]

Future Value [FV] , Future Cash Flow [ CFi ], Nominal Value of a Cash Flow

Rate of Return vs. Discount Rate vs. Implied Return [ I ]

Term [N]

Payment [PMT]

Lump Sum, Uneven Stream

Annuity, Annuity Due, Deferred Annuity

Perpetuity, Growing Perpetuity

Formulas for the A2 Lecture

r = rf + rp

PV = FV / (1+r)n or PV = CF / (1+r)n …. Lump Sum …or FV = PV(1+r)n

PV = ∑ [ CFi / (1+r)i ] …..General Formula and Uneven Stream

PV = (PMT / r) (1 - 1/(1+r)n) …Annuity (ordinary)

PV of an annuity due = (1+r) * (PV of an equivalent ordinary annuity)

PV = PMT / r …Perpetuity

PV = PMT1 / (r-g) …Growing Perpetuity

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Solutions to A2StudyPack2012