Time value of money II: valuing streams of cash flows

Most financial assets and projects produce not one cash flow but many: a bond pays coupons, a share pays dividends, a machine saves costs every year, a loan requires monthly payments. This lesson extends the valuation principle to streams of cash flows. It reviews the annuity and perpetuity formulas from BUS-110 lesson 8, adds growing annuities and growing perpetuities, which are the building blocks of share and firm valuation, and applies them to mortgages, loans with fees and the choice between leasing and buying. The emphasis is on choosing the right formula for the shape of the cash flows and on checking answers with a direct calculation.

Any stream is a sum of single cash flows

The value of a stream of cash flows is simply the sum of the present values of each cash flow, because present values add: owning the whole stream is the same as owning each piece. For an uneven stream, that sum is the only way. For regular streams, formulas save time, but each formula is just a shortcut for the sum. Drawing a timeline, with cash flows placed at the date they occur, is the most reliable way to avoid errors about timing.

Annuities and perpetuities

An annuity is a stream of equal payments at equal intervals for a fixed number of periods; a perpetuity continues forever. If payments occur at the end of each period (an ordinary annuity), the present value of \( n \) payments of \( C \) at rate \( r \) per period, and of a perpetuity, are given below. An annuity due, with payments at the start of each period, is worth \( (1 + r) \) times as much, because each payment arrives one period sooner. The perpetuity formula also shows why an annuity is the difference between two perpetuities: one starting now and one starting after \( n \) periods.

Growing perpetuities and growing annuities

Many cash flows are expected to grow: dividends, rents, salaries and the free cash flow of a successful business. If the first payment \( C_1 \) arrives one period from now and payments grow at a constant rate \( g \) forever, with \( g < r \), the present value is the growing perpetuity formula. It is the basis of the dividend discount model in lesson 6 and of terminal values in business valuation. If growth continues for only \( n \) periods, the growing annuity formula applies.

Two warnings. First, \( C_1 \) is next period's cash flow, not this period's; if you are told the payment just made, multiply by \( (1 + g) \). Second, the formula only works when \( g < r \). As \( g \) approaches \( r \) the value explodes, which is mathematically correct but economically implausible: no business can grow faster than the economy forever. In practice, long-run growth rates in perpetuity formulas are usually kept at or below the expected long-run growth of nominal GDP.

# Check the growing annuity formula by summing each year's savings
r, g, c1, n = 0.08, 0.03, 18_000, 15
direct = sum(c1 * (1 + g) ** (t - 1) / (1 + r) ** t for t in range(1, n + 1))
formula = c1 / (r - g) * (1 - ((1 + g) / (1 + r)) ** n)
level = c1 / r * (1 - (1 + r) ** -n)
print(f"direct sum {direct:,.0f}  formula {formula:,.0f}  level annuity {level:,.0f}")

direct sum 183,191  formula 183,191  level annuity 154,071

Mortgages and amortization

A loan repaid in equal instalments is an annuity from the lender's point of view: the loan amount equals the present value of the payments at the loan rate. Solving the annuity formula for \( C \) gives the payment. Early payments are mostly interest, because the balance is large; later payments are mostly principal. The outstanding balance at any date is the present value of the remaining payments. BUS-110 lesson 9 builds full schedules; here the focus is on how the conventions of each country change the numbers.

Canadian mortgages are usually amortized over 25 years, or up to 30 years for some insured first-time buyers and new builds under rules introduced in 2024, but the interest rate is fixed only for a shorter term, commonly five years, after which the loan is renewed at the rate then prevailing. That exposes Canadian borrowers to interest-rate risk at each renewal, as many found when rates rose sharply in 2022 and 2023. Rates are compounded semi-annually, so the monthly rate is derived as in lesson 3.

Loans with fees: the effective cost

Lenders often charge fees up front: origination fees, points, legal costs or arrangement fees. The borrower receives less than the face amount but repays the full amount, so the true cost is higher than the quoted rate. The way to measure it is to find the discount rate that makes the present value of the payments equal to the net amount received. That rate is the loan's internal rate of return from the lender's side, and the effective cost from the borrower's. It has no closed-form solution, so it is found by trial and error or with a spreadsheet's RATE or IRR function; the code below uses bisection, the same method used for bond yields in lesson 5.

# Effective cost of a loan with an up-front fee, found by bisection
def annuity_pv(payment, rate, n):
    return payment * (1 - (1 + rate) ** -n) / rate

principal, fee, n = 200_000, 6_000, 60
i = 0.07 / 12
payment = principal * i / (1 - (1 + i) ** -n)
lo, hi = 1e-6, 0.05
for _ in range(100):
    mid = (lo + hi) / 2
    if annuity_pv(payment, mid, n) > principal - fee:
        lo = mid
    else:
        hi = mid
print(f"payment {payment:,.2f}  monthly cost {mid * 100:.4f} %  EAR {((1 + mid) ** 12 - 1) * 100:.2f} %")
print(f"EAR without the fee {((1 + i) ** 12 - 1) * 100:.2f} %")

payment 3,960.24  monthly cost 0.6905 %  EAR 8.61 %
EAR without the fee 7.23 %

Leasing or buying

A firm that needs equipment can buy it, usually financed with a loan, or lease it, paying regular amounts for its use. Leasing is a form of borrowing: the lease payments are a fixed obligation, like debt service. A simple comparison, ignoring taxes for now, discounts the costs of each option at the firm's borrowing rate, because the lease payments are as certain as loan payments. Buying costs the purchase price minus the present value of the asset's resale value at the end; leasing costs the present value of the lease payments, which are often made at the start of each period. Taxes change the comparison, because the owner claims depreciation (capital cost allowance in Canada) and the lessee deducts lease payments; lesson 8 shows how to include those tax effects. Accounting standards now put most leases on the balance sheet (IFRS 16 for public Canadian companies and ASC 842 under US GAAP), so leasing no longer hides debt from analysts.

Choosing the right tool