Question

1. You are planning to retire in 40-years. You plan on putting aside $300 each month and increase that saving by 0.5% each month. Suppose your investments earn 1% per month, what will you accumulate after 40 years if you start saving one month from and stop after a last installment in 40 years.

2. What is the Effective Annual rate for a housing loan at 6% APR with monthly compounding? What is the APR for a loan when the EAR is 6% with monthly compounding?

3. Compute the outstanding balance on a home mortgage at 4% APR (30 years term at the origination date, and for an original balance of $100,000, monthly payments) after you have already made payments for 10 years.   What is the principal and interest component of your next payment?

4. Compute the present value of payments of $300 after 1 year and $500 after 2 years if the rates of interest are 5% and 7% for corresponding maturities of 1 year and 2 years?

5. A 10% coupon bond with semi-annual coupon payments, $1000 face value matures in 10 years. (a) Compute its price if the YTM is 8% quoted as an APR with semi-annual compounding.   (b) At what YTM will it be a par bond? (c) If the YTM increases to 10%, compute the % change in price.

  

6. Compute the price of a zero coupon bond with maturity 30 years and par value $1000 and an annual YTM of 12%? If the YTM remains unchanged what will its price be 20 years from today?

7. A constant growth stock has the first period dividend at t=1 of $4. If required rate of return is 10% and growth in dividends is 5%- (a) compute the price of the stock, (b) its capital gains yield and its dividend yield? (c) what is the expected stock price in 4 years (at t=4)?

8. A non-constant growth stock pays dividends of $1, $3, $3 and $4 in the next 4 years. After the fourth dividend, dividend growth will be 5%. If the required return on the stock is 10% per year, compute its price today?

9. A project pays $500 at t=1 and 700 at t=7. Compute the NPV, IRR, Payback period of the project if it costs $700 today. Assume cost of capital is 10% where required.

Answer 1

a). Future Value (FV) of a growing annuity factor = [(1+r)^n - (1+g)^n]/(r-g) where

r = 1%; g = 0.5%; n = number of payments = 40*12 = 480

FV factor = [(1+1%)^480 - (1+0.5%)^480]/(1% - 0.5%) = 107.69/0.5% = 21,538.05

FV of the growing annuity = payment per month * FV factor = 300*21,538.05 = 6,461,416.29

b). Monthly interest rate r = 6%/12 = 0.5%

EAR = (1+r)^(12) -1 = (1+0.5%)^12 -1 = 6.17%

If EAR = 6% with monthly compounding then monthly interest rate r will be:

(1+r)^12 = (1+6%)

r = (1+6%)^(1/12) -1 = 0.487%

APR = r*12 = 0.487%*12 = 5.84%

c). APR = 4% so monthly interest rate r = 4%/12 = 0.333%

PV = 100,000; rate = 0.333%; n (number of payments) = 30*12 = 360, CPT PMT.

PMT = 477.42 (This is the monthly payment)

Loan balance after 10 years or 20 years from the end of the loan: PMT = 477.42; rate = 0.333%; n = 20*12 = 240, CPT PV.

PV = 78,783.96 (This is the outstanding balance on the original balance of 100,000)

For the next payment, interest amount will be 78,783.96*0.333% = 262.61

Balance amount will be PMT - interest amount = 477.42 - 262.61 = 214.80

d). PV = payment in year 1/(1+ interest rate)^1 + payment in year 2/(1+ interest rate)^2

= 300/(1+5%) + 500/(1+7%)^2 = 285.71 + 436.72 = 722.43

e). Coupon payment = face value*coupon rate/2 = 1000*10%/2 = 50 (as it is paid semi-annually)

FV = 1,000; PMT = 50; N (number of payments) = 10*2 = 20; rate = 8%/2 = 4% (semi-annual rate), CPT PV.

PV of the bond = 1,135.90

For a bond to sell at par, its YTM has to equal its coupon rate so at an annual YTM of 10%, it will be a par bond.

Price at 10% YTM will be 1,000

%change in price = (1,000 - 1,135,90)/1,135.90 = -11.96%

f). Zero coupon bond will pay no coupons and will be redeemed at face value at maturity.

PV = FV/(1+YTM)^N = 1000/(1+12%)^30 = 33.38

Price in 20 years = 1000/(1+12%)^10 = 321.97