Financial Concepts Review
Installment Loans and Amortization
Installment loan structure:
You agree to a fixed monthly payment (the payment you signed up for with the bank).
Each payment contains a portion that covers interest and a portion that reduces the principal.
The cash outflow each month equals the contractual monthly payment.
First-month example (car loan or mortgage):
Given loan balance: $320{,}000; annual interest rate: 6%; monthly rate: i = rac{0.06}{12} = 0.005.
Monthly payment: $1{,}918.56 (as stated in the transcript).
Interest portion in month 1: Interest1 = Balance0 imes i = 320{,}000 imes 0.005 = 1{,}600.00.
Principal portion in month 1: Principal1 = Payment - Interest1 = 1{,}918.56 - 1{,}600.00 = 318.56.
New balance after month 1: Balance1 = Balance0 - Principal_1 = 320{,}000 - 318.56 = 319{,}681.44.
Journal entry for month 1:
Debit Interest Expense: 1{,}600.00
Debit Notes Payable (principal reduction): 318.56
Credit Cash: 1{,}918.56
Second month (illustrative continuation):
Balance before second payment: 319{,}681.44
Interest: Interest_2 = 319{,}681.44 imes 0.005 \,( ext{per period}) \,=\ 1{,}598.41
Principal: Principal_2 = 1{,}918.56 - 1{,}598.41 = 320.15
New balance: Balance_2 = 319{,}681.44 - 320.15 = 319{,}361.29
Key takeaway: the interest portion decreases over time and the principal portion increases slowly as the balance falls.
Mortgage-specific observations:
Monthly payments are typically constant, but the split between interest and principal changes over time.
Early payments are mostly interest; later in the term, more of each payment goes to principal.
After about year 15 in a 30-year mortgage, principal payoff accelerates as the balance shrinks.
Extra payments to principal:
If you make two extra payments per year, those payments go entirely to reducing the principal.
Result: the loan is paid off faster because the principal balance is reduced more quickly, which reduces future interest in a compounding way.
Journal entries outline for each regular payment (example 6% scenario):
Debit Interest Expense for the interest portion.
Debit Note Payable (or Loans Payable) for the principal portion.
Credit Cash for the total monthly payment.
Notes about balance updating:
After each payment, the new loan balance is reduced by the principal portion of that payment.
Formula to track balance over time:
Interestt = Balancet imes rac{r}{m}
Principalt = Payment - Interestt
Balance{t+1} = Balancet - Principal_t
How to think about the numbers for a real loan:
The monthly payment is fixed by the loan agreement.
The interest expense each period is computed on the outstanding balance using the periodic rate.
The portion of the payment that goes to interest vs principal shifts over the life of the loan as the balance changes.
Bond Issuance, Discount, and Premium; Effective-Interest Amortization
Key concepts:
Bonds can be issued at par (face value), at a discount, or at a premium relative to their face value.
Interest payments are fixed by the stated coupon rate, paid at each period (e.g., semiannually).
The carrying value (book value) of the bond changes over time due to amortization of any discount or premium.
In the accounting entries, the actual cash paid is the coupon payment; the interest expense is determined by the market (effective) rate applied to the carrying value.
Basic notation:
Face value (par) of bond: F
Stated annual coupon rate: c
Market (effective) annual rate at issue: r
Periods per year: m
Coupon payment per period: C = F imes rac{c}{m}
Interest (effective) expense per period: It = CarryingValuet imes rac{r}{m}
Number of periods: n = ext{years} imes m
Case A: Bond issued at par
Issuance entry: Debit Cash F; Credit Bonds Payable F
Each period:
Cash payment: C = F imes rac{c}{m}
Interest expense: It = CarryingValuet imes rac{r}{m}
Amortization amount: At = It - C
Journal effects depend on whether there is a discount or premium; at par, there is no amortization effect.
Case B: Bond issued at a discount
Example values (illustrative):
Face value: F = 80{,}000
Sold for cash: Cash = 73{,}900
Discount on Bonds Payable: Discount = F - Cash = 6{,}100
Initial entry: Debit Cash 73{,}900; Debit Discount on Bonds Payable 6{,}100; Credit Bonds Payable 80{,}000
Ongoing amortization (effective-interest method):
Each period uses carrying value; period rate = market rate / m.
Example first period (semiannual, market rate 8% → per-period rate 0.04):
Carrying value at start: BV_0 = 73{,}900
Interest expense: I1 = BV0 imes 0.04 = 2{,}956
Cash interest (coupon): C = F imes rac{c}{m} = 80{,}000 imes rac{0.07}{2} = 2{,}800
Amortization (increase in carrying value): A1 = I1 - C = 2{,}956 - 2{,}800 = 156
Journal: Debit Interest Expense 2{,}956; Debit Discount on Bonds Payable 156; Credit Cash 2{,}800
New carrying value: BV1 = BV0 + A_1 = 73{,}900 + 156 = 74{,}056
Next period uses updated carrying value: I2 = BV1 imes 0.04 = 2{,}962.24; Amortization A2 = I2 - C = 2{,}962.24 - 2{,}800 = 162.24; New carrying value updates accordingly.
