Financial instruments and insurance contracts as cashflow models: formula sheet

Key formulas

Discrete cashflow

C_t = amount paid at time t (t in years from time 0)

Use positive for receipts and negative for payments, from one stated viewpoint.

Continuous cashflow

Total paid from t1 to t2 = ∫ ρ(t) dt, integrated from t1 to t2

ρ(t) is the rate of payment per year at time t.

Expected value of a contingent payment

E[payment] = C × P(payment is made)

For a single payment C at a fixed time, due only if an event occurs.

Net cashflow at time t

Net_t = Income_t − Outgo_t

Combine all flows at the same time before valuing.

Present value of a set of certain flows

PV = Σ C_t × v^t, where v = 1 ÷ (1 + i)

Assumes a constant effective annual rate i. Covered fully in the equation of value topics.

Price of a coupon bond

P = F·r·aₙ + C·vⁿ

F = nominal, r = coupon rate per period, C = redemption amount, n = number of coupon periods, v = 1/(1+i) at the yield per period. The annuity and the power of v must use the same period.

Price when redeemed at par

P = F(r·aₙ + vⁿ)

Use when C = F. This is the same formula with C replaced by F.

Makeham's formula

P = K + (g ÷ i)(C − K), where K = C·vⁿ and g = F·r ÷ C

g is the modified coupon rate: annual coupon per unit of redemption. The rate i and the n in K must be per coupon period. It is quick when you already know K.

Zero-coupon bond price

P = C·(1 + i)⁻ⁿ

A single cashflow C at time n. i is the effective annual yield when n is in years.

Index-linked payment

Payment at t = Nominal payment × I(t − lag) ÷ I(base)

Without a lag, use I(t) ÷ I(0). Check the question for the base date and lag.

Money and real yields

1 + i = (1 + r)(1 + e)

Exact when inflation e is constant. i = money yield, r = real yield. An index-linked bond can be valued at the real yield using the real (unindexed) cashflows.

Premium or discount test

g > i: premium; g < i: discount; g = i: price = redemption

Use as a reasonableness check on your answer. It holds for a bond redeemed at C with modified coupon rate g.

Present value of dividends with constant growth (Gordon model)

P₀ = D₁ ÷ (i − g), for i > g

D₁ is the dividend due one year from now, g is the constant annual growth rate, i is the effective annual discount rate. Needs dividends to start in one year and continue forever.

Dividend growth

Dₜ = D₀ × (1 + g)ᵗ

Dividend in year t if growth is a constant g each year.

Present value of equity over a holding period

P₀ = Σ Dₜ vᵗ + S × vⁿ, where v = 1 ÷ (1 + i)

Dividends for t = 1 to n plus the expected sale price S at time n.

Net rental income

Net rent = gross rent − costs (maintenance, management, voids)

Value property on net income, not on gross rent.

Present value of property over a holding period

P₀ = Σ Rₜ vᵗ + S × vⁿ, where Rₜ is net rent

Rent review changes Rₜ in steps, not smoothly, so list the years that apply.

Rent in advance

Rent paid at start of each period is valued like an annuity-due

Check payment timing carefully; property rent is often paid in advance.

Forward price, asset with no income

K = S₀ × (1 + i)^T, or K = S₀ × e^(δT)

S₀ is the spot price now, T is the time to expiry in years, i is the effective annual risk-free rate and δ the force of interest. Assumes no income, no storage costs and no arbitrage.

Forward price, asset with known income

K = (S₀ − I) × (1 + i)^T, where I = present value of income paid before expiry

Discount each income payment to time 0 at the risk-free rate before subtracting.

Forward price, continuous dividend yield

K = S₀ × e^((δ − d)T)

d is the continuously compounded dividend yield. Use only when the question gives a yield.

Long forward payoff at expiry

S_T − K

Short forward payoff is K − S_T.

Long call payoff

max(S_T − K, 0)

Writer's payoff is −max(S_T − K, 0).

Long put payoff

max(K − S_T, 0)

Writer's payoff is −max(K − S_T, 0).

Option profit for the holder

Payoff − premium × (1 + i)^T

Accumulate the premium only if the question asks for profit including interest.

Value of an existing forward at time t

(F_t − K) × v^(T − t)

F_t is the current forward price for the same expiry. This is the value of the long position, discounted at the risk-free rate.

Annuity-immediate present value

a_n̄| = (1 − vⁿ) ÷ i, where v = 1 ÷ (1 + i)

Payments of 1 at the end of each period for n periods. i is the effective rate per payment period.

Annuity-immediate accumulation

s_n̄| = ((1 + i)ⁿ − 1) ÷ i = (1 + i)ⁿ × a_n̄|

Value of the payments just after the last one.

Annuity-due present value

ä_n̄| = (1 − vⁿ) ÷ d = (1 + i) × a_n̄|

Payments at the start of each period. d = i ÷ (1 + i).

