Debt - Topic 3 (Principles of Finance)
3.1 Amortising loans
Think of borrowing a lump sum to purchase something.
Key questions to answer:
What is the regular monthly payment required?
How much will I still owe at different points of the loan’s life?
What impact will a change in interest rates have on my regular payment? What if I keep the payment and vary the loan term?
What is the total amount of interest I can expect to pay over the life of the loan?
Core formula (annuity/loan payment):
Present value of an annuity (regular payments) with payment CF, periodic rate r, and n periods:
PV = CF \, \frac{1 - (1+r)^{-n}}{r}
Solve for CF if PV, r, n are known: CF = PV \cdot \frac{r}{1 - (1+r)^{-n}}
Takeaway: loan payments are governed by the annuity formula; the amount owing at any time is the PV of the remaining payments; ensure consistency of frequency (r and n) with the cash-flow frequency.
3.1.0 Amortising loans – examples and interpretation
Example setup (Example 1):
Borrow $600,000 for 20 years.
Interest rate: 4.8% p.a. compounded monthly; monthly payments required.
What is the regular monthly payment?
How much is owed after 5 years? How much interest has been paid?
If after 5 years rate rises to 6% p.a. monthly, what is the new monthly payment?
If rates rise and you keep the payment constant but extend the term, what happens?
Worked result for Example 1 (monthly payments):
PV = $600{,}000,
r = 0.048/12 = 0.004 (per month),
n = 20 × 12 = 240 months.
Solve for CF (monthly payment):
CF = PV \cdot \frac{r}{1 - (1+r)^{-n}}
CF ≈ 3{,}893.75\$ per month
Example 2 (amount owing after 5 years):
After 5 years, remaining payments = 15 years = 180 months.
PV of remaining payments at the original rate:
PV = CF \cdot \frac{1 - (1+r)^{-n}}{r} with CF = 3{,}893.75, r = 0.004, n = 180.
PV ≈ 498{,}933.60\$
Example 2 (interest paid in first 5 years):
Total payments in 5 years = 3{,}893.75 \times 60 = 233{,}625
Principal repaid in first 5 years = 600{,}000 - 498{,}933.60 = 101{,}066.38
Interest paid = total payments − principal repaid = 233{,}625 - 101{,}066.38 = 132{,}558.62
Example 3 (new payment after rate rise to 6% p.a., 5 years in):
Amount owed at that point: PV = 498{,}933.60
New rate: r = 0.06/12 = 0.005 per month; n = 180 remaining months.
Solve for new CF: 498{,}933.60 = CF \cdot \frac{0.005}{1 - (1+0.005)^{-180}}
New monthly payment CF ≈ 4{,}210.29\$" per month
Example 4 (keeping original payment but extending term after rate rise):
Solve for n in: 498{,}933.60 = 3{,}893.75 \cdot \frac{0.06/12}{1 - (1+0.06/12)^{-n}}
Result: n ≈ 205.2 periods (months, i.e., about 17 years).
Takeaways (3.1.4):
Working with loans reduces to the ordinary annuity formula: PV = CF \frac{1 - (1+r)^{-n}}{r}
The amount owing at any time is the PV of the remaining payments.
Always express r and n consistently with the cash-flow frequency.
3.2 Short-term debt securities
Definition: a loan that can be traded in the (secondary) market; debt securities include bonds, debentures, notes, and bills of exchange.
Distinguishing features:
Short-term vs long-term;
Single vs multiple future cash flows;
Quoted and traded using simple vs compound interest;
Secured vs unsecured.
In practice, two broad groups split by term and cash-flows:
Short-term debt securities: term < 1 year (usually < 6 months), single future cash flow, traded on yield basis using simple interest.
Long-term debt securities: term > 1 year, more than one future cash flow, traded using compound interest.
Issuers: both government and private entities.
3.2.0 Short-term debt securities – Australian context
In Australia, common short-term debt instruments include:
Treasury notes (government debt)
Bills of exchange (private debt; usually bank-guaranteed)
Promissory notes (private debt; not bank-guaranteed)
Common features:
Term is 6 months or less;
One face-value payment at maturity;
Face values often round numbers (e.g., $500{,}000);
Traded in secondary market using simple interest on a yield basis.
