ACTL10001 Chap.4 Bonds and Loans
Bonds and Loans
Bond prices combine the present value of coupons with the present value of redemption. Coupon rates determine cash amounts, while market yield determines discounting. Because promised cashflows are fixed, price falls as yield rises. Loan valuation uses the same equation of value: the amount advanced equals the present value of repayments.
An outstanding balance after a payment can be calculated prospectively from remaining payments or retrospectively from accumulated past cashflows. This chapter also separates interest and principal within a repayment and uses trial values and interpolation to estimate yields when a closed form is unavailable.
What this chapter covers
- 01
Coupon and redemption cashflows
- 02
Bond equations of value
- 03
Price and yield direction
- 04
Yield interpolation
- 05
Nominal yield reporting
- 06
Level loan repayments
- 07
Prospective balances
- 08
Interest and principal splits
Worked example · free
A level payment loan
- 1Payment X satisfies 100,000 = Xa10 at 6%.
- 1a10 = 7.360087, so X = $13,586.80.
- 1After payment four, six payments remain, so balance = 13,586.80a6.
- 1The balance is $66,720.67 approximately.
Key terms
- Bond Coupon
- A periodic interest cashflow specified by the bond contract and face value.
- Redemption Value
- The amount repaid to the bondholder at maturity.
- Bond Yield
- The discount rate that equates promised bond cashflows with market price.
- Equation Of Value
- An equality expressing all cashflows at a single valuation date.
- Loan Balance
- The value of the borrower's remaining repayment obligation at a stated time.
- Principal Repayment
- The part of an instalment that reduces outstanding debt after interest is paid.
Bonds and Loans FAQ
Why does bond price fall when yield rises?
The bond cashflows stay fixed while a higher market yield discounts each future payment more heavily, reducing their total present value at the current pricing date.
What rate goes into a half yearly bond equation?
Use the half year effective yield. Convert the resulting rate afterward if the question requests a nominal or annual effective quotation for the reported answer.
How do I find a loan balance?
Immediately after a payment, value only the remaining repayments at that date. This prospective method is usually the shortest route to the outstanding loan obligation.
How is each loan payment divided?
Interest equals the period rate times the opening balance. The remainder of the fixed payment reduces principal and produces the closing balance for that repayment period.
How do I know whether a bond should sell above or below face value?
Compare the yield per coupon period with the coupon rate per period. A higher market yield implies a discount price, a lower yield implies a premium, and equality implies par, assuming redemption at face value.
What checks should follow a calculated loan balance?
Confirm the timing is immediately before or after a repayment, verify the number of instalments still outstanding and reconcile one subsequent period. Interest on the opening balance plus the opening balance, less the payment, should equal the closing balance. A prospective value from remaining payments should also agree with a retrospective calculation at the same date.
State the periodic rate used so that the reconciliation is reproducible. Round only the reported balance after completing both checks.
Exam move
For bonds, draw coupons and redemption separately, state the per coupon yield and test whether discount or premium price matches the final direction. Practise two trial yields and interpret the residual sign before interpolating. For loans, solve the initial payment, then calculate balances at several dates from remaining payments. Reconcile one period by opening balance plus interest minus payment equals closing balance.
This links annuity formulas to the economic movement of debt. Divide revision into two connected ledgers: security cashflows and loan cashflows. For a bond, calculate the coupon amount from the coupon rate and face value, then discount that stream separately from redemption. Match the yield period to coupon frequency before using an annuity factor.
Predict premium, par or discount before calculating price, and investigate any result that contradicts the prediction. For yield questions, choose trial rates that bracket the market price and record the residual sign beside each trial. Interpolate only after the root is visibly bracketed, then substitute the estimated yield back into the price equation.
For loans, label whether each balance is immediately before or after payment. Use the prospective method from remaining repayments and the retrospective method from past cashflows at the same date; agreement is a powerful verification. Construct a short amortisation row showing opening balance, interest, payment, principal reduction and closing balance.
Confirm that principal reduction plus interest equals the instalment and that debt falls when payment exceeds interest. Include one exercise with a changed repayment or refinancing date so the valuation-date choice stays explicit. Retain the periodic rate and remaining payment count in every balance formula.
Conclude with an economic sentence: bond yield is the discount rate equating price to promised cashflows, while a loan balance is the value of obligations still outstanding.
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