Level loan repayment

R = L ÷ a_n̄|

Loan L repaid by n payments of R at the end of each period.

Interest in a payment

Iₜ = i × B_{t−1}

B_{t−1} is the balance just after the previous payment.

Capital in a payment

Cₜ = R − Iₜ

For a level loan, Cₜ = R × v^(n−t+1), so capital repaid grows by a factor (1 + i) each period.

Balance (prospective)

Bₜ = R × a_{n−t}̄|

Balance just after the t-th payment, with n − t payments remaining.

Balance (retrospective)

Bₜ = L(1 + i)ᵗ − R × s_t̄|

Gives the same value as the prospective method.

Perpetuity

a_∞ = 1 ÷ i

Payments of 1 at the end of each period, forever.

Present value of a cashflow

PV = C × v^t, where v = 1 ÷ (1 + i)

C is the amount paid at time t. i is the effective annual interest rate.

Expected present value of a contingent cashflow

EPV = C × v^t × P(event occurs at or by time t as specified)

Use the probability of the exact condition for payment, for example survival to time t.

Survival-contingent payment at time t

EPV = C × v^t × tpx

tpx is the probability that a life now aged x survives t years. Use for annuity payments and premiums.

Death benefit paid at end of year of death

EPV = S × Σ v^(k+1) × kpx × qx+k, summed from k = 0 to n − 1

This is a term assurance with sum assured S and term n years. Payment is at the end of the year of death.

Equivalence principle

EPV(premiums) = EPV(benefits) + EPV(expenses)

Gives the premium when there is no profit loading. Use the same basis for all three terms.

Net cashflow at time t

NCFt = premiums − benefits − expenses (at time t)

Used for projecting cashflows. The expected value allows for the probability the policy is in force.

Quick revision

• A cashflow model lists amount, timing and certainty of each payment.
• Fix a sign convention: inflows positive, outflows negative, and stay with it.
• Bond cashflows are coupons plus redemption, fixed in advance unless the bond is index-linked or has default risk.
• Bond price equals the present value of its cashflows at the investor's required yield.
• Loan repayments are interest on the outstanding balance plus capital repaid.
• Equity and property cashflows are uncertain, so the model needs assumptions about growth and dividends or rent.
• A forward or futures payoff depends on the underlying price at maturity compared with the agreed price.
• An option payoff cannot be negative for the holder, because the holder can choose not to exercise.
• Insurance premiums are usually certain in amount, while benefits depend on a contingent event.
• Always state assumptions on interest rate, timing and contingency in written answers.
• Check units and timing: annual rate versus payment frequency is a common slip.
• Method marks come from a correct cashflow set-up, so show the timeline.

Common mistakes

• Mixing viewpoints, so the signs of some flows are reversed. Fix: Write the viewpoint at the top and check each sign against it.
• Putting a flow at the wrong time, for example a payment in advance placed at the end of the period. Fix: Underline the timing words and convert them to exact time points on the timeline.
• Using the annual coupon with a half-yearly discount rate Fix: Convert coupon, term and yield to the same period before you start. Write 'per half-year' next to each figure.
• Applying the coupon rate to the redemption value instead of the nominal value Fix: The coupon is F × r. In Makeham's formula use g = F·r ÷ C, not r.
• Using D₀ instead of D₁ in the Gordon formula. Fix: Compute D₁ = D₀ × (1 + g) first, then divide by (i − g).
• Treating equity cashflows as guaranteed like bond coupons. Fix: Say that dividends and sale prices are uncertain and that you are valuing expected amounts under assumptions.
• Giving an option a negative payoff for the holder. Fix: The holder will not exercise at a loss. The payoff is at least 0. The premium is the maximum loss.
• Confusing delivery price with the spot price at expiry. Fix: K is fixed at the start in the contract. S_T is the market price at expiry. The payoff compares the two.
• Using the annual rate when payments are monthly. Fix: Convert to the effective rate per payment period first. If the rate is a nominal 12% convertible monthly, use i = 1% per month.
• Mixing up annuity-immediate and annuity-due. Fix: Mark the first payment on a timeline. If it is one period after the loan, use a_n̄|. If it is at the loan date, use ä_n̄|.

Exam tips

• Always state the viewpoint and sign convention in the first line of a written answer. Examiners award marks for clear assumptions.
• Draw a timeline even for short questions. It helps catch missing flows and wrong timing.
• In life contingency questions, check that the premium is paid only while the life survives and the benefit only on the stated event.
• In computer-based papers, lay out the cashflows in a column by time before you apply any discount formula.
• In MCQs, check the timing words: in arrear, in advance, continuously. They change the answer.
• Draw a timeline first. In written questions, a clear cashflow list earns method marks even if your arithmetic slips.
• State your assumptions: coupon dates, redemption at par unless told otherwise, no default, no tax, and whether an index lag is ignored.
• Always say which rate you are using (per half-year, nominal or effective). Examiners look for the conversion.