Learning Module 2
Time Value of Money
Key Outcomes Summary & Practice Problems
What you must be able to do
Curriculum Year: 2026
Calculate and interpret the present value (PV) of fixed-income and equity instruments based on expected future cash flows.
Calculate and interpret the implied return of fixed-income instruments (YTM) and the implied required return and implied growth rate of equity instruments given the PV and cash flows.
Explain the cash flow additivity principle, its importance for the no-arbitrage condition, and its use in calculating implied forward rates, forward exchange rates, and option values.
1 ยท Core TVM Equations
Present value (PV) and future value (FV) are linked through the discount rate r and number of periods t. These relationships underpin every valuation in this module.
Discrete: FV_t = PV(1 + r)^t
Continuous: FV_t = PVยทe^(rt)
Solving for PV:
Discrete: PV = FV_t ยท (1 + r)^(-t)
Continuous: PV = FV_t ยท e^(-rt)
Level periodic payment A from principal PV over t periods at rate r:
A = r ร PV / [1 โ (1 + r)^(-t)]
Excel equivalent: PMT(rate, nper, -PV, 0)
r and t must match the same compounding frequency.
Critical rule: r (rate) and t (periods) must always use the same compounding frequency. For a semiannual coupon bond with annualized YTM of 6%, use r = 3% and t = (years ร 2). Mismatching frequency is the most common bond pricing error.
2 ยท Fixed-Income Instruments โ PV & Implied YTM
DISCOUNT BOND
Principal only at maturity. No coupons. PV = FV/(1+r)^t. Price < par. Return = spread between price and face value.
COUPON BOND
Periodic coupons + par at maturity. PV = ฮฃ PV(coupons) + PV(par). When coupon rate = YTM, PV = par.
LEVEL-PAYMENT
Uniform payments (A) cover both interest and principal. Mortgages & amortising loans. Use annuity formula.
PERPETUAL BOND
Coupon paid forever, no principal repaid. PV = PMT/r. Higher r โ lower PV (strict inverse relationship).
PV = ฮฃ[PMT/(1+r)^t] + FV/(1+r)^N
Semiannual: r = YTM/2, N = years ร 2, PMT = annual coupon/2
Price-yield inverse: YTM โ โ Price โ | YTM โ โ Price โ
Discount: coupon rate < YTM โ PV < par
Premium: coupon rate > YTM โ PV > par
Par: coupon rate = YTM โ PV = par
Discount bond (zero-coupon): r = (FV/PV)^(1/t) โ 1
Coupon bond: solve for r iteratively or use RATE(nper, pmt, -PV, FV)
YTM is the single internal rate of return that equates the
bond price with the PV of ALL future cash flows โ regardless
of their timing. It is the market's uniform discount rate.
Price accretion: a discount bond's PV rises toward par as maturity approaches (t decreases), even if r is unchanged. This rise represents the implied interest earned each period.
Negative yield bonds: PV > FV. Investors pay more than they receive at maturity. Observed in German Bunds (2016) โ demand for safe assets exceeds concern for return of capital.
Semiannual compounding note: to compare a semiannual YTM with an annual rate, you must convert โ (1 + r_semi)ยฒ โ 1 gives the annual equivalent effective yield.
3 ยท Equity Instruments โ DDM & Implied Returns
Constant perpetual dividend D, required return r:
PV = D / r
Implied r: r = D / PV
Constant growth g, next dividend D_(t+1) = D_t(1+g), where r > g:
PV = D_(t+1) / (r โ g)
Solve for implied required return: r = D_(t+1)/PV + g
Solve for implied growth rate: g = r โ D_(t+1)/PV
r โ g = dividend yield spread. Higher g โ higher PV (lower denominator).
High growth g_s for n periods, then long-run growth g_l forever:
Step 1: PV of Stage 1 = ฮฃ [D_t(1+g_s)^i / (1+r)^i] for i=1 to n
Step 2: Terminal value at t=n: TV_n = D_(n+1) / (r โ g_l)
Step 3: PV of TV = TV_n / (1+r)^n
Total PV = Stage 1 PV + PV of terminal value
Forward P/E = payout ratio / (r โ g)
Solving for implied g: g = r โ payout ratio / (P/E)
Valuation signal:
Model PV > market price โ underpriced (consider buying)
Model PV < market price โ overpriced (consider selling/shorting)
Three equity growth models: (1) No growth โ constant dividend perpetuity; (2) Constant growth โ Gordon Growth Model; (3) Changing growth โ two-stage (high growth then stable) or multi-stage DDM.
Dโ vs Dโ trap: the GGM formula uses next period's dividend (Dโ = Dโ ร (1+g)), not the most recent payment. Using Dโ understates the value.
P/E ratio: price-to-earnings is a relative valuation metric. A stock trading at P/E = 20 means investors pay 20 times earnings per share โ reflecting combined expectations of future return and growth.
4 ยท Cash Flow Additivity & No-Arbitrage
Cash Flow Additivity Principle: The PV of a combined cash flow stream equals the sum of the PVs of its individual components โ measured at the same point in time, using the same discount rate. This means two strategies with identical cash flows must have the same price. Any price discrepancy creates a riskless arbitrage profit, which markets quickly eliminate.
No-arbitrage: 2-year spot vs 1-year + implied 1-year forward:
(1 + rโ)ยฒ = (1 + rโ)(1 + Fโ,โ)
Fโ,โ = (1 + rโ)ยฒ / (1 + rโ) โ 1
Fโ,โ = the breakeven reinvestment rate starting in 1 year.
If actual future rate deviates, arbitrage profits are possible.
Spot Sโ (domestic per foreign unit); r_d = domestic; r_f = foreign rate
Fโ,T = Sโ ร e^((r_d โ r_f)รT) (continuous compounding)
If r_d > r_f โ domestic currency depreciates on forward basis
(more domestic currency needed to buy 1 foreign unit in future)
This is covered interest rate parity โ arbitrage-free condition.
Hedge ratio (delta): ฮ = (c_u โ c_d) / (S_u โ S_d)
Risk-neutral probability: ฯ = [(1+r) โ d] / [u โ d]
Option price: cโ = [ฯยทc_u + (1โฯ)ยทc_d] / (1+r)
where u = up factor, d = down factor
Replicating portfolio: ฮ units of stock โ bond = call payoff
Portfolio is risk-free (same value in both up & down scenarios)