Question

Suppose an investor with a 7-year investment horizon is considering the purchase of a 4.50% APR, monthly payment, mortgage with 22 years (264 months) remaining until maturity. The mortgage currently has an outstanding balance of $245,000 and is selling to offer a YTM of 4.8% on the secondary market. The investor expects to be able to reinvest the first 36 monthly cashflows at 4.8% (over their entire reinvestment interval), but expects to be able to reinvest the last 48 monthly payments at only 4.5%. At the end of her investment horizon, she expects to be able to sell the mortgage at a YTM of 4.5%. What is the total/expected (effective) return offered by this security?

Please show all steps and thought processes with a FINANCIAL CALCULATOR so that I can understand!

Answer 1

First we have to find the monthly payment.

I/Y = 4.5/12 = 0.375 (we divide by 12 because it is a monthly payment)

PV = 245,000 (because it is the outstanding balance)

N = 264

FV = 0 (as the loan would have been paid down by now)

Computing PMT on a financial caluclator, we get PMT = $1,463.596

This is the cashflow we would get everymonth if we were to purchase the mortgage.

To find the price of the mortgage we would pay to get a YTM of 4.8%, I/Y = 4.8/12 = 0.4

Keeping the same PMT, FV and N, we calculate PV = $238,354.723

To find the value of the reinvested cash flows,

For the first 36 months,

N = 36

PV = 0

I/Y = 4.8/12 = 0.4

PMT = $1,463.596

Computing FV = $56,550.585

For the next 48 months,

N = 48,

I/Y = 4.5/12 = 0.375

PMT = $1,463.596

PV = -56,550.585 (because this is the amount of money we already have gained, which will now grow by the new interest rate)

FV = $144,495.69 (this is the amount our cashflows will grow to at the end of the 7 years)

We have to find the price we will sell the mortgage for at the end of 7 years.

N = 264 - (7 x 12) = 180 (because 7 years have passed, there are now 180 more months left for the mortgage)

I/Y = 4.5/12 = 0.375

FV = 0

PMT = $1,463.596

Computing PV = $191,321.433 (this is what we will sell the bond for after 7 years)

Total return = (Price of selling the bond - Price of buying the bond + Gains from cash flow)/ Price of buying the bond

= (191,321.433 - 238,354.723 + 144,495.69)/238,353.723 = 0.4089 or 40.89%

To find effective annual return,

N = 7

PV = -238,354.723

FV = 191,321.433 + 144,495.69 = 341,817.123

PMT = 0

Compute I/Y = 5.285%