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44

3. The Federal National Mortgage Association (FNMA) issues MBS call] "Fannie Maes." FNMA is chartered by die U. S. Government and co oration (but not the government) guarantees payment of principal interest.

Collateralized mortgage obligations

Collateralized mortgage obligations (CMOs) are a newer type of financ instrument designed to improve upon the traditional MBS. CMOs involve i same type of investment in real estate mortgages as MBS, but involve compl( structuring of the portfolios designed to lessen the uncertainty in cash flow i is typical in traditional MBS.

I However, they do not have a specific maturity date and are sold on a basis ch quotes their "average life." The average life is the average amount of diat each dollar of principal is expected to be outstanding, assuming a enable prepayment schedule for the mortgages in the pool. CMOs are uctured with a wide range in average life in order to provide more choice to I investor.

CMOs typically offer higher yields than corporate bonds in order to ate for the uncertainly of their repayment schedule and the attendant vestment risk. CMOs make payments monthly or quarterly rather than niannually as with bonds. CMOs are quite liquid, since there is an active rket in existing issues. Prices in the market for existing CMOs vary versely with prevailing interest rates, similarly to bond prices.

tions

lOptions are financial instruments which give the owner the right to buy or I security on a future date at a fixed price, called the exercise price or price. There are two types of options: calls and puts. A call gives the er die right to purchase the security at the exercise price, while a put gives iiowner the right to sell the security at the exercise price. With European the exercise of the option must occur on a fixed date at which time the rtion expires. With American options the exercise can occur on any future i through the expiry date.

Investors can either buy or sell options. Thus, if an investor thinks the rity price is likely to rise, then a call should be purchased or a put should old. Conversely, if an investor thinks the security price is likely to fall, a put should be purchased or a call should be sold. IfeThere are two primary motivations for buying or selling options, which are asite ends of the spectrum. One motivation is speculation. Option prices g;highly volatile, which leads to a large potential for profit and a large risk ss. For example, consider a security selling at $50 with a call option to Jthe security at $45. The option is worth at least $5 (it will sell for more $5 prior to expiry). If the security increases in price to $55 the option is I at least $10. Thus, a 10% increase in the price of the underlying security is approximately to a 100% increase in the value of the option. This effect ailed leverage.

slThe second motivation is quite the opposite. Options offer great flexibility developing hedging strategies to reduce investment risk. Hedging was ussed briefly in Section 8.7. Many of the option hedging investment ategies have become quite complex, and require computer analysis to plement effectively.

Mutual funds are pooled investment accounts in which the indivic investor buys shares. Their primary advantage is to offer a greater degree i diversification than individuals could achieve on their own.

Originally, mutual funds were primarily invested in common stocks. recently, mutual funds offering a broad array of other investment options hav appeared. Even within the traditional category of common stock mutual fur considerable variation exists among various funds in their relative emphases i safety of principal, income, and growth.

Mortgage backed securities

Mortgage backed securities (MBS) are securities which are created out i a defined pool of real estate mortgages. They are issued by governmentally owned or chartered corporations holding large numbers mortgages. Investors receive periodic payments consisting of both principal i interest. The rate at which principal is repaid is variable depending on the i at which the underlying mortgages are paid off The tiiree co orations are j follows:

1. The Government National Mortgage Association (GNMA) issues called "Ginnie Maes." GNMA is owned by the U. S. Government payment of principal and interest is guaranteed by the full faith and ct of the U. S. Government.

2. The Federal Home Loan Mortgage Co oration (FHLMC) issues called "Freddie Macs." FHLMC is chartered by the U. S. Government i the co oration (but not the government) guarantees payment of princip and interest.



Show that the following series of identities is valid: - =A-AO +i)-" = C

8.3 Real estate mortgages

1000

111.

112. F

r III

V 113.

A family purchases a house for $ 160,000. They agree to put 25 % down and finance the balance with a 30-year fixed rate mortgage at 9 %. In order to secure this loan they must pay 2 points at setdement. They go to settiement on September 16. Find the total amount of interest paid during the first calendar year of the mortgage, if 1 112 points is deemed to be interest and the family makes payments on the first day of each month.

Find the APR on the mortgage loan in Exercise 10, assuming that the other 1/2 point and all other closing costs do not have to be reflected in die APR.

A 15-year mortgage has monthly payments of $1000 with interest convertible monthly. At the end of each month, die borrower makes a $1000 payment. In addition to the regular monthly payment, the borrower makes an additional payment equal to the amount of principal that would have been repaid in the next regular monthly payment. Under this arrangement the loan will be completely repaid after 90 payments. Show that the amount of interest saved over the Ufe of the loan is equal to

90,000-1000113-s

where the annuity symbols are computed at the monthly rate of interest on the loan.

A builder took out a $2,000,000 construction loan disbursed in three installments. The first installment of $1,000,000 is disbursed immediately and this is followed by two $500,000 installments at six month intervals. The interest on the loan is calculated at a rate of 15% convertible semiannually and accumulated to die end of the second year. At that time, the loan and accumulated interest will be replaced by a 30-year mortgage at 12% convertible monthly. The amount of the monthly mortgage payment for the first five years will be one-half of the payment for the sixth and later years. The first monthly mortgage payinent is due exacdy two years after the initial disbursement of the construction loan. Find the amount of the 12th mortgage payment. Answer to the nearest dollar.

A $100,000 loan is to be repaid by 30 equal payments at die end of each year. Interest on the loan is at 8% effective. In addition to the annual payments, the borrower must pay an origination fee at the time the ban is made. This fee is 2% of die loan but does not reduce die loan balance. When die second payment is due, die borrower pays die remaining loan balance in fiill Determine die yield to die lender considering the origination fee and the early pay-off of die loan.

