Question

Prob Set 2.7

6. (7 points) There is a commercial bank whose only assets are a loan portfolio of $100 million of30-year fixed-rate mortgages(with monthly payments), and whose only liability is a single, $90 million 1-year certificate of deposit (CD),interest payable annually. The current value of the bank is therefore $10m ($100m in Assets - $90m in Liabilities). The 30-year mortgages have an average annual fixed interest rate of 5.60%, and the CD has an interest rate of 3.20%. Show all work.  

Assume now that the bank uses present value (PV), market-based accounting to value its assets/loan portfolio, and current 30-year mortgage rates have risen by 4.00% to a new rate of 9.60%. For example, if the bank wanted to sell some (or all) of its mortgage portfolio, the market value (PV) of the mortgages would be determined by using the current, new interest rate of 9.60%. The bank continues to pay the current market rate on its deposits, so we can assume that the market value of the deposits (liabilities) remain constant at $90m.

f. Calculate the new PV of the bank’s mortgage loan portfolio, assuming 29 years to maturity, a market interest rate of 9.60% for mortgages, and no change in the bank’s monthly mortgage paymentsfrom part a above.  

g. Calculate the new value of the bank, assuming the new PV (market value) of the loan portfolio from part f (market value of the bank’s assets), and assuming the bank’s liabilities remain at $90m.

Answer 1

Part f) We first calculate the periodic monthly mortgage payments for the bank's $100 million mortgage loan asset.

This is done with the formula : M = P x [ r x (1 + r)ⁿ / (1 + r)ⁿ - 1 ]

(Note : Alternative way is to use the PMT formula in MS Excel)

where, M is Monthly mortgage payment, P is Prinicipal loan amount, r is interest rate per month and n is number of periods.

here, P = $100 million

r = 5.6% / 12 = 0.467% = 0.00467

n = 30 years x 12 months = 360

$\therefore$ M = 100 million x [ 0.00467 x (1 + 0.00467)³⁶⁰ / (1 + 0.00467)³⁶⁰ - 1 ]

= 100 million x [ 0.00467 x 5.351 / 5.351 - 1 ]

= 100 million x [ 0.025 / 4.351 ]

= 100 million x 0.0057

= $ 0.57 million

Now we find the new PV of the Mortgage loan asset with 29 years to maturity and a market interest rate of 9.6%. This will be nothing but the Present Value of an Annuity, with periodic payments = $ 0.57 million.

PV = A x [1 - 1 / (1 + r)ⁿ ] / r    (where A is the periodic monthly payment)

here, r = 9.6% / 12 = 0.8% = 0.008

n = 29 years x 12 = 348

$\therefore$ PV = 0.57 million x [ 1 - 1 / (1 + 0.008)³⁴⁸ ] / 0.008

= 0.57 million x [ 1 - 1 / 16.005 ] / 0.008

= 0.57 million x [ 1 - 0.0625 ] / 0.008

= 0.57 million x 0.9375 / 0.008

= 0.57 million x 117.1875

= $ 66.797 million (rounded off)

Part g) New Value of the Bank = PV of Mortgage Loan Assets - Liabilities

= $ 66.797 million - 90 million

= $ -23.203 million   (negative value)