3.2.1 Treasury Notes
Example: 13-week Treasury note issued by the government via the Reserve Bank.
Face value: $10{,}000{,}000;
Yield: 7.650% p.a.; term = 91 days.
Price (present value) calculation uses simple interest:
P_0 = \frac{FV}{1 + r \cdot \frac{d}{365}} where d = 91 days.
P_0 = \frac{10{,}000{,}000}{1 + 0.0765 \cdot \frac{91}{365}} = 9{,}812{,}844
After 3 weeks (21 days), market yields 7.550% p.a. and the note is sold with 70 days to maturity:
Price = P_0 = \frac{FV}{1 + r \cdot \frac{d}{365}} with r = 0.0755, d = 70.
P_0 = \frac{10{,}000{,}000}{1 + 0.0755 \cdot \frac{70}{365}} = 9{,}857{,}272
Relationship between price and yield:
Price-yield relation: P_0 = \frac{FV}{1 + r \cdot \frac{d}{365}}
Inverse relationship: as required yield r increases, price P decreases; conversely, as r decreases, P increases.
3.2.2 Promissory notes and Bills of exchange
Trading and pricing are similar to Treasury notes, but issued by corporations (higher yields due to default risk).
Bills of exchange:
Issued with bank guarantees; more complex institutional details.
Market mechanics (endorsements):
Bills can be traded multiple times in the secondary market; the seller typically endorses the bill.
Example: A $100{,}000 90-day commercial bill purchased at a yield of 9.120% p.a. and sold after 30 days at yield 8.925% p.a.
Purchase price: P_0 = \frac{FV}{1 + r \cdot \frac{d}{365}} with d = 90 days, r = 0.09120.
Purchase price: P_0 = 100{,}000 / (1 + 0.09120 \cdot 90/365) = 97{,}800.69
Selling price after 30 days with d = 60 days, r = 0.08925:
P_0 = 100{,}000 / (1 + 0.08925 \cdot 60/365) = 98{,}554.09
30-day return (percentage):
Return = (Selling price − Purchase price) / Purchase price
= (98{,}554.09 − 97{,}800.69) / 97{,}800.69 = 0.77034% per 30 days
Convert to annual return:
Return_p.a. = (1 + 0.0077034)^{365/30} - 1 = 9.786\%
Takeaways (3.2.3):
Key features: < 1 year, single cash flow, discounted price with yield quoted on simple interest basis.
Issued by both governments and non-government borrowers.
Non-government issuers include Promissory Notes and Bills of Exchange; may be backed by banks.
Pricing exhibits a negative relationship between price and yield to maturity.
3.3 Long-term debt securities
Types of long-term debt securities in Australia:
Commonwealth Government securities (bonds)
Semi-government bonds
Debentures (private sector) – often secured; corporate bonds may be unsecured.
Coupon-paying bonds typically pay coupons annually or semi-annually (twice a year).
Many long-term bonds promise fixed coupons and repayment of face value at maturity.
3.3.1 Long-term debt securities – features (example bond)
Example bond features (illustrative):
A first mortgage bond with a coupon rate of 7% per annum, paying coupons semi-annually.
Face value (par) $500.00; maturity terms and indenture details included.
Coupons and principal are fixed per the indenture; payments may be in gold coin or currency terms depending on issue.
The key is ownership transfer via coupons and bearer rights and the secured claim against collateral.
3.3.2 Long-term debt securities – pricing
General pricing formula for a coupon-paying bond (annual coupons):
Let FV = face value, C = coupon payment per period, r = yield to maturity per period, n = number of periods.
Coupon payment: C = c \, FV where c is the coupon rate per period.
Present value of coupons plus face value:
PV = \frac{C}{r} \left(1 - (1+r)^{-n}\right) + \frac{FV}{(1+r)^n}
Determination of the coupon rate at issuance:
The coupon rate c is set so that the price equals par (PV = FV) when issued, i.e., the yield to maturity equals the coupon rate at issuance.
At issuance: PV = \frac{C}{r} \left(1 - (1+r)^{-n}\right) + \frac{FV}{(1+r)^n} = FV when C = c FV and r = c.
Example (illustrative): pricing a bond with annual coupons:
Suppose coupon rate c = 0.085 (8.5%), FV = $100, n = 5, r = 0.085.
