Econ 134A Test 1Spring 2012

Solution sketches

LyricMusic National Bank Lyric loans

$360,000 from Music National Bank today. The stated annual interest rate for the loan is 3.6%, compounded monthly. The first payment will be made one month from today.

(a) The loan is amortized over 30 years with monthly payments, and Lyric pays an equal amount of principal each month of repayment. How much will each of the first two payments be?

Principal paid each month: $360K/360 = $1K

Interest 1st month: $360K(.036/12) = $1,080 total payment is $2,080

Interest 2nd month: $359K(.036/12) = $1,077 total payment is $2,077

LyricMusic National Bank (b) Suppose that instead the loan is amortized

such that equal monthly payments of $1,000 are made over 35 years. Will the loan be paid off after 35 years? Will money still be owed after 35 years? If so, how much will be owed?

Use the annuity formula to find the PV of all payments

PV = (C/r)(1 – 1/(1+r)T), plug in C = $1,000, r = .003, T = 420 PV = $238,603.48

PV owed is $360K – PV of payments = $121,396.52 FV owed in 35 years is $121,396.52(1.003)420 =

$427,167.41 Answer: Loan is not paid off after 35 years, so

money will still be owed; amount owed is $427,167.41

LyricMusic National Bank (c) Suppose that instead the loan is

amortized with monthly payments made over 40 years. The first payment of $X will be made one month from today. Each subsequent payment will be 0.1% higher than the previous payment. Find X.

Use the growing annuity formula from the formula sheet, PV=$360,000, r=.003, g=.001, T=480 Solving for C, you get $1,168.12 (this is X)

Dexter

Dexter has just purchased a zero-coupon bond that will mature five years from today, for $3,200. The face value of the bond is $4,000. For this problem, assume that all yields to maturity are expressed as yearly rates.

(a) What is the yield to maturity? 3,200(1+r)5 = 4,000 r = .04564 =

4.564%

Dexter

(b) Suppose that later today, the yield to maturity changes to 4%. How much does the value of the bond change. (Please make sure to clearly state if the bond goes up or down in value.) 4,000/(1.04)5 = $3,287.71 The value has gone up by $3,287.71 -

$3,200, or $87.71

6700 block of DP There are two investments that could go on an

empty lot of the 6700 block of Del Playa Dr. Due to the small size of the lot, only one of the investments can be implemented. This lot cannot be used for any other purposes in the next five years. Assume an effective annual discount rate of 8%.

To implement an investment called “Deltopia Falls,” $50,000 must be invested today, and $100,000 will be received five years from today.

To implement an investment called “Halloween DP,” $20,000 must be invested today, and $39,000 will be received one year from today.

6700 block of DP

(a) Whatis the internal rate of return of each investment? Deltopia Falls: $50,000(1 + IRR)5 =

$100,000 IRR = .14870 = 14.870% Halloweentown: $20,000(1+IRR) =

$39,000 IRR = .95 = 95%

6700 block of DP (b) Which investment should be chosen?

You must justify your answer in 40 words or less. You can also show additional work below, if needed. Deltopia Falls

NPV = 100,000/1.085 – 50,000 = $18,058.32 Halloweentown

NPV = 39,000/1.08 – 20,000 = $16,111.11 Choose the project with the highest NPV, if

positive Choose Deltopia Falls

Solve each of the following

(a) Ava will receive $500 three months from today and $1000 nine months from today. The effective annual interest rate is 9%. Find the total of the present value of the two payments. PV = $500/1.090.25 + $1,000/1.090.75

PV = $1,426.75

Solve each of the following (b) Amanda will receive $900 every two years

forever, starting five years from now. The effective annual interest rate is 14%. Find the total of the present value of all payments.

The interest rate every two years is 1.142 – 1 = 29.96%

I can treat this like receiving a perpetuity every two years, starting three years from now, with interest rate every two years of 29.96%

I can use the perpetuity formula to find the future value 3 years from now (remember that our time frame is every two years)

FV = 900 / .2996 = $3,004.01 Discount by 3 years to get PV = $3,004.01/1.143 =

$2,027.62

Solve each of the following (c) Elena will take out a loan for $5,000 later

today. The effective annual interest rate for the loan is 17%. She will make yearly payments (starting one year from today) of $1,500 for 3 years. She will also make a balloon payment 3 years from today to pay off the loan. How much will the balloon payment have to be in order to pay off the loan?

PV of payments is $1500/1.17 + $1500/1.172 + $1500/1.173 = $3,314.38

PV of balloon payment is $5,000 - $3,314.38 = $1,685.62 FV of balloon payment 3 years from now is

$1,685.62(1.17)3 = $2,699.72

Savannah For this problem, assume that the effective annual

interest rate (EAIR) is 8%: Savannah is set to receive five payments such that the total of all of the payments add up to a present value of $600. The first payment is to be made today, and each subsequent payment will be made one year after the previous. (Note that the final payment will be received by Savannah four years from today.) In nominal terms, each payment is set to be the same amount. How much does each payment need to be?

Note first payment is made today, so we can use the annuity formula with T = 4, and add an additional undiscounted payment today

PV = (C/r)(1 – 1/(1+r)T) + C Plug in PV = $600, r = .08, T = 4 to get…

600 = 4.3121C C = $139.14