3. A loan of $12,000 is to be repaid in one year according to one of the following arrangements:

A - $1000 payable at the end of each month in addition to a finance charge] of $1000 payable when the loan is approved. ."i

- Repayment at the end of each month according to an amortization schedule] with i<2> = 12%.

Find the difference in the amount of interest paid under options A and B.

4. Find the APR for both options A and in Exercise 3.

5. A borrower visits three banks to obtain quotes on a car loan repayable with 241 monthly installments. The first bank quotes a monthly payment of X based on a I carrying charge equal to the product of the initial balance, the number of ye repay, and 6.5%. The second bank quotes a monthly payment of Y based annual effective interest rate of 12.6%. The third bank quotes a monthly pay of Z based on an interest rate of 12% covertible monthly. Rank the values of X, and Z.

6. A loan of $8000 at an interest rate of 12% per annum is to be repaid with payments:

A payment of $2000 at the end of 3 months.

A payment of $4000 at the end of 9 months.

A payment of $X at the end of 12 months. Find X:

a) Using the United States Rule.

b) Using the Merchants Rule.

7. A borrows $10,000 for two years at an effective rate of interest of 10%. A ; to pay interest at the end of each year and to repay the principal at the end of I years. At die end of one year A is able to pay only $500. Find die necessary to completely repay the loan at the end of two years:

a) Using the actuarial method.

b) Using die United States Rule.

8. A borrower deposits $200 immediately for the guarantee to be able to borrow $1(1 at the end of one year. The borrower must repay the $1000 at the end of two;

a) Find the two positive yield rates for this transaction.

b) Truth in lending does not resolve this situation since there is only one adv Compute the APR using the method of equated time on the payments rather 1 on the advances.

9. The tables promulgated under truth in lending for handling irregular define three terms as follows:

A - number of regular payments.

- equivalent single payment point (exact, not by die method of equated I - finance charge per $1000 of payment (related to payments by borrower, not to advances).



8.7 Short sales

In die illustration in diis section die yield rate was 36% when considering bodi die interest on margin and the payment of dividends. Find the yield rate if the margin requirement had been 60% instead of 50%.

jM. Pursuing Exercise 43 one step fiirtfier, if die margin requirement is ni, where 0 < m < 1, find a general formula for die yield rate as a function of m.

8.6 Capitalized cost

37. A machine sells for $10,000 and has a salvage value of $1000 at die end of 10 \ years. The annual maintenance expense of the machine is $500. Assuming 5% interest:

a) Calculate the periodic charge of the asset.

b) Calculate the capitahzed cost of the asset.

38. Machine 1 sells of $1000 widi a salvage value of $50 at die end of nine years. Machine 2 sells for $1100 widi a salvage value of $200 at die end of nine years. At what rate of interest would a purchaser be indifferent between the two machines? Assume equal maintenance expenses for die two machines.

39. Plastic hays last 8 years and cost $20 each. Metal hays last 24 years and cost $X,.i each. Trays are needed for 48 years, and inflation will increase die cost of die ttays 5% per year. At 10.25% interest, determine X so diat die buyer is indifferent between purchasing plastic or metal trays.

40 Rework Example 8.12 assuming diat the cost of poles is expected to increase by 2% per year indefinitely into the future.

41 A construction firm buys $1000 worth of lumber. Find the maximum amount the firm would be willing to pay to tteat die lumber to extend its life from 10 to 15 years if die salvage value in eidier case is $50. The effective rate of interest is 3 1/2%.

42. Machine 1 sells for $100,000, has an annual maintenance expense for die first year of $3000, and has a usefiil Ufe of 20 years widi no salvage value. Machine 2 has an annual maintenance expense for the first year of $10,000 and has a useful life of 15 years widi no salvage value. It is anticipated diat die cost of die machines and tiie annual maintenance expenses will increase by 4% per year indefinitely into die fiitute. Machine 2 produces output twice as fast as Machine 1. Maintenance expenses are paid at die beginning of each year. Assuming die effective rate of interest is 8%, find die price of Machine 2 at which a buyer would be indifferent between the two machines. Answer to the nearest $100.

32. piece of equipment tiiat was purchased for $15,000 will have a salvage value of $2000 after 15 years. Its book value has been determined by depreciation in accordance widi die compound interest mediod, using an interest rate of 5% pgr annum. At the end of the 10th year, the depreciation method is changed to the straight Une method for the remaining 5 years. Determine the book value at the end of the 12th year. Answer to the nearest dollar.

33. A company buys two machines. Both machines are expected to last 14 years, and each has a salvage value of $1050. Machine A costs $2450, while Machine costs $y. The depreciation method used for Machine A is the straight line method, while the depreciation mediod used for Machine is the sum of the years digits method. The present value of the depreciation charges made at the end of each year for Machines A and are equal. If die effective rate of interest is 10%, calculate Y.

34. A new machine which costs $11, has a salvage value of $900. The machine has a useful lifetime of 100 years. Let:

BVSL, = die book value of die asset at die end of year t under die sttaight line

depreciation method. BVSD, = die book value of die asset at die end of year t under die sum of die

years digits depreciation method. For what value of t is BVSL, - BVSD, a maximum?

35. Assume that book values of an asset are continuous functions of t. Find an expression for that value of t at which the excess of B, by the straight line meUiod over S, by the constant percentage method is a maximum.

36. An asset costing $5000 has a salvage value of $2000 at die end of n years. If die depreciation charge for die 12di year is $100 using die sum of die years digits method, find n.



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