# How are loan repayments calculated?

Hi, I have a bit of coursework to work on, and there is this section where I am told… I am borrowing \$42,000,000, 18 year term, at 6.5% interest I used an online calculator to calculate loan repayment amount, they all agree: \$330,000 each month. I also used this PMT function in microsoft excel, and again, it comes out with \$330,000. The thing I need to know is, what were the calculations that actually went on behind the scenes? How much of this, each month, is interest, and how much is towards the principal? Thanks!

And as an aside: I am also asked… “If the borrower took out an endowment policy repaying the whole loan at the end of the term what would be the monthly interest payments?” I’m not sure how to interpret this, or answer it, any ideas would be great.

amortization schedule

lookup annuities (i.e. discounting multiple future streams of cash flow so you get one present value – in this case, your PV = \$42M, so solve for your X (cash flow) for a given i (6.5% rate) it’s a basic formula: PV = X * A (sub n) where A (sub n) = [1 - (1+i)^-1] / i Once you get your cash flow (monthly payments), then P&I is determined from an amortization scheduled (go look that up): http://en.wikipedia.org/wiki/Amortization_schedule

My guess is with the endowment policy the loan payments become interest only or \$227,500 a month. \$42MM*(6.5%/12) then the policy repays at loan maturity…that does not take into account the endowment policy payments And like mwvt said amoritization schedule with monthly payments of \$329,923.97 Month Year Payment Interest Principal Balance 11 2008 329,923.97 0.00 0.00 42,000,000.00 12 2008 329,923.97 230,817.71 99,106.26 41,900,893.74 1 2009 329,923.97 230,273.05 99,650.92 41,801,242.82 2 2009 329,923.97 229,725.41 100,198.56 41,701,044.26 . . . 10 2026 329,923.97 9,692.54 320,231.43 1,443,440.36 11 2026 329,923.97 7,932.66 321,991.31 1,121,449.04 12 2026 329,923.97 6,163.10 323,760.87 797,688.18 1 2027 329,923.97 4,383.82 325,540.15 472,148.03 2 2027 329,923.97 2,594.76 327,329.21 144,818.82 3 2027 145,614.70 795.87 144,818.82 0.00

One way to think about it without all the computer formulas is this: If an interest rate r=5% (or whatever), and you receive a yearly payment pmt=\$1000 for the rest of eternity, the present value of all those payments is: PV(payments) = pmt/r = \$1000 / 0.05 = \$20,000 This comes from the mathematics of converging series. A financial instrument that pays out this way is called a perpetuity (brits call it a console loan, I think) Now, if you have a mortgage for 30 years, that can be broken down into two perpetuities, one that starts today, where you are paying, plus another one that starts after 30 years, which has the opposite cash flows (so it cancels out the remaining “infinite stream” of payments, so you can stop paying). So the present value of the mortgage is: PV(30y mortgage) = PV(\$1000 payments for eternity) - PV(\$1000 perpetuity, starting 30 years from now) That equals PV(mortgage) = \$1000/0.05 - [(1+0.05)^(-30)]*(\$1000/0.05) = 15372 PV(mortgage) = pmt/r - [(1+r)^(-years)]*(pmt/r) What a lender does is the reverse calculation… they ask what payment over 30 years will make the PV equal the amount they are lending you today. Algebraically, that’s more difficult to solve, so most people use a calculator/computer/amortization tables to work out the number. From knowing the payment, one can work back how much of each payment is interest, and from there figure that the remainder of the payment is principal, which can be used to figure out the next interest payment. Also, this calculation was based on an annual payment of \$1000. Most mortgages are monthly, so you’d take 1/12 the interest rate, and you’d use 30*12=360 as the final payment number.

bchadwick Wrote: ------------------------------------------------------- > PV(\$1000 payments for eternity) Sounds like part of an alimony calculation to me.