Question 1
Sam receives payments at the end of each year for 10 years. The first payment is $1,000 and each later payment is $100 more than the one before. The annual effective interest rate is 6.0%.
Calculate the present value of the payments.
Exam FM formulas, and when to use each
No formula sheet is handed out for Exam FM. These are the results worth knowing cold, with what each one is for.
| Annuity-immediate | a(n) = (1 − vⁿ) ÷ i |
|---|---|
| Bond price | P = Fr·a(n) + Cvⁿ, at the yield per coupon period |
| Loan balance | Payment × a(n − k) just after payment k |
| Modified duration | Macaulay duration ÷ (1 + i) |
Rates
- Nominal to effective: 1 + i = (1 + i⁽ᵐ⁾ ÷ m)ᵐ.
- Discount: d = i ÷ (1 + i), and v = 1 − d.
- Force of interest: 1 + i = e^δ. With a varying force, accumulate by e to the integral of δ.
- Real rate: (1 + i) ÷ (1 + inflation) − 1.
Annuities
| Payments | Present value |
|---|---|
| Level, at period ends, n payments | a(n) = (1 − vⁿ) ÷ i |
| Level, at period starts | ä(n) = (1 + i)·a(n) |
| Level, forever | 1 ÷ i |
| 1, 2, ..., n | (Ia)(n) = (ä(n) − nvⁿ) ÷ i |
| n, n − 1, ..., 1 | (Da)(n) = (n − a(n)) ÷ i |
| Growing by g each period, n payments | (1 − ((1 + g) ÷ (1 + i))ⁿ) ÷ (i − g) |
| Growing by g, forever | 1 ÷ (i − g) |
Loans
- Level payment: loan ÷ a(n).
- Balance after payment k: payment × a(n − k), or loan(1 + i)ᵏ − payment × s(k).
- Interest in payment k: i × balance after payment k − 1.
- Principal repaid grows by the factor 1 + i from one payment to the next.
Bonds
- Price: Fr·a(n) + Cvⁿ. Coupon rate above yield gives a premium; below gives a discount.
- Premium or discount when redeemed at par: F(r − j)·a(n).
- Book value after coupon t: the price of the remaining payments at the original yield.
- Callable at par: assume the earliest call for a premium bond and the latest date for a discount bond.
Duration and immunization
- Macaulay duration: Σ t·PV(t) ÷ Σ PV(t).
- Price change ≈ −price × modified duration × change in yield.
- First-order Macaulay approximation: P(i₁) ≈ P(i₀) × ((1 + i₀) ÷ (1 + i₁)) to the power of the Macaulay duration.
- Redington immunization: equal present values, equal durations, asset convexity greater than liability convexity.
- Forward rate from spot rates: (1 + s₂)² = (1 + s₁)(1 + f).
Try 3 questions
Question 2
A 10-year bond has a face amount of $1,000 and pays semiannual coupons at an annual coupon rate of 8.0%. It is bought to yield a nominal annual rate of 6.0% convertible semiannually.
Calculate the price of the bond.
Question 3
A bond is priced at $1,040.00 at an annual effective yield of 5.00%, where its modified duration is 6.50.
Using the first-order modified duration approximation, estimate the price if the yield moves to 5.40%.
Next step
Questions people ask
Is there a formula sheet on Exam FM?
No. Nothing is provided, so the annuity, loan, bond and duration formulas have to be known. A financial calculator does the arithmetic.
What is the difference between Macaulay and modified duration?
Macaulay duration is the present-value-weighted average time of the cash flows. Modified duration is Macaulay divided by 1 + i, and measures the percent change in price for a change in yield.
When is a bond priced at a premium?
When its coupon rate per period is higher than the yield per period. The buyer pays more than the redemption amount for the larger coupons.