Spot-Rate Valuation Problem Workshop

Putting Spot-Rate Valuation Together

A spot-rate valuation problem may require several decisions before the arithmetic begins: identify the cash-flow dates, interpret each rate quotation, and match each payment to the discount factor for its maturity. The central discipline is to discount each payment using the rate or zero-coupon price that applies to that payment date. A quoted rate is not automatically an effective annual rate, and one maturity’s spot rate does not generally apply to another maturity.

A zero-coupon bond makes the connection between prices and discount factors direct. It pays a single redemption amount at one future date. Its price today is the redemption amount multiplied by the discount factor for that date. Once the relevant discount factors are known, the present value of a schedule of payments follows by adding the present values of its individual cash flows.

Definition: The maturity-$t$ effective annual spot rate, denoted here by $i_t$, is the annual-effective rate that discounts a single payment at time $t$ to its value at time 0. Its discount factor is $v(0,t)=(1+i_t)^{-t}$. A zero-coupon bond that pays $F$ at time $t$ has price $P=Fv(0,t)$. The rate $i_t$ may differ from spot rates at other maturities.

For a cash-flow schedule with payments $C_t$ at dates $t$, the present value is the sum of each payment multiplied by the discount factor for its own date. The dates and units must agree: if $t$ is measured in years, the rate conversion and exponent must also describe a period measured in years.

\operatorname{PV}=\sum_t C_t v(0,t)

A rate quotation must be put on the appropriate basis before it is used. For a nominal annual rate $i^{(m)}$ convertible $m$ times per year, the rate per conversion period is $i^{(m)}/m$. When $mt$ is an integer, the discount factor over $t$ years is $\left(1+i^{(m)}/m\right)^{-mt}$. Its equivalent effective annual rate is $\left(1+i^{(m)}/m\right)^m-1$. Under a constant force of interest $\delta$, the discount factor over $t$ years is $e^{-\delta t}$. These formulas require that the stated compounding convention applies over the period being valued. Do not substitute a nominal or continuous quotation directly into the effective-annual formula without conversion.

Why Cash-Flow Values Add

The sum formula is justified by the value of each dated payment considered separately. For every payment $C_t$, a zero-coupon bond paying $C_t$ at time $t$ has price $C_t v(0,t)$. A portfolio consisting of those zero-coupon payments reproduces the original schedule date by date. The portfolio’s value is therefore the sum of those individual prices, provided the quoted discount factors are the factors used for valuation.

\text{Value of the replicated schedule} =\sum_t \text{price of a zero-coupon payment at }t =\sum_t C_t v(0,t)

This argument also explains why payments at different dates generally cannot be discounted with one arbitrarily selected spot rate. Each payment is a separate dated claim, and its matching zero-coupon price determines its discount factor.

Worked Examples

Worked Example: Infer Spot Rates from Zero-Coupon Prices and Value a Schedule

At time 0, hypothetical zero-coupon bonds each promise a redemption of $1,000. Their prices are $961.5385 for payment at year 1, $889.9964 for payment at year 2, and $863.8376 for payment at year 3. The unrounded prices are, respectively, $1000/1.04$, $1000/(1.06)^2$, and $1000/(1.05)^3$. Find the three effective annual spot rates and the value of a schedule paying $300 at year 1, $500 at year 2, and $800 at year 3.

Method. Divide each zero-coupon price by its redemption amount to obtain its discount factor. Then use that maturity’s factor for the cash flow on the same date.

Year 1. The discount factor is $961.5385/1000$, or $1/1.04$ using the unrounded price. Thus the annual-effective spot rate is $i_1=4\%$, and the year-1 payment has present value

300\left(\frac{961.5385}{1000}\right)\approx 300\left(\frac{1}{1.04}\right)=288.4615.

Year 2. The discount factor is $889.9964/1000=1/(1.06)^2$ using the unrounded price. Therefore

i_2=\left(\frac{1000}{889.9964}\right)^{1/2}-1\approx 6\%, \qquad \operatorname{PV}_2=500\left(\frac{1}{1.06^2}\right)=444.9982.

Year 3. The discount factor is $863.8376/1000=1/(1.05)^3$ using the unrounded price. Therefore

i_3=\left(\frac{1000}{863.8376}\right)^{1/3}-1\approx 5\%, \qquad \operatorname{PV}_3=800\left(\frac{1}{1.05^3}\right)=691.0701.

Add the three present values: $288.4615+444.9982+691.0701=1{,}424.5298$. The schedule is worth $1,424.53, rounded to the nearest cent. As a check, the three discount factors are approximately 0.9615385, 0.8899964, and 0.8638376; each is multiplied by the payment at its own date. The differing rates are not interchangeable.

Common mistake avoided: Applying the year-1 rate of 4% to all three payments would ignore the year-2 and year-3 zero-coupon prices. The maturity-specific discount factors, not a single convenient rate, produce the stated value.

Worked Example: Convert a Nominal Quotation Before Valuing Cash Flows

Assume a flat spot curve quoted at a nominal annual rate of 10% convertible semiannually. A hypothetical two-year zero-coupon bond pays $1,000 at maturity. Find its price, then value a separate schedule paying $250 at year 1 and $600 at year 2 under the same flat-curve assumption.

Method. Convert the nominal quotation to its semiannual periodic rate, count the conversion periods to each payment date, and discount using those periods. The flat-curve assumption is important: this example specifies that the same quoted rate basis applies at both maturities.

The semiannual rate is $0.10/2=0.05$, or 5% per half-year. The equivalent effective annual rate is

i=(1+0.10/2)^2-1=1.05^2-1=0.1025=10.25\%.

