Financial Markets & Expectations

Expected Present Discounted Value

coreExam · high

The expected present discounted value (PDV) is the value today of an expected sequence of future payments, discounted at the relevant interest rates. PDV = z_t + z_{t+1}/(1+i_t) + z_{t+2}/[(1+i_t)(1+i_{t+1})] + … With a constant rate i and constant payment z forever, PDV = z/i (perpetuity). This gives the inverse link between bond prices and yields.

Derivation

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The Core Idea

Because 1 euro today can be lent out at rate $i$ , it grows to $1 + i$ next year. Working backward:

\text{1 euro next year is worth } \frac{1}{1 + i} \text{ today.}

Apply this repeatedly:

PDV_t = z_t + \frac{E_t[z_{t+1}]}{1 + i_t} + \frac{E_t[z_{t+2}]}{(1 + i_t)(1 + i_{t+1})} + \cdots

The Perpetuity

If both $z$ and $i$ are constant forever, the formula collapses to a geometric series:

PDV = z \sum_{k=1}^{\infty} \frac{1}{(1+i)^k} = \frac{z}{i}

This is the consol formula — one of the most-used shortcuts in finance. It also reveals the inverse relation between bond prices and yields.

Bond Prices and Yields

For a finite bond paying coupon $c$ for $T$ periods plus face value $F$ :

P = \sum_{k=1}^{T} \frac{c}{(1+i)^k} + \frac{F}{(1+i)^T}

Every term shrinks as $i$ rises → $P$ falls. The longer the maturity, the more sensitive.

Why It Matters

Investment decisions: accept a project if $PDV(\text{inflows}) > PDV(\text{outflows})$ .
Asset pricing: stocks = PDV of expected dividends; bonds = PDV of coupons + face value.
Monetary policy transmission: a CB rate change shifts the whole discount path, repricing everything — a key channel in Ch 17.

Worked Example

Perpetuity pays 20 per year forever. Initial yield i = 5%. (a) Find the price. (b) Rates rise to i = 7%. Find new price and % loss.

(a) P = z/i = 20/0.05 = 400.
(b) New P = 20/0.07 ≈ 285.71. Loss = (400 − 285.71)/400 ≈ 28.6%.
The approximation Δ%P ≈ −Δi/i would give −40% — the exact formula gives 28.6% because the relationship is convex.

Initial P = 400. After +200 bp yield rise: P = 285.71, a 28.6% capital loss. Bond prices are very sensitive at low yields.

Common Mistakes

—Dropping the +1 in the discount factor: the formula is 1/(1+i), not 1/i for single-period discounting.
—Confusing the perpetuity formula z/i (only for constant-z, constant-i perpetuity) with the general discrete PDV sum.
—Forgetting that PDV uses expected future payments and rates — uncertainty is handled via the expectation.
—Applying 1-period discount factors to multi-period payments without compounding.

Exam Cues

→Perpetuity price: P = z/i. Memorise and know its derivation (geometric series).
→Bond prices and yields: inverse relationship. Write the DCF formula to see it.
→Multi-period discounting: product of (1+i_j) in denominator — not (1+i)^k unless i is constant.
→Real vs nominal: adjust zs and is consistently. Nominal payments → nominal yields; real → real.