Case C: Bond issued at a premium
Example values (illustrative):
Face value: F = 80{,}000
Sold for cash: Cash = 85{,}951
Premium on Bonds Payable: Premium = Cash - F = 5{,}951
Initial entry: Debit Cash 85{,}951; Debit Premium on Bonds Payable 5{,}951; Credit Bonds Payable 80{,}000
Ongoing amortization (effective-interest method):
Use carrying value and per-period rate from market.
Example with market rate 6% → per-period rate 0.03 (semiannual):
Carrying value start: BV_0 = 85{,}951
Interest expense: I1 = BV0 imes 0.03 = 2{,}578.53
Cash interest (coupon): C = 2{,}800
Premium amortization (reduces premium): A1 = C - I1 = 2{,}800 - 2{,}578.53 = 221.47
Journal: Debit Interest Expense 2{,}578.53; Debit Premium on Bonds Payable 221.47; Credit Cash 2{,}800
New carrying value: BV1 = BV0 - A_1 = 85{,}951 - 221.47 = 85{,}729.53
Note on sign convention: when amortizing a premium, you debit Premium on Bonds Payable (reducing it); when amortizing a discount, you debit Discount on Bonds Payable (increasing it).
Important takeaway about the carry value in bonds:
The carrying value changes each period based on amortization, which in turn affects future interest expense calculations.
Over time, the carrying value moves toward the face value as the bond approaches maturity.
Present Value Concepts and Bond Pricing Tools
Core idea:
Bond price today equals the present value of all future cash flows: the periodic coupon payments plus the maturity principal.
Present value depends on the market rate and the number and size of payments.
Present value formula (two components):
Present value of an ordinary annuity (coupons):
PV_{ ext{coupons}} = C imes rac{1 - (1 + r)^{-n}}{r}Present value of the principal (lump-sum at maturity):
PV_{ ext{principal}} = F imes (1 + r)^{-n}Bond price:PV = PV{ ext{coupons}} + PV{ ext{principal}}
Where:
F = face value
C = F imes rac{c}{m} = coupon per period
r = ext{market rate per period} = rac{ ext{annual market rate}}{m}
n = ext{total number of periods} = ext{years} imes m
Example setup (illustrative numbers):
Face value F = 100{,}000, annual coupon rate c = 7 ext{%}, semiannual payments (m = 2), market rate per year r_{ ext{annual}} = 8 ext{%}
Per-period coupon: C = F imes rac{c}{m} = 100{,}000 imes rac{0.07}{2} = 3{,}500; per-period market rate: r = rac{0.08}{2} = 0.04
Number of periods: n = 10 ext{ years} imes 2 = 20
Present value (two-component): use the formulas above to compute PV of coupons and PV of principal.
Excel and calculator tools for present value:
Excel function: PV(rate, nper, pmt, fv, type)
For bonds, you typically use: ext{PV} = -PV(r, n, C, F, 0) (type = 0 means payments at period end, and the negative sign reflects cash outflow).
Example: per-period rate r = 0.04, periods n = 20, coupon pmt = -3{,}500, face value fv = 100{,}000.
Using a financial calculator or spreadsheet, you can quickly vary rate, pmt, and n to see the present value correspond to market price.
Practical note: With present value, the book shows a computed result around a value like ext{PV} ext{(at 8%)}
eq F; the numeric result depends on the inputs. The key is understanding the method, not just the final number.
Present value in corporate finance:
Present value concepts underpin equity valuation and company valuations; projections over multiple years are discounted to today using a discount rate.
The practice of discounting cash flows is fundamental to evaluating investments, projects, and financing.
Practical alternatives to calculators:
Present value tables (older method): locate factors for a given rate and period count and multiply by cash flows; still conceptually useful.
Excel/Sheets: highly recommended for efficiency and less error-prone, especially for long streams of cash flows.
Real-world side note in the lecture:
Inflation and the time value of money are a backdrop for why discounting matters; money today is worth more than money tomorrow due to earning potential and inflation.
The narrative touches on practical benefits of understanding present value: it informs pricing, investment decisions, and long-term financial planning.
Summary of Practical Applications and Takeaways
For installment loans and mortgages:
Always start with the balance and the periodic rate to compute Interest_t.
The remainder of the payment reduces the principal.
Two common actionable tips:
Make extra payments toward principal to reduce the balance faster and shorten the loan term.
Expect the interest portion to decline over time and the principal portion to increase with each payment.
For bonds:
Distinguish between par, discount, and premium issuance scenarios.
Use the effective-interest method to amortize any discount or premium.
The amortization entry differs depending on whether you are amortizing a discount or a premium (discount increases carrying value; premium decreases carrying value).
For present value and valuation:
Bond prices reflect the present value of coupon payments plus principal, discounted at the market rate per period.
Tools (Excel/calculator/tables) help compute present value quickly; the underlying math is the same across tools.
Understanding PV is essential for investment valuation, business valuation, and long-range financial planning.
Conceptual connections:
Time value of money underpins both loan amortization and bond pricing.
The mechanics of amortization link current cash payments to changes in balance or carrying value over time.
Real-world relevance:
Car loans, student loans, and mortgages use these same principles; recognizing the split between interest and principal helps you plan payoff strategies.
Present value concepts are foundational in corporate finance and valuation practice.
Small but important clarifications from the lecture:
When payments include interest at a fixed rate, the cash outflow remains the same, while the interest expense and principal reduction change month to month.
Carrying value changes with each amortization, which in turn affects future interest expense calculations.