There are four half-year periods in two years. Thus the zero-coupon price is

P=1000(1.05)^{-4} =\frac{1000}{1.21550625} \approx 822.7025.

The price is $822.70, rounded to the nearest cent. The one-year payment has two half-year periods to discount, while the two-year payment has four:

\begin{aligned} \operatorname{PV} &=250(1.05)^{-2}+600(1.05)^{-4}\\ &=\frac{250}{1.1025}+\frac{600}{1.21550625}\\ &=226.7574+493.6215\\ &=720.3789. \end{aligned}

The schedule is worth $720.38, rounded to the nearest cent. A check using the effective annual rate gives the same factors: $(1.1025)^{-1}=(1.05)^{-2}$ for year 1 and $(1.1025)^{-2}=(1.05)^{-4}$ for year 2.

Common mistake avoided: Treating 10% nominal convertible semiannually as 10% effective annually gives a two-year zero price of $1000/(1.10)^2\approx \$826.45$. That is not the stated convention. The correct effective annual rate is 10.25%, giving a price of $822.70.

Worked Example: Value Multiple Dates with Different Quotation Conventions

A hypothetical liability schedule pays $400 at year 1, $700 at year 2, and $1,000 at year 3. The corresponding spot quotations are: 4% effective annually at year 1; 6% nominal annually convertible semiannually at year 2; and a constant force of interest of 0.05 per year at year 3. Find the present value. Treat each quotation as applying only to its stated maturity.

Method. Convert each quoted basis into the discount factor for its own maturity. An equivalent effective annual rate is useful for interpretation, but calculating the discount factors directly avoids mixing conventions.

At year 1, the effective annual rate is already given:

v(0,1)=(1.04)^{-1}=0.9615385, \qquad \operatorname{PV}_1=400(0.9615385)=384.6154.

At year 2, the nominal rate gives a semiannual rate of $0.06/2=0.03$. Four half-year periods elapse:

v(0,2)=(1.03)^{-4}=\frac{1}{1.12550881}\approx 0.8884870, \qquad \operatorname{PV}_2=700(0.8884870)\approx 621.9409.

At year 3, the force of interest gives

v(0,3)=e^{-(0.05)(3)}=e^{-0.15}\approx 0.8607080, \qquad \operatorname{PV}_3=1000(0.8607080)\approx 860.7080.

Add the date-specific values: $384.6154+621.9409+860.7080=1{,}867.2643$. The present value is $1,867.26, rounded to the nearest cent. The result is checked by confirming that each payment was multiplied by the discount factor generated from the quotation attached to its own maturity.

Common mistake avoided: The 6% nominal quotation is not a 6% effective rate, and the force of interest is not inserted into $(1+i)^t$. Converting each quotation according to its stated convention gives consistent discount factors.

Checks and Common Traps

A reliable solution makes the timing and rate basis visible. Before calculating, write down each payment date and identify the corresponding quoted rate, zero-coupon price, or discount factor. Then check that the exponent or number of conversion periods matches the elapsed time. For example, a nominal rate convertible semiannually has $2t$ periods over $t$ years, not $t$ periods.

A useful numerical check is to inspect the discount factors. For positive rates, a later payment’s discount factor is often smaller, but a non-flat spot curve means this pattern is not guaranteed in every maturity comparison. Do not impose a monotonicity assumption that the data do not provide. Instead, verify each factor from its own quotation and use the same factor consistently when checking a zero-coupon price and the associated cash flow.

Given information	Discount-factor calculation	Typical trap
Effective annual spot rate $i_t$	$(1+i_t)^{-t}$	Using a different maturity’s spot rate
Nominal annual rate $i^{(m)}$, convertible $m$ times yearly	$(1+i^{(m)}/m)^{-mt}$	Using $i^{(m)}$ as though it were effective
Constant force of interest $\delta$	$e^{-\delta t}$	Using an annual-effective discount formula without converting

Key takeaway: For a dated cash-flow schedule, value each payment with the discount factor for its date and quotation convention, then add the present values. A zero-coupon price is a direct check on the discount factor used for its maturity.

Check Your Understanding

For each problem, state the rate basis or timing used and round monetary answers to the nearest cent.

A one-year zero-coupon bond pays $1,000 and costs $960. Find its effective annual spot rate and the present value of a $480 payment due in one year using that rate. (Answer: spot rate $4.1667\%$; present value $460.80.)
A flat spot curve is quoted at 8% nominal annually, convertible semiannually. Find the effective annual rate and the price of a two-year zero-coupon bond paying $1,000. (Answer: effective annual rate $8.16\%$; price $1000/(1.04)^4=$$854.80.)
The one-year effective spot rate is 5%, and the two-year effective spot rate is 6%. Find the present value of $300 due in one year and $700 due in two years. (Answer: $300/1.05+700/(1.06)^2=$$908.71.)
A liability pays $100 at year 1, $200 at year 2, and $300 at year 3. Its maturity-specific quotations are 4% effective annually at year 1, 8% nominal annually convertible semiannually at year 2, and force of interest 0.03 per year at year 3. Find its present value. (Answer: $100/1.04+200/(1.04)^4+300e^{-0.09}=$$541.29.)

Given information	Discount-factor calculation	Typical trap
Effective annual spot rate \(i_t\)	\((1+i_t)^{-t}\)	Using a different maturity’s spot rate
Nominal annual rate \(i^{(m)}\), convertible \(m\) times yearly	\((1+i^{(m)}/m)^{-mt}\)	Using \(i^{(m)}\) as though it were effective
Constant force of interest \(\delta\)	\(e^{-\delta t}\)	Using an annual-effective discount formula without converting

What You'll Learn