PV = \frac{0.085 \cdot 100}{0.085} \left(1 - (1+0.085)^{-5}\right) + \frac{100}{(1+0.085)^5}
Numerically: PV = 33.50 + 66.50 = 100
One-year later (rates unchanged):
PV after 1 year with 4 remaining coupons: PV = \frac{8.5}{0.085} (1 - (1.085)^{-4}) + \frac{100}{(1.085)^4} = 100
Valuing a bond holding for 1.5 years (with rates unchanged):
Use: P_{-0.5} = \frac{8.50}{0.085} (1 - (1.085)^{-4.5}) + \frac{100}{(1.085)^{4.5}}
Then forward pricing: present value = P_{-0.5} × (1 + r)^{0.5}
Yield to maturity (YTM) concept:
YTM is the rate of return required by the investor, reflecting risk and opportunity cost.
It is the discount rate that equates the bond price with its present value; it changes with market conditions.
Semi-annual coupons:
If coupons are paid semi-annually, use a half-yearly yield: if annual yield is R, then half-year yield is R/2.
Example structure: 4-year bond, 8.5% p.a. coupon, semi-annual payments, current semi-annual yield is 9% p.a. (i.e., 4.5% per half-year).
PV calculation uses the half-year periods and half-year rates accordingly.
3.3.5 Takeaways – Long-term debt securities
Issued by both government and companies.
Often include fixed coupons and principal repayment; pricing uses standard PV approach.
Bond prices move inversely with yields to maturity; discount vs premium depends on coupon rate relative to yield.
Relationship of price and time to maturity manifests as a yield curve across maturities.
3.4 Yields – some final points
Yields reflect the risk attached to promised cash flows; higher risk → higher yield (risk premium).
Ratings agencies assign ratings that influence perceived risk and thus yield.
Yields can change due to rating changes or shifts in demand/supply; however, yields ultimately reflect market pricing.
A key driver is the risk-free rate (government bond rate). When plotting yield spreads, we separate the effect of the risk-free rate from the additional risk premium.
Yield curve (term structure of interest rates): plot yields against term to maturity for a single borrower; conveys market expectations about future short-term rates.
Takeaways (3.4):
The spread (difference in yields) across risks captures compensation for risk differences.
The yield curve informs expectations about future short-term rates.
3.5 Green bonds
Green bonds share many features with non-green bonds:
Tend to be fixed rate and long-term; coupons plus repayment at maturity.
Funds are earmarked for environmentally beneficial projects.
Distinguishing feature: use of proceeds for environmentally friendly projects.
3.5.1 Green bonds – Australian issue
Global and Australian data show increasing green bond issuance.
In June 2024, the Australian Government issued $7 billion of Green Treasury Bonds.
Issue was over-subscribed with bids around $22 billion; evidence of strong demand.
The pricing effect can include a GREENIUM, i.e., a small yield premium (lower yield) due to demand for green bonds.
3.5.2 Green bonds – International evidence
German government has issued twin bonds since 2020: identical terms (same maturity, same coupon) except one is green and one is not.
3.5.3 Takeaways – Green bonds
Green bonds are issued to fund environmentally positive projects.
Issuance has grown dramatically in recent years.
In Australia, the initial green issuance showed a small greenium (~3 basis points at issuance); since then, the greenium has averaged around 0.1 basis points.
International evidence supports the existence of a greenium, though it is typically very small and highly context-specific.
Formulae (quick reference)
Ordinary annuity solving for n (when CF, PV, r are known):
n = \log\left( \frac{CF}{CF - PV \cdot r} \right) / \log(1 + r)
Value of a short-term security (single cash flow):
P_0 = \frac{FV}{1 + r \cdot \frac{d}{365}}
Value of a coupon-paying bond (annual coupons):
Let C be the coupon payment per period (C = c × FV), r the yield per period, n the number of periods, and FV the face value.
PV = \frac{C}{r} \left(1 - (1+r)^{-n}\right) + \frac{FV}{(1+r)^n}
Semi-annual coupon bonds (adjusted for half-year periods):
Use half-year rate r/2 and 2n periods (if yields are quoted on a semi-annual basis).
General note on bond pricing and yields:
Bond prices move inversely with yields; discount vs premium determined by coupon rate